% Time-stamp: <2000-05-14 15:54:30 whitcher> % % BIBLIOGRAPHY OF WAVELET, TIME-SERIES, AND RELATED DOCUMENTS % % Brandon J Whitcher % EURANDOM, P.O. Box 513 % 5600 MB Eindhoven, The Netherlands % whitcher@eurandom.tue.nl % http://www.eurandom.tue.nl/whitcher % % % Please don't remove this header if you copy or transfer this file. % % @String{AandA = "Astronomy and Astrophysics"} @String{AAP = "Annals of Applied Probability"} @String{AASS = "Astronomy \& Astrophysics Supplement Series"} @String{ACHA = "Applied and Computational Harmonic Analysis"} @String{AG = "Annales Geophysicae"} @String{AML = "Applied Mathematical Letters"} @String{AMM = "American Mathematical Monthly"} @String{AMS = "American Mathematical Society"} @String{AnJ = "The Astronomical Journal"} @String{AofMS = "The Annals of Mathematical Statistics"} @String{AofS = "Annals of Statistics"} @String{ApJ = "The Astrophysical Journal"} @String{AS = "Applied Statistics"} @String{BAMathS = "Bulletin of the American Mathematical Society (N.S.)"} @String{BAMetS = "Bulletin of the American Meteorological Society"} @String{Ber = "Bernoulli"} @String{BKA = "Biometrika"} @String{BLM = "Boundary-Layer Meteorology"} @String{CE = "Computational Economics"} @String{CG = "Chemical Geology"} @String{CS = "Computational Statistics"} @String{CSA = "Communications in Statistics A"} @String{CSB = "Communications in Statistics B"} @String{CSDA = "Computational Statistics \& Data Analysis"} @String{CSS = "Computing Science and Statistics"} @String{ECA = "Econometrica"} @String{ECJ3 = "Electronics and Communications in Japan 3"} @String{EE = "Empirical Economics"} @String{EI = "Earth Interactions"} @String{EL = "Economics Letters"} @String{EP = "Environmental Pollution"} @String{ET = "Econometric Theory"} @String{FS = "Financial Stochastics"} @String{GJI = "Geophysical Journal International"} @String{GJRAS = "Geophysical Journal of the Royal Astronomical Society"} @String{GP = "Geophysical Prospecting"} @String{GRL = "Geophysical Research Letters"} @String{IEICETFECCS = "IEICE Transactions on Fundamentals of Electronics Communications and Computer Science"} @String{IEEEAPM = "IEEE Antennas and Propagation Magazine"} @String{IEEECSE = "IEEE Computational Science and Engineering"} @String{IEEEP = "IEEE Proceedings"} @String{IEEES = "IEEE Spectrum"} @String{IEEESPL = "IEEE Signal Processing Letters"} @String{IEEESPM = "IEEE Signal Processing Magazine"} @String{IEEESPS = "IEEE Signal Processing Society"} @String{IEEETAC = "IEEE Transactions on Automatic Control"} @String{IEEETAE = "IEEE Transactions on Audio and Electroacoustics"} @String{IEEETAES = "IEEE Transactions on Aerospace and Electronic Systems"} @String{IEEETASSP = "IEEE Transactions on Acoustics, Speech, and Signal Processing"} @String{IEEETC = "IEEE Transactions on Communications"} @String{IEEETCS = "IEEE Transactions on Circuits and Systems"} @String{IEEETCS2 = "IEEE Transactions on Circuits and Systems--II"} @String{IEEETGRS = "IEEE Transactions on Geoscience and Remote Sensing"} @String{IEEETIM = "IEEE Transactions on Instrumentation and Measurement"} @String{IEEETIP = "IEEE Transactions on Image Processing"} @String{IEEETIT = "IEEE Transactions on Information Theory"} @String{IEEETMI = "IEEE Transactions on Medical Imaging"} @String{IEEETPAMI = "IEEE Transactions on Pattern Analysis and Machine Intelligence"} @String{IEEETSP = "IEEE Transactions on Signal Processing"} @String{IJC = "International Journal of Climatology"} @String{IJE = "International Journal of Electronics"} @String{IJF = "International Journal of Forecasting"} @String{JAcSA = "Journal of the Acoustical Society of America"} @String{JAM = "Journal of Applied Meteorology"} @String{JAOT = "Journal of Atmospheric and Oceanic Technology"} @String{JAS = "Journal of Atmospheric Science"} @String{JASA = "Journal of the American Statistical Association"} @String{JASS = "Journal of Applied Statistical Science"} @String{JAT = "Journal of Approximation Theory"} @String{JBES = "Journal of Business and Economic Statistics"} @String{JC = "Journal of Climate"} @String{JCGS = "Journal of Computational and Graphical Statistics"} @String{JCty = "Journal of Climatology"} @String{JEcol = "Journal of Ecology"} @String{JEcon = "Journal of Econometrics"} @String{JEDC = "Journal of Economic Dynamics and Control"} @String{JETE = "Journal of Economic Theory and Econometrics"} @String{JF = "Journal of Forecasting"} @String{JFAA = "Journal of Fourier Analysis and Applications"} @String{JFM = "Journal of Fluid Mechanics"} @String{JGR = "Journal of Geophysical Research"} @String{JGRA = "Journal of Geophysical Research -- Atmospheres"} @String{JISS = "Journal of the Italian Statistical Society"} @String{JMP = "Journal of Mathematical Physics"} @String{JMS = "Journal of the Meteorological Society"} @String{JNS = "Journal of Nonparametric Statistics"} @String{JPA = "Journal of Physics A"} @String{JPO = "Journal of Physical Oceanography"} @String{JRSSA = "Journal of the Royal Statistical Society A"} @String{JRSSB = "Journal of the Royal Statistical Society B"} @String{JSCS = "Journal of Statistical Computation and Simulation"} @String{JTSA = "Journal of Time Series Analysis"} @String{JVA = "Journal of Vibration and Acoustics"} @String{JVCIR = "Journal of Visual Communication and Image Representation"} @String{KKA = "Kybernetika"} @String{MCM = "Mathematical and Computer Modelling"} @String{MKA = "Metrika"} @String{MWR = "Monthly Weather Review"} @String{NJ = "New Jersey"} @String{NY = "New York"} @String{OE = "Optical Engineering"} @String{PEM = "Probability Engineering Mechanics"} @String{PIEEE = "Proceedings of the IEEE"} @String{PR = "Pattern Recognition"} @String{PRE = "Physical Review E"} @String{PRL = "Physical Review Letters"} @String{PRSLA = "Proceedings of the Royal Society of London, Series A"} @String{PTRSLA = "Philosophical Transactions of the Royal Society of London A"} @String{QJRMS = "Quarterly Journal of the Royal Meteorological Society"} @String{RES = "Review of Economic Studies"} @String{SC = "Statistics and Computing"} @String{SD = "San Diego"} @String{SF = "San Francisco"} @String{SIAM = "Society for Industrial and Applied Mathematics"} @String{SIAMJMA = "SIAM Journal of Mathematical Analysis"} @String{SIAMJSC = "SIAM Journal of Scientific Computing"} @String{SIAMJSSC = "SIAM Journal on Scientific and Statistical Computing"} @String{SiG = "Surveys in Geophysics"} @String{SINUM = "SIAM Journal of Numerical Analysis"} @String{SIREV = "SIAM Review"} @String{SJS = "Scandanavian Journal of Statistics"} @String{SN = "Statistica Neerlandica"} @String{SNDE = "Studies in Nonlinear Dynamics and Economics"} @String{SP = "Signal Processing"} @String{SPIE = "The International Society for Optical Engineering"} @String{SPL = "Statistics \& Probability Letters"} @String{SPTA = "Stochastic Processes and Their Applications"} @String{SS = "Statistical Science"} @String{SSin = "Statistica Sinica"} @String{TAS = "The American Statistician"} @String{TECH = "Technometrics"} @String{TIEICEA = "Transactions of the Institute of Electronics, Information and Communication Engineers A"} @String{TPIA = "Theory of Probability and Its Applications"} @String{WRR = "Water Resources Research"} %%%%%----------%%%%%----------%%%%%----------%%%%%----------%%%%% @Article{abr-ben:adaptive, title = "Adaptive thresholding of wavelet coefficients", author = "Felix Abramovich and Y. Benjamini", journal = CSDA, volume = "22", pages = "351--361", year = "1996", URL = "http://www.math.tau.ac.il/~felix/ltx/JCSDA.ps.gz", email = "felix@math.tau.ac.il", } @InCollection{abr-ben:thresholding, title = "Thresholding of wavelet coefficients as multiple hypotheses testing procedure", author = "Felix Abramovich and Y. Benjamini", pages = "5--14", crossref = "ant-opp:wavelets", URL = "http://www.math.tau.ac.il/~felix/Papers.html", email = "felix@math.tau.ac.il", abstract = "", } @Unpublished{abr-etal:scaling, title = "Wavelets for the analysis, estimation and synthesis of scaling data", author = "P. Abry and P. Flandrin and M. S. Taqqu and D. Veitch", year = "1998", note = "preprint", URL = "http://www.serc.rmit.edu.au/~darryl/wwbook.ps", } @Article{abr-fla:initial, title = "{O}n the initialization of the discrete wavelet transform algorithm", author = "P. Abry and P. Flandrin", journal = IEEESPL, volume = "1", number = "2", year = "1994", pages = "32--34", abstract = "The authors show that making use of the discrete wavelet transform to analyse data implies performing a preliminary initialization of the fast pyramidal algorithm. An approximation enabling easy performance of such an initialization is proposed.", keywords = "initialization. discrete wavelet transform algorithm. fast pyramidal algorithm. performance.", } @InProceedings{abr-gon-fla:based, title = "Wavelet-based spectral analysis of $1/f$ processes", author = "P. Abry and P. Gon\c{c}alv{\`e}s and P. Flandrin", booktitle = "Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing", volume = "3", year = "1993", pages = "237--240", note = "Minneapolis, MN, USA", keywords = "1/f processes, time-scale-based spectral estimation, matched tilings, time-frequency plane, probability density function, discrete wavelet scheme, minimum variance estimator, weighted least-squares", abstract = "The authors attempt to show how and why a time-scale-based spectral estimation naturally suits the nature of $1/f$ processes, characterized by a power spectral density proportional to $|\nu|^{-\alpha}$. They show that a time-scale approach allows an unbiased estimation of the spectral exponent alpha and interpret this result in terms of matched tilings of the time-frequency plane. They derive explicitly the probability density function of the estimated value of $\alpha$. From this analysis, they find that there exists an optimum number of scales to use in a discrete wavelet scheme for obtaining a minimum variance estimator and that an improved procedure can be designed by making use of weighted least-squares in the estimation.", } @InProceedings{abr-gon-fla:spectrum, title = "Wavelets, spectrum analysis and 1/f processes", author = "P. Abry and P. Gon\c{c}alv{\`e}s and P. Flandrin", pages = "15--29", crossref = "ant-opp:wavelets", keywords = "", abstract = "The purpose of this paper is to evidence why wavelet-based estimators are naturally matched to the spectrum analysis of $1/f$ processes. It is shown how the revisiting of classical spectral estimators from a time-frequency perspective allows to define different wavelet-based generalizations which are proved to be statistically and computationally efficient. Discretization issues (in time and scale) are discussed in some detail, theoretical claims are supported by numerical experiments and the importance of the proposed approach in turbulence studies is underlined.", } @Unpublished{abr-sap-sil:bayesian, title = "Wavelet thresholding via a {B}ayesian approach", author = "Felix Abramovich and T. Sapatinas and Bernard Silverman", year = "1996", note = "Submitted", URL = "http://www.math.tau.ac.il/~felix/ltx/Bayes.ps.gz", email = "felix@math.tau.ac.il", } @Article{abr-sel:synthesis, title = "The wavelet-based synthesis for fractional {B}rownian motion - {P}roposed by {F}. {S}ellan and {Y}. {M}eyer: {R}emarks and fast implementation", author = "P. Abry and F. Sellan", journal = ACHA, volume = "3", number = "4", pages = "377--383", year = "1996", } @Article{abr-sil:inverse, title = "Wavelet decomposition approaches to statistical inverse problems", author = "F. Abramovich and B. W. Silverman", journal = BKA, volume = "85", number = "1", pages = "115--129", year = "1998", keywords = "exact risk analysis. near-minimax estimation. singular value decomposition. spatially adaptive estimation. statistical linear inverse problem. vaguelette. wavelet.", abstract = "A wide variety of scientific settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise. Primary interest is in some function f(t) but data are accessible only about some linear transform corrupted by noise; The usual linear methods for such inverse problems do not perform satisfactorily when f(t) is spatially inhomogeneous. One existing nonlinear alternative is the wavelet-vaguelette decomposition method, based on the expansion of the unknown f(t) in wavelet series. In the vaguelette- wavelet decomposition method proposed here, the observed data are expanded directly in wavelet series. The performances of various methods are compared through exact risk calculations, in the context of the estimation of the derivative of a function observed subject to noise. A result is proved demonstrating that, with a suitable universal threshold somewhat larger than that used for standard denoising problems, both the wavelet-based approaches have an ideal spatial adaptivity property.", } @Article{abr-vei-fla:revisiting, title = "Long Range Dependence: {R}evisiting Aggregation with Wavelets", author = "Patrice Abry and Darryl Veitch and Patrick Flandrin", journal = JTSA, volume = "19", number = "3", pages = "253--266", year = "1998", URL = "http://www.serc.rmit.edu.au/~darryl/A2.ps", keywords = "long-range dependence, self-similarity, aggregation, multiresolution analysis, wavelet transform, parameter estimation", abstract = "The aggregation procedure is a natural way to analyse signals which exhibit long-range dependent features and has been used as a basis for estimation of the Hurst parameter, H. In this paper it is shown how aggregation can be naturally rephrased within the wavelet transform framework, being directly related to approximations of the signal in the sense of a Haar-multiresolution analysis. A natural wavelet based generalisation to traditional aggregation is then proposed: ``a-aggregation''. It is shown that a-aggregation cannot lead to good estimators of H, and so a new kind of aggregation, ``d-aggregation'', is defined, which is related to the details rather than the approximations of a multiresolution analysis. An estimator of H based on d-aggregation has excellent statistical and computational properties, whilst preserving the spirit of aggregation. The estimator is applied to telecommunications network data.", } @Article{abr-vei:traffic, title = "Wavelet analysis of long-range-dependent traffic", author = "P. Abry and D. Veitch", journal = IEEETIT, volume = "44", number = "1", pages = "2--15", year = "1998", URL = "http://www.serc.rmit.edu.au/~darryl/A1.ps", keywords = "Hurst parameter long-range dependence packet traffic parameter estimation stationarity telecommunications networks time-scale analysis wavelet decomposition", abstract = "A wavelet-based tool for the analysis of long-range dependence and a related semi-parametric estimator of the Hurst parameter is introduced, The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as wed as allowing their detection and identification. Statistical, computational, and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long-range dependence in Ethernet traffic traces, New features are found with important implications for the choice of valid models for performance evaluation, A study of mono versus multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.", } @Book{ack:real-time, title = "Real-Time Signal Processing: Design and Implementation of Signal Processing Systems", booktitle = "Real-Time Signal Processing: Design and Implementation of Signal Processing Systems", author = "John G. Ackenhusen", publisher = "Prentice Hall", address = "Upper Saddle River, NJ", year = "1999", pages = "461", } @TechReport{agu:evaluating, title = "Wavelet and Autoregressive Decompositions for Evaluating Frequency Compositions in Time Series", author = "Omar Aguilar", institution = "Institute of Statisics and Decision Sciences, Duke University", year = "1996", note = "Discussion Paper 96-22", URL = "ftp://ftp.isds.duke.edu/pub/WorkingPapers/96-22.ps", } @InProceedings{al-li:application, title = "Application of shift-invariant wavelet transform to video coding", author = "Mohammed A. Al-Mohimeed and Ching-Chung Li", pages = "64--75", booktitle = "Video Techniques and Software for Full-Service Networks", editor = "Tzi-cker Chiueh and Andrew G. Tescher", volume = "2915", series = "Proceedings of the SPIE", year = "1997", abstract = "The standard discrete wavelet transform lacks translation invariance in 1-D signals and 2-D images. The down-sampling at each coarser scale accentuates the undesirable effects of the shift-variance, in particular, on the motion estimation from decomposed subimages in video coding. In this paper, we present a study of applying the Chui-Shi shift-invariant wavelet transform using 'oversampling frames' to video compression. Further, we present an algorithm for approximating the motion fields at different scales and different frequency bands by utilizing the multiresolution structure of wavelet decomposition. Motion vectors at a higher resolution are predicted by the motion vectors at a lower resolution through a proper scaling. Experimental results on a salesman video sequence show that the use of the 2-D oversampling algorithm of a biorthogonal spline wavelet has reduced the required number of motion vectors while maintaining an acceptable prediction error when compared to the classical block matching technique using the standard wavelet transform. The proposed approach will advance the video compression methodology for applications to HDTV and video conferencing.", } @Book{ald-uns:medicine, title = "Wavelets in Medicine and Biology", booktitle = "Wavelets in Medicine and Biology", author = "Akram Aldroubi and Michael Unser", publisher = "CRC Press Inc.", address = "Boca Raton", pages = "608", year = "1996", ISBN = "0-8493-9483-X", URL = "http://www.crcpress.com/prods/9483.htm", abstract = "Considerable attention from the international scientific community is currently focused on the wide ranging applications of wavelets. For the first time, the field's leading experts have come together to produce a complete guide to wavelet transform applications in medicine and biology. Wavelets in Medicine and Biology provides accessible, detailed, and comprehensive guidelines for all those interested in learning about wavelets and their applications to biomedical problems. The book consists of four main sections: Theory and Implementation of Wavelet Transforms, Wavelets in Medical Imaging and Tomography, Wavelets and Biomedical Signal Processing, Wavelets and Mathematical Models in Biology. The introductory material is written for non-experts and includes basic discussions of the theoretical and practical foundations of wavelet methods. The background and introduction is followed by contributions from the most prominent researchers in the field, giving the reader a complete survey of the use of wavelets in biomedical engineering. An international perspective is provided throughout the book, with contributions from experts from Germany, France, America, Belgium, Holland, Turkey, and Switzerland.", } @TechReport{all-tet:checking, title = "Checking for Model Consistency in Optimal Fingerprinting", author = "M. R. Allen and S. F. B. Tett", number = "RAL-TR-97-040", institution = "Council for the Central Laboratory of the Research Councils", year = "1997", } @Article{and-tre:trends, title = "Using Wavelets to Detect Trends", author = "Edgar L. Andreas and George Trevi{\~n}o", journal = JAOT, volume = "14", number = "3", year = "1997", pages = "554--564", keywords = "", abstract = "Wavelets are a new class of basis functions that are finding wide use for analyzing and interpreting time series data. This paper describes a new use for wavelets--identifying trends in time series. The general signal considered has a quadratic trend. The inverted Haar wavelet and the elephant wavelet, respectively, provide estimates of the first-order and second-order coefficients in the trend polynomial. Unlike usual wavelet applications, however, this analysis requires only one wavelet dilation scale L, where L is the total length of the time series. Error analysis shows that wavelet trend detection is roughly half as accurate as least squares trend detection when accuracy is evaluated in terms of the mean-square error in estimates of the first-order and second-order trend coefficients. But wavelet detection is more than twice as efficient as least squares detection in the sense that it requires fewer than half the number of floating-point operations of least squares regression to yield the three coefficients of the quadratic trend polynomial. This paper demonstrates wavelet trend detection using artificial data and then various turbulence data collected in the atmospheric surface layer, and last, provides guidelines on when linear and quadratic trends are ``significant'' enough to require removal from a time series.", } @Article{and:magnitude, title = "A Wavelet Magnitude Analysis Theorem", author = "James C. Anderson", journal = IEEETSP, volume = "41", number = "12", year = "1993", pages = "3541--3543", abstract = "Wavelet transform is the constant-Q special case of the generalized short time Fourier transform (GSTFT), and is useful for wavelet analysis. Scalograms are analyzed using specific types of filter/detector banks. GSTFT results are universally applicable to wavelet theory and are useful tools for scalogram sampling for computation and data reduction functions.", } @Article{ans-gui-kai:lagrange, title = "Wavelet Construction Using Lagrange Halfband Filters", author = "R. Ansari and C. Guillemot and J. F. Kaiser", journal = IEEETCS, volume = "38", number = "9", year = "1991", pages = "1116--1118", abstract = "Using the approach described by M.J.T. Smith and T.P. Barnwell (1986) for obtaining exact-reconstruction filter banks, the authors present conjugate-quadrature and linear-phase solutions for two-channel filter banks using Lagrange halfband filters. It is shown that the wavelet solutions obtained by I. Daubechies (1988) under certain regularity conditions are the same as the conjugate-quadrature solutions derived from Lagrange halfband filters using the above approach. The linear-phase solution that is described provides filters with simple coefficients.", } @Article{ant-gij-gre:model, title = "Model selection using wavelet decomposition and applications", author = "A. Antoniadis and I. Gijbels and G. Gr{\'e}goire", journal = BKA, volume = "84", number = "4", pages = "751--763", year = "1997", keywords = "consistency hypothesis testing minimum description length criterion model selection nonparametric regression wavelet decomposition", abstract = "In this paper we discuss how to use wavelet decompositions to select a regression model. The methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients. Consistency properties of the selection rule are established and simulation studies reveal information on the distribution of the minimum description length selector. We then apply the selection rule to specific problems, including testing for pure white noise. The power of this test is investigated via simulation studies and the selection criterion is also applied to testing for no effect in nonparametric regression.", email = "Anestis.Antoniadis@imag.fr, gijbels@stat.ucl.ac.be, Gerard.Gregoire@imag.fr", } @Article{ant-gij-mac:hazard, title = "Nonparametric estimation for the location of a change-point in an otherwise smooth hazard function under random censoring", author = "A. Antoniadis and I. Gijbels and B. MacGibbon", journal = SJS, volume = "", number = "", pages = "???--???", year = "2000", URL = "http://www.stat.ucl.ac.be/dp/dp98/dp9820.ps", keywords = "", abstract = "" } @Article{ant-gre-mck:curve, title = "Wavelet Methods for Curve Estimation", author = "A. Antoniadis and G. Gr{\'e}goire and I. W. McKeague", journal = JASA, volume = "89", number = "428", pages = "1340--1353", year = "1994", keywords = "Nonparametric regression", } @Article{ant-gre-nas:density, title = "Density and hazard rate estimation for right censored data using wavelet methods", author = "Anestis Antoniadis and G{\'e}rard Gr{\'e}goire and Guy P. Nason", journal = JRSSB, volume = "61", number = "1", year = "1999", pages = "63--84", URL = "http://www.stats.bris.ac.uk:81/pub/reports/Wavelets/whf03.ps.gz", keywords = "hazard rate survival data wavelet estimation", abstract = "This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right-censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators have pointwise and global mean-square consistency, obtain the best possible asymptotic mean integrated squared error convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show that these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The first uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases. The second is concerned with times of unemployment for women and the wavelet estimate, through its flexibility, provides a new and interesting interpretation.", } @TechReport{ant-gij:abrupt, title = "Detecting abrupt changes by wavelet methods", author = "Anestis Antoniadis and Ir{\`e}ne Gijbels", number = "9716", institution = "Institute de Statistique, Universit{\'e} Catholique de Louvain", year = "1997", URL = "http://www.stat.ucl.ac.be/dp/dp97/dp9716.ps", note = "submitted to the {\em J. Nonpar. Stat.}", } @Article{ant-gus:wavelets, title = "Wavelets and stochastic processes", author = "Antoniou, I. and Gustafson, K.", journal = "Mathematics and Computers in Simulation", volume = "49", number = "1-2", year = "1999", pages = "81--104", abstract = "Wavelets are known to have intimate connections to several other parts of mathematics, notably phase-space analysis of signal processing, reproducing kernel Hilbert spaces, coherent states in quantum mechanics, spline approximation theory, windowed Fourier transforms, and filter banks. Here, we establish and survey a new connection, namely to stochastic processes. Key to this link are the Kolmogorov systems of ergodic theory.", } @Article{ant-pha:irregular, title = "Wavelet regression for random or irregular design", author = "Anestis Antoniadis and Dinh Tuan Pham", journal = CSDA, volume = "28", number = "4", year = "1998", pages = "353--369", abstract = "In this paper, wavelet regression estimators are introduced, both in the random and the irregular design cases and without the restriction that the sample size be a power of two. A fast computational algorithm for approximating the empirical counterpart of the scaling and wavelet coefficients is developed. The convergence rate of the estimator is established. The method is illustrated by some simulations and by a real example.", } @InCollection{ant:change-point, title = "Wavelet estimators for change-point regression models", author = "Anestis Antoniadis", booktitle = "Spline Functions and the Theory of Wavelets", editor = "Serge Dubuc and Gilles Deslauriers", volume = "18", series = "CRM Proceedings \& Lecture Notes", publisher = "American Mathematical Society", year = "1999", pages = "335--346", description = "This work is based on a series of thematic workshops on the theory of wavelets and the theory of splines. Important applications are included. The volume is divided into four parts: Spline Functions, Theory of Wavelets, Wavelets in Physics, and Splines and Wavelets in Statistics. Part one presents the broad spectrum of current research in the theory and applications of spline functions. Theory ranges from classical univariate spline approximation to an abstract framework for multivariate spline interpolation. Applications include scattered-data interpolation, differential equations and various techniques in CAGD. Part two considers two developments in subdivision schemes; one for uniform regularity and the other for irregular situations. The latter includes construction of multidimensional wavelet bases and determination of bases with a given time frequency localization. In part three, the multifractal formalism is extended to fractal functions involving oscillating singularites. There is a review of a method of quantization of classical systems based on the theory of coherent states. Wavelets are applied in the domains of atomic, molecular and condensed-matter physics. In part four, ways in which wavelets can be used to solve important function estimation problems in statistics are shown. Different wavelet estimators are proposed in the following distinct cases: functions with discontinuities, errors that are no longer Gaussian, wavelet estimation with robustness, and error distribution that is no longer stationary. Some of the contributions in this volume are current research results not previously available in monograph form. The volume features many applications and interesting new theoretical developments. Readers will find powerful methods for studying irregularities in mathematics, physics, and statistics.", } @Article{ant:review, title = "Wavelets in statistics: a review (with discussion)", author = "Anestis Antoniadis", journal = JISS, volume = "6", number = "2", year = "1999", pages = "???--???", } @Article{arf:subdiurnal, title = "On subdiurnal effects in earth rotation", author = "Arfa-Kaboodvand, K. and Groten, E.", journal = "Studia Geophysica et Geodaetica", volume = "43", number = "3", pages = "275--283", year = "1999", } @TechReport{ari-vid:scalograms, title = "On Wavelet Scalograms and Their Applications in Economic Time Series", author = "Miguel A. {Ari\~{n}o} and Brani Vidakovic", number = "95-21", year = "1995", institution = "Institute of Statisics and Decision Sciences, Duke University", URL = "ftp://ftp.isds.duke.edu/pub/Users/brani/papers/WavTS.ps", } @TechReport{ari:forecasting, title = "Time Series Forcasts Via Wavelets: {A}n Application to Car Sales in the {S}panish Market", author = "Miguel A. {Ari\~{n}o}", number = "95-30", year = "1995", institution = "Institute of Statisics and Decision Sciences, Duke University", URL = "ftp://ftp.isds.duke.edu/pub/WorkingPapers/95-30.ps", } @Article{aro:two-normal, title = "The probability function of the product of two normally distributed variables", author = "Leo A. Aroian", journal = AofMS, volume = "18", pages = "265--271", year = "1947", } @Article{bai-sap-pow-krz:underwater, title = "Signal detection in underwater sound using wavelets", author = "T. C. Bailey and T. Sapatinas and K. J. Powell and W. J. Krzanowski", journal = JASA, volume = "93", pages = "73--83", year = "1998", URL = "http://www.stats.bris.ac.uk/~mafs/jasa98_draft.ps.gz", keywords = "Multivariate density estimation; Segmentation; Short-time Fourier transform; Signal detection; Signal processing; Thresholding; Underwater sounds; Wavelet decomposition", abstract = "This article considers the use of wavelet methods in relation to a common signal processing problem, that of detecting transient features in sound recordings that contain interference or distortion. In this particular case, the data are various types of underwater sounds, and the objective is to detect intermittent departures (potential `signals') from the background sound environment in the data (`noise'), where the latter may itself be evolving and changing over time. We develop an adaptive model of the background interference, using recursive density estimation of the joint distribution of certain summary features of its wavelet decomposition. Observations considered to be outliers from this density estimate at any time are then flagged as potential `signals.' The performance of our method is illustrated on artificial data, where a known `signal' is contaminated with simulated underwater `noise' using a range of different signal-to-noise ratios, and a `baseline' comparison is made with results obtained from a relatively unsophisticated, but commonly used, time-frequency approach. A similar comparison is then reported in relation to the more significant problem of detecting various types of dolphin sound in real conditions.", } @Article{bal-oli-bau:discovery, title = "Discovery of the near 158 day periodicity in group sunspot numbers during the eighteenth century", author = "Ballester, J. L. and Oliver, R. and Baudin, F.", journal = ApJ, volume = "522", number = "2", year = "1999", pages = "L153--L156", abstract = "A new record of solar activity, made by compiling the daily number of sunspot groups visible on the Sun's surface between 1610 and 1995, has recently been made available by Hoyt and Schatten. Wavelet analysis of this record shows that an episode of the periodicity near 158 days occurred during the eighteenth century, around the maximum of solar cycle 2, and that episodes of the periodicity, much weaker than that in solar cycle 2, have appeared around the maxims of solar cycles 16-21 (covering the interval 1923-1986). The presence of the periodicity in the group sunspot number confirms that it is caused by a periodic emergence of magnetic flux. On the other hand, periodogram analysis allows one to compare the behavior of the periodicity in both sunspot groups and sunspot areas, and the results suggest that, at least during the twentieth century, the periodic emergence of magnetic flux has adopted two different forms. In solar cycles 16 and 17, new sunspot groups were periodically formed, simultaneously increasing the number of sunspot groups and the total sunspot area on the Sun's surface, while during solar cycles 18, 19, 20, and 21 the periodicity has occurred within already formed sunspot groups, increasing sunspot areas only. We point out that this second type of emergence, which enhances the magnetic complexity of sunspot groups, is responsible for the appearance of the periodicity in high-energy solar flares as detected by the Solar Maximum Mission during solar cycle 21.", } @InProceedings{bao-erd-che:scale, title = "Scale-translation filtering for wideband correlated noise attenuation", author = "F. Bao and N. Erdol and Z. Chen", pages = "652--660", crossref = "szu:wavelet2", abstract = "A novel idea of scale-translation filtering based on the orthonormal wavelet transform is developed and demonstrated.", keywords = "scale-translation filtering. wideband correlated noise attenuation. orthonormal wavelet transform.", } @InProceedings{bao-erd:discrete, title = "{O}n the discrete wavelet transform and shiftability", author = "Bao. Feng and N. Erdol", booktitle = "Conference Record of the Twenty-Seventh Asilomar Conference on Signals, Systems and Computers", volume = "2", editor = "A. Singh", year = "1993", pages = "1442--1445", note = "1-3 Nov. 1993, Pacific Grove, CA, USA", abstract = "We analyze the relationship between the change that is observed in the wavelet coefficients when a signal is time shifted and the time and frequency distributions of the wavelet functions. We address the effects of shift variance and show how it can be useful.", keywords = "discrete wavelet transform. wavelet coefficients. time shifted signal. frequency distribution. time distribution. wavelet functions. shift variance.", } @InProceedings{bao-erd:optimal, title = "{T}he optimal wavelet transform and translation invariance", booktitle = "IEEE International Conference on Acoustics, Speech and Signal Processing", volume = "3", pages = "13--16", year = "1994", author = "F. Bao and N. Erdol", note = "19-22 April 1994, Adelaide, SA, Australia", abstract = "Orthonormal wavelet representations are known to be time-variant. With shifting of the input signal, the energy distribution in time-scale plane also changes. We define the `separability' of a wavelet function both in the scale and translation domains as a measure of its localization with respect to translation. We derive a criterion for the optimal initial phase and then develop an algorithm for its choice in the case of stationary and nonstationary signals.", keywords = "optimal wavelet transform. orthonormal wavelet representations. input signal shifting. energy distribution. wavelet function separability. translation domain. scale domain. translation invariance. optimal initial phase. algorithm. nonstationary signals. stationary stationary signals. time-frequency transform. time-variant representation.", } @Article{bas-etal:modeling, title = "Modeling and estimation of multiresolution stochastic processes", author = "M. Basseville and A. Benveniste and K. C. Chou and S. A. Golden and R. Nikoukhah and A. S. Willsky", journal = IEEETIT, volume = "38", number = "2", year = "1992", pages = "766--784", keywords = "multiscale statistical signal processing, data fusion, estimation, multiresolution stochastic processes, image processing, wavelet transform, modeling paradigm, dynamic models, multiscale stationarity, homogeneous trees, covariance kernels", abstract = "An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. A natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image. In particular, it is shown how the wavelet transform directly suggests such a modeling paradigm. This perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time-like variable. The investigation of models on homogeneous trees is emphasized. The framework examined here allows for consideration, in a very natural way, of the fusion of data from sensors with differing resolutions. Also, thanks to the fact that wavelet transforms do an excellent job of 'compressing' large classes of covariance kernels, it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect.", } @InProceedings{bay-bar:multiple, title = "Multiple Window Time-Frequency Analysis", author = "Metin Bayram and Richard G. Baraniuk", booktitle = "Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis", pages = "173--176", year = "1996", keywords = "multiple window time-frequency analysis robust method time-varying spectrum estimation nonstationary random process time-scale planes Thomson's method wavelet functions statistical test chirping line components stationary signals", abstract = "We propose a robust method for estimating the time-varying spectrum of a non-stationary random process. Our approach extends Thomson's powerful multiple window spectrum estimation scheme to the time-frequency and time-scale planes. The method refines previous extensions of Thomson's method through optimally concentrated window and wavelet functions and a statistical test for extracting chirping line components.", } @TechReport{bay-bar:multiple2, title = "Multiple Window Time-Varying Spectrum Estimation", author = "Metin Bayram and Richard G. Baraniuk", institution = "Isaac Newton Institute program on Nonlinear and Nonstationary Signal Analysis", year = "1999", abstract = "We overview a new non-parametric method for estimating the time-varying spectrum of a non-stationary random process. Our method extends Thomson's powerful multiple window spectrum estimation scheme to the time-frequency and time-scale planes. Unlike previous extensions of Thomson's method, we identify and utilize optimally concentrated Hermite window and Morse wavelet functions and develop a statistical test for extracting chirping line components. Examples on synthetic and real-world data illustrate the superior performance of the technique.", } @Article{bel-wan:symmetric, title = "Compactly supported orthogonal symmetric scaling functions", author = "Belogay, E. and Wang, Y.", journal = ACHA, volume = "7", number = "2", year = "1999", pages = "137--150", abstract = "Daubechies (1988, Comm. Pure Appl. Math. 41, 909-996) showed that, except for the Hear function, there exist no compactly supported orthogonal symmetric scaling functions for the dilation q = 2. Nevertheless, such scaling functions do exist for dilations q > 2 (as evidenced by Chui and Lien's construction (1995, Appl. Comput. Harmon. Anal. 2, 68-84) for q = 3); these functions are the main object of this paper. We construct new symmetric scaling functions and introduce the ``Batman'' family of continuous symmetric scaling functions with very small supports. We establish the exact smoothness of the ```Batman'' scaling functions using the joint spectral radius technique.", } @Book{ben-fra:wavelets, title = "Wavelets: Mathematics and Applications", editor = "John J. Benedetto and Michael W. Frazier", publisher = "CRC Press", address = "Boca Raton", year = "1994", pages = "575", loc = "QA403.3 .W4 1994", } @Article{ben-hoc:fdr, title = "Controlling the False Discovery Rate: {A} Practical and Powerful Approach to Multiple Testing", author = "Yoav Benjamini and Yosef Hochberg", journal = JRSSB, volume = "57", number = "1", year = "1995", pages = "289--300", abstract = "The common approach to the multiplicity problem calls for controlling the familywise error rate (FWER). This approach, though, has faults, and we point out a few. A different approach to problems of multiple significance testing is presented. It calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate. This error rate is equivalent to the FWER when all hypotheses are true but is smaller otherwise. Terefore, in problems where the control of the false discovery rate rather than that of the FWER is desired, there is potential for a gain in power. A simple sequential Bonferroni-type procedure is proved to control the false discovery rate for independent test statistics, and a simulation study shows that the gain in power is substantial. The use of the new procedure and the appropriateness of the criterion are illustrated with examples.", keywords = "Bonferroni-type procedures Familywise error rate Multiple-comparison procedures p-values", } @InProceedings{bey-etal:SAR, title = "{SAR} imaging and multiresolution analysis", author = "Beylkin, G. and Gorman, J. D. and Li-Fliss, S. and Ricoy, M. A.", booktitle = "Algorithms for Synthetic Aperture Radar Imagery II", series = "Proceedings of the SPIE", volume = "2487", year = "1995", pages = "144--152", keywords = "SAR imaging multiresolution analysis synthetic aperture radar image formation algorithms unequally-spaced FFT USFFT algorithm complexity computational cost multiresolution SAR imaging image reconstruction", abstract = "Many synthetic aperture radar (SAR) image formation algorithms require the computation of a multidimensional Fourier transform of irregularly-sampled or unequally-spaced data samples. We apply a recently developed algorithm, the unequally-spaced FFT (USFFT) (Beylkin, 1995), to SAR image formation and compare its accuracy and complexity to a conventional algorithm. We find that the USFFT algorithm allows comparable accuracy to traditional approaches at a slightly reduced computational cost. We briefly discuss extensions of the USFFT algorithm to multiresolution SAR imaging.", } @InProceedings{bey-sai:autocorrelation, title = "Wavelets, their autocorrelation functions, and multiresolution representation of signals", author = "Gregory Beylkin and Naoki Saito", booktitle = "Intelligent Robots and Computer Vision XI: Biological, Neural Net and 3-D Methods", series = "Proceedings of the SPIE", volume = "1826", year = "1992", pages = "39--50", keywords = "compactly supported wavelets, autocorrelation functions, multiresolution signal representation, iterative interpolation schemes, dilations, translations, auto-correlation shell, signal analysis", URL = "ftp://amath-ftp.colorado.edu/pub/wavelets/papers/spie.ps.Z", abstract = "We summarize the properties of the auto-correlation functions of compactly supported wavelets, their connection to iterative interpolation schemes, and the use of these functions for multiresolution analysis of signals. We briefly describe properties of representations using dilations and translations of these auto-correlation functions (the auto-correlation shell) which permit multiresolution analysis of signals.", } @Article{bey:representation, title = "On the representation of operators in bases of compactly supported wavelets", author = "G. Beylkin", journal = SINUM, volume = "29", year = "1992", pages = "1716--1740", } @Book{bic-dok:basic-ideas, title = "Mathematical Statistics: Basic Ideas and Selected Topics", author = "Peter J. Bickel and Kjell A. Doksum", publisher = "Holden-Day, Inc.", address = SF, pages = "492", year = "1977", } @TechReport{bie-vid:adaptive, title = "Time Adaptive Wavelet Denoising", author = "Concha Bielza and Brani Vidakovi\'{c}", year = "1996", institution = "Institute of Statisics and Decision Sciences, Duke University", URL = "ftp://ftp.isds.duke.edu/pub/Users/brani/papers/Timeadapt.ps", } @Article{bij-sle-rue-leg:universe, journal = PIEEE, volume = "84", number = "4", year = "1996", pages = "670--679", title = "{W}avelets and the study of the distant universe", author = "A. Bijaoui and E. Slezak and F. Rue and E. Lega", abstract = "The large-scale distribution of galaxies in the Universe exhibits structures at various scales, these so-called groups, clusters, and superclusters of galaxies being more or less hierarchically organized. A specific vision model is needed in order to detect, describe, and classify each component of this hierarchy. To do so rue have developed a multiscale vision model based on an unfolding into a scale space allowing us to detect structures of different sizes. A discrete wavelet transform is done by the a trous algorithm. The algorithm is implemented for astronomical images and also for lists of object positions, currently called catalogues in astronomical literature. Some applications on astrophysical data of cosmological interest are briefly described: 1) inventory procedures for galaxy counts on wide-field images, 2) processing of X-ray cluster images, leading to the analyses of the total matter distribution, and 3) detection of large-scale structures from galaxy counts. From the analyses of n-body simulations we show that the vision model from the wavelet transform provides a new statistical indicator on cosmological scenarios.", keywords = "transform. clusters", } @Unpublished{bij-sta-mur:atrous, title = "Restauration des Images Multi-Echelles par l'{A}lgorithme \`{a} Trous", author = "Albert Bijaoui and Jean-Luc Starck and Fionn Murtagh", year = "1994", note = "In French", URL = "http://http.hq.eso.org/~fmurtagh/papers/trait-sig-94.ps", } @Book{bil:probability68, title = "Convergence of Probability Measures", author = "P. Billingsley", year = "1968", publisher = "John Wiley \& Sons", address = NY, } @Article{bra-mci:climate, title = "Determining climate-induced patterns using wavelet analysis", author = "G. A. Bradshaw and B. A. McIntosh", journal = EP, volume = "83", year = "1994", pages = "133--142", abstract = "A method using wavelet analysis is introduced for the purpose of identifying and isolating inferred climatic components of the hydrologic record. This method affords an informed procedure for choosing filter dimensions for the purpose of signal decomposition.", } @Article{bra-spi:canopy, title = "Characterizing canopy gap structure in forests using wavelet analysis", author = "G. A. Bradshaw and Thomas A. Spies", journal = JEcol, volume = "80", number = "2", year = "1992", pages = "205--215", keywords = "forest ecology, plant canopies, spatial analysis, statistics", abstract = "1. The wavelet transform is introduced as a technique to identify spatial structure in transect data. Its main advantages over other methods of spatial a nalysis are its ability to preserve and display hierarchical information while allowing for pattern decomposition. 2. Two applications are presented: simple one-dimensional simulations and a set of 200-m transect data of canopy opening measurements taken in 12 stands dominated by Pseudotsuga menziesii ranging over three developmental stages. 3. The calculation of the wavelet variance, derived from the transform, facilitates comparison based on dominant scale of pattern between multiple datase ts such as the stands described. 4. The results of the analysis indicate that while canopy pattern trends follow stand development, small to intermediate disturbances significantly influence canopy structure.", } @Unpublished{bre-liu-tor:monsoon, title = "Intraseasonal oscillations off Monterey, CA", author = "L. Breaker and P. Liu and C. Torrence", year = "1998", note = "National Center for Atmospheric Research", URL = "http://www.cgd.ucar.edu/~torrence/liu/", abstract = "", } @Article{bre:heuristics, title = "Heuristics of instability and stabilization in model selection", author = "L. Breiman", journal = AofS, volume = "24", year = "1996", pages = "2350--2383", } @Unpublished{bri-chi-iri-mor:markov-chain, title = "Some Wavelet-Based Analyses of Markov Chain Data", author = "David R. Brillinger and Chang Chiann and Rafael A. Irizarry and Pedro A. Morettin", year = "1998", note = "Version 5.0", } @Article{bri-hen:multigrid, title = "Wavelets and Multigrid", author = "William L. Briggs and Van Emden Henson", journal = SIAMJSC, volume = "14", number = "2", year = "1993", pages = "506--510", URL = "http://www-math.cudenver.edu/~wbriggs/psfiles/briggs_henson_93.ps", } @Article{bri:cumulants, title = "Some Uses of Cumulants in Wavelet Analysis", author = "David R. Brillinger", journal = "Nonparametric Statistics", volume = "6", year = "1996", pages = "93--114", } @InProceedings{bri:point-processes, title = "Some Wavelet Analysis of Point Process Data", author = "David R. Brillinger", booktitle = "Thirty-First Asilomar Conference on Signals, Systems and Computers", year = "1997", pages = "93--114", } @Article{bro:distribution, title = "The Distribution Function of Positive Definite Quadratic Forms in Normal Random Variables", author = "Robert H. Brown", journal = SIAMJSSC, volume = "7", pages = "689--695", year = "1986", keywords = "Analysis of variance; Unbalanced ANOVA design; Laguerre polynomial", } @Article{bro:quantitative, title = "Quantitative convergence assessment for {M}arkov chain {M}onte {C}arlo via cusums", author = "S. P. Brooks", journal = SC, volume = "8", number = "3", year = "1998", pages = "267--274", } @Article{bru-don-gao:wavelet, title = "Wavelet analysis [for signal processing]", author = "Andrew Bruce and David Donoho and Hong-Ye Gao", journal = IEEES, volume = "33", number = "10", year = "1996", pages = "26--35", abstract = "As every engineering student knows, any signal can be portrayed as an overlay of sinusoidal waveforms of assorted frequencies. But while classical analysis copes superbly with naturally occurring sinusoidal behavior-the kind seen in speech signals-it is ill-suited to representing signals with discontinuities, such as the edges of features in images. Latterly, another powerful concept has swept applied mathematics and engineering research: wavelet analysis. In contrast to a Fourier sinusoid, which oscillates forever, a wavelet is localized in time-it lasts for only a few cycles. Like Fourier analysis, however, wavelet analysis uses an algorithm to decompose a signal into simpler elements. Here, the authors describe how localized waveforms are powerful building blocks for signal analysis and rapid prototyping-and how they are now available in software toolkits.", } @Book{bru-gao:book, title = "Applied Wavelet Analysis with {S-PLUS}", author = "Andrew Bruce and Hong-Ye Gao", year = "1996", publisher = "Springer", address = NY, ISBN = "0-387-94714-0", URL = "http://www.springer-ny.com/catalog/np/jul96np/DATA/0-387-94714-0.html", abstract = "This book introduces applied wavelet analysis through the S-PLUS software system. Using a visual data analysis approach, wavelet concepts are explained in a way that is intuitive and easy to understand. In addition to wavelets, a whole range of related signal processing techniques such as wavelet packets, local cosine analysis, and matching pursuits are covered. Applications of wavelet analysis are illustrated, including nonparametric function estimation, digital image compression, and time-frequency signal analysis. The book and software is intended for a broad range of data analysts, scientists, and engineers. While most textbooks on wavelet analysis require advanced training in mathematics, this book minimizes reliance on formal mathematical methods. Readers should be familiar with calculus and linear algebra at the undergraduate level.", } @Article{bru-gao:waveshrink, title = "Understanding {W}ave{S}hrink: {V}ariance and Bias Estimation", author = "Andrew Bruce and Hong-Ye Gao", journal = BKA, volume = "83", number = "4", year = "1996", URL = "ftp://ftp.statsci.com/pub/gao/varbias.ps.Z", keywords = "Bias and Variance Estimation; Confidence Interval; Hard and Soft Shrink; Non-parametric Regression; Signal De-noising; Threshold Selection; Wavelet Transform; WaveShrink", abstract = "Donoho and Johnstone's WaveShrink procedure has proven valuable for signal de-noising and non-parametric regression. WaveShrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise. WaveShrink has very broad asymptotic near-optimality properties. In this paper, we derive computationally efficient formulas for computing the exact bias, variance and $L_2$ risk of WaveShrink estimates in finite sample situations. These formulas provide a new way of understanding how WaveShrink works, what its limitations are, and the pros and cons of the shrinkage schemes: {\em soft} shrink vs. {\em hard} shrink. It complements the tools of simulation and asymptotic analysis. We use these formulas to estimate the bias, the variance and the $L_2$ risk for WaveShrink. Variance estimates are used to construct approximate pointwise confidence intervals and applied to synthetic and real examples. We also address the problem of threshold selection, computing minimax thresholds and ideal thresholds for both hard and soft shrinkage.", } @Article{bru-gao-stu:subset, title = "Subset-selection and ensemble methods for wavelet de-noising", author = "Andrew Bruce and Hong-Ye Gao and Werner Stuetzle", journal = SSin, volume = "9", number = "1", year = "1999", pages = "167--182", keywords = "cycle spinning model combination nonparametric regression stepwise regression wavelet shrinkage", abstract = "Many nonparametric regression procedures are based on ``subset selection'': they choose a subset of carriers from a large or even infinite set, and then determine the coefficients of the chosen carriers by least squares. Procedures which can be cast in this framework include Projection Pursuit, Turbo, Mars, and Matching Pursuit. Recently, considerable attention has been given to ``ensemble estimators'' which combine least squares estimates obtained from multiple subsets of carriers. In the parametric regression setting, such ensemble estimators have been shown to improve on the accuracy of subset selection procedures in some situations. In this paper we compare subset selection estimators and ensemble estimators in the context of wavelet de-noising. We present simulation results demonstrating that a certain class of ensemble wavelet estimators, based on the concept of ``cycle spinning'', are significantly more accurate than subset selection methods. These advantages hold even when the subset selection procedures can rely on an oracle to select the optimal number of carriers. We compute ideal thresholds for translation invariant wavelet shrinkage and investigate other approaches to ensemble wavelet estimation.", } @Book{bur-gop-guo:book, title = "Introduction to Wavelets and Wavelet Transforms: A Primer", author = "C. Sidney Burrus and Ramesh A. Gopinath and Haitao Guo", year = "1998", publisher = "Prentice Hall", address = NJ, ISBN = "0-13-489600-9", URL = "http://www-dsp.rice.edu/wavebook", abstract = "This primer presents a well balanced blend of the mathematical theory underlying wavelet techniques and a discussion that gives insight into why wavelets are successful in signal analysis, compression, dection, numerical analysis, and a wide variety of other theoretical and practical applications. It fills a gap in the existing wavelet literature with its unified view of expansions of signals into bases and frames, as well as the use of filter banks as descriptions and algorithms.", } @Article{bur-etal:spatio, title = "{A} wavelet multiresolution analysis for spatio-temporal signals", author = "T. J. Burns and S. K. Rogers and M. E. Oxley and D. W. Ruck", journal = IEEETAES, volume = "32", number = "2", year = "1996", pages = "628--649", abstract = "The wavelet filters of the conventional 3D multiresolution analysis possess homogeneous spatial and temporal frequency characteristics which Limits one's ability to match filter frequency characteristics to signal frequency behavior. Also, the conventional 3D multiresolution analysis employs an oct-tree decomposition structure which restricts the analysis of signal details to identical resolutions in space and time. This paper presents a 3D wavelet multiresolution analysis constructed from nonhomogeneous spatial and temporal filters, and an orthogonal sub-band coding scheme that decouples the spatial and temporal decomposition processes.", } @Article{bur-wil-nas:impact, title = "Impact during equine locomotion: {T}echniques for measurement and analysis", author = "J. F. Burn and A. M. Wilson and G. P. Nason", journal = "Equine Veterinary Journal", volume = "23", year = "1997", pages = "9--12", } @TechReport{bus-len:numerical, title = "Numerical tests for bivariate wavelet schemes", author = "R. Buschini and L. Lenarduzzi", number = "IAMI 97.5", year = "1997", institution = "Instituto per le Applicazioni della Matematica e dell'Informatica", } @Article{cai-bro:nonequispaced, title = "Wavelet Shrinkage for Nonequispaced Samples", author = "T. Tony Cai and Lawrence D. Brown", journal = AofS, volume = "26", number = "5", year = "1998", pages = "1783--1799", URL = "http://www.stat.purdue.edu/people/tcai/paper/noneq.ps", keywords = "wavelets multiresolution approximation nonparametric regression minimax adaptivity piecewise Holder class", abstract = "Standard wavelet shrinkage procedures for nonparametric regression are restricted to equispaced samples. There, data are transformed into empirical wavelet coefficients and threshold rules are applied to the coefficients. The estimators are obtained via the inverse transform of the denoised wavelet coefficients. In many applications, however, the samples are nonequispaced. It can be shown that these procedures would produce suboptimal estimators if they were applied directly to nonequispaced samples. We propose a wavelet shrinkage procedure for nonequispaced samples. We show that the estimate is adaptive and near optimal. For global estimation, the estimate is within a logarithmic factor of the minimax risk over a wide range of piecewise Holder classes, indeed with a number of discontinuities that grows polynomially fast with the sample size. For estimating a target function at a point, the estimate is optimally adaptive to unknown degree of smoothness within a constant. In addition, the estimate enjoys a smoothness property: if the target function is the zero function, then with probability tending to 1 the estimate is also the zero function.", } @Article{cai-hur-tsa:score, title = "Score tests for heteroscedasticity in wavelet regression", author = "Z. W. Cai and C. M. Hurvich and C. L. Tsai", journal = BKA, volume = "85", number = "1", pages = "229--234", year = "1998", keywords = "de-noising. signal extraction. thresholding", abstract = "We consider two Score tests for heteroscedasticity in the errors of a signal;plus-noise model, where the signal is estimated;by wavelet thresholding methods. The error variances are assumed to depend on observed covariates, through a parametric relationship of known form. The tests are based on the approaches of Breusch & Pagan (1979) and Koenker (1981). We establish the asymptotic validity of the tests and examine their performance in a simulation study. The Koenker test is found to perform well, in terms of both size and power.", } @TechReport{cai-sil:incorporating, title = "Incorporating Information on Neighboring Coefficients into Wavelet Estimation", author = "T. Tony Cai and Bernard W. Silverman", institution = "Department of Statistics, Purdue University", number = "98-13", year = "1998", URL = "http://www.stat.purdue.edu/people/tcai/NeighBlock/neighblock.ps", } @TechReport{cai:adaptivity, title = "On Adaptivity of {B}lock{S}hrink Wavelet Estimator Over {B}esov Spaces", author = "T. Tony Cai", institution = "Department of Statistics, Purdue University", number = "97-05", year = "1997", URL = "http://www.stat.purdue.edu/people/tcai/paper/blockbesov.ps", } @TechReport{cai:minimax, title = "Minimax Wavelet Estimation Via Block Thresholding", author = "T. Tony Cai", institution = "Department of Statistics, Purdue University", number = "96-41", year = "1996", URL = "http://www.stat.purdue.edu/people/tcai/paper/blockmini.ps", } @Article{cam-mas:approximation, title = "Wavelet approximation of deterministic and random signals: convergence properties and rates", author = "S. Cambanis and Elias Masry", journal = IEEETIT, volume = "40", number = "4", year = "1994", pages = "1013--1029", keywords = "convergence properties, convergence rates, deterministic signals, wavelet approximation, signal resolution, correlation function, stationary random signals, nonstationary random signals, finite mean energy, scale function, signal smoothness", abstract = "The multiresolution decomposition of deterministic and random signals and the resulting approximation at increasingly finer resolution is examined. Specifically, an nth-order expansion is developed for the error in the wavelet approximation at resolution $2^{-l}$ of deterministic and random signals. The deterministic signals are assumed to have n continuous derivatives, while the random signals are only assumed to have a correlation function with continuous nth-order derivatives off the diagonal-a very mild assumption. For deterministic signals square integrable over the entire real line, for stationary random signals over finite intervals, and for nonstationary random signals with finite mean energy over the entire real line, the smoothness of the scale function can be matched with the signal smoothness to substantially improve the quality of the approximation. In sharp contrast, this is feasible only in special cases for nonstationary random signals over finite intervals and for deterministic signals which are only locally square integrable.", } @Unpublished{cap:denoised, title = "Wavelet de-noised financial time series", author = "Enrico Capobianco", year = "1997", note = "Department of Mathematical Modelling, Technical University of Denmark", URL = "http://eivind.imm.dtu.dk/staff/enrico/duke_enrico.ps", } @Unpublished{cap:feature-detection, title = "Feature Detection with Matching Pursuit in High Frequency Nonstationary Time Series", author = "Enrico Capobianco", year = "1999", note = "Department of Mathematical Modelling, Technical University of Denmark", URL = "ftp://eivind.imm.dtu.dk/pub/300699a.ps.Z", } @Unpublished{cap:high-frequency, title = "High Frequency Stock Returns Volatility: Feature Extraction and Pattern Recognition via Wavelet Transforms", author = "Enrico Capobianco", year = "1999", note = "Department of Mathematical Modelling, Technical University of Denmark", URL = "", } @Unpublished{cap:volatility, title = "Wavelet Transforms for the Statistical Analysis of Returns Generating Stochastic Processes", author = "Enrico Capobianco", year = "1998", note = "Department of Mathematical Modelling, Technical University of Denmark", URL = "http://eivind.imm.dtu.dk/staff/enrico/jedc.ps.Z", } @Unpublished{cap:wavelets, title = "Wavelets for High Frequency Financial Time Series", author = "Enrico Capobianco", year = "1999", note = "Department of Mathematical Modelling, Technical University of Denmark", email = "enrico@eivind.imm.dtu.dk", URL = "http://eivind.imm.dtu.dk/staff/enrico/int99.ps.Z", } @InProceedings{car-hud:EEG-signals, title = "Wavelet denoising of {EEG} signals and identification of evokedresponse potentials", author = "R{\'e}ne A. Carmona and Lonnie H. Hudgins", pages = "91--104", crossref = "lai-uns:wavelet2", abstract = "The purpose of this study is to apply a recently developed wavelet based de-noising filter to the analysis of human electroencephalogram (EEG) signals, and measure its performance. The data used contained subject EEG responses to two different stimuli using the `odd-ball' paradigm. Electrical signals measured at standard locations on the scalp were processed to detect and identify the Evoked Response Potentials (ERP's). First, electrical artifacts emitting from the eyes were identified and removed. Second, the mean signature for each type of response was extracted and used as a matched filter to define baseline detector performance for the noisy data. Third, a nonlinear filtering procedure based on the wavelet extrema representation was used to de-noise the signals. Overall detection rates for the de-noised signals were then compared to the baseline performance. It was found that while the filtered signals have significantly lower noise than the raw signals, detector performance remains comparable. We therefore conclude that all of the information that is important to matched filter detection is preserved by the filter. The implication is that the wavelet based filter eliminates much of the noise while retaining ERP's.", URL = "", } @Book{car-hwa-tor:book, title = "Practical Time-Frequency Analysis: Gabor and Wavelet Transforms with an Implementation in S", booktitle = "Practical Time-Frequency Analysis: Gabor and Wavelet Transforms with an Implementation in S", author = "Ren{\'e} A. Carmona and Wen L. Hwang and Brun Torr{\'e}sani", publisher = "Academic Press", address = "San Diego", series = "Wavelet Analysis and Its Applications", volume = "9", year = "1998", } @Article{car-hwa-tor:characterization, title = "Characterization of Signals by the Ridges of Their Wavelet Transforms", author = "Ren{\'e} A. Carmona and Wen L. Hwang and Brun Torr{\'e}sani", journal = IEEETSP, volume = "45", number = "10", pages = "2586--2590", year = "1997", URL = "ftp://chelsea.princeton.edu/pub/outgoing/signal/cht1.ps", abstract = "We present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of one-dimensional (1-D) signals, These detection procedures are shown to be robust to additive white noise, We also derive and test a new reconstruction procedure, The latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge. This provides a very efficient way to code the information contained in the signal.", } @Article{car-etal:absolute, title = "Absolute optimal time-frequency basis -- a research tool", author = "C{\^a}rsteanu, A. and Sapozhnikov, V. B. and Venugopal, V. and Foufoula-Georgiou, E.", journal = JPA, volume = "30", number = "20", pages = "7133--7146", year = "1997", URL = "", abstract = "The paper presents a method for finding the absolute best basis out of the library of bases offered by the wavelet packet decomposition of a discrete signal. Data-adaptive optimality is achieved with respect to an objective function, e.g. minimizing entropy, and concerns the choice of the Heisenberg rectangles tiling the time-frequency domain over which the energy of the signal is distributed. It is also shown how optimizing a concave objective function is equivalent to concentrating maximal energy into a few basis elements. Signal- adaptive basis selection algorithms currently in use do not generally find the absolute best basis, and moreover have an asymmetric time-frequency adaptivity-although a complete wavepacket decomposition comprises a symmetric set of tilings with respect to time and frequency. The higher adaptivity in frequency than in time can lead to ignoring frequencies that exist over short time intervals (short as compared to the length of the whole signal, not to the period corresponding to these frequencies). Revealing short-lived frequencies to the investigator can bring up important features of the studied process, such as the presence of coherent ('persistent') structures in a time series.", } @InProceedings{car:transients, title = "Wavelet identification of transients in noisy time series", author = "Ren{\'e} A. Carmona", pages = "392--400", crossref = "lai:wavelet1", URL = "http://www.isds.duke.edu/~brani/wp/CarmonaTSTrans.ps", keywords = "transients detection, wavelet identification, noisy time series, signal analysis, denoising procedure, wavelet transform, bootstrap, direct Monte Carlo simulations, reconstruction algorithm", abstract = "The detection of transients in noisy time series is an important part of modern signal analysis because of the importance of its civil and military applications. The author presents a new denoising procedure, the output of which gives a very reasonable guess for the component of the input signal which was buried in noise. The algorithm has two main components. The first one concerns the identification of the main characteristics of the noise component and of the typical effects it has on the wavelet transform of the input signal. This information is used to identify the points in the time-scale space which cannot be extrema of the wavelet transform, unless something else than noise was present in the input signal. This is done by bootstrap in general but direct Monte Carlo simulations can be used when parametric knowledge on the distribution of the noise is available. The second part deals with the actual reconstruction of what is believed to be the component of the input which is to be identified. This part of the algorithm uses the reconstruction procedure of Mallat and Zhong (1992) as revised by the author (1992) the main novelty being the fact that this procedure is fed with the set of points in the time-scale plane which passed the trimming test of the extrema of the wavelet transform. The author illustrates the efficiency of the reconstruction algorithm using the examples of transients used previously by the author (1992).", } @Article{cha-ho:mra, title = "Multiresolution analysis, its link to the discrete parameter wavelet transform, and its initialization", author = "Y. T. Chan and K. C. Ho", journal = IEEETSP, volume = "44", number = "4", year = "1996", pages = "1001--1007", keywords = "Filtering, Electronics, Signal processing", abstract = "Two-scale wavelet equations are derived for equivalent multiresolution analysis (MRA) detail parameters and the discrete parameter (DP) wavelet transform coefficients for a signal s(t). MRA initialization by prefiltering its input signal s(n) obtains the equivalence between the DP and MRA coefficients. MRA gives the DP of a signal s(t) when s(n) are samples of the inner product of s(t) and the scaling function. A simulation example is presented to discuss the prefiltering procedure's effectiveness.", } @TechReport{cha-lon:k-stationarity, title = "K-stationarity and wavelets", author = "Bing Cheng and Howell Tong", number = "96-28", institution = "Institute of Mathematics and Statistics, University of Kent at Cantebury", year = "1996", URL = "ftp://ftp.ukc.ac.uk/pub/maths/reports/1996/28/28.ps.gz", } @Book{cha:basics, title = "Wavelet Basics", author = "Y. T. Chan", publisher = "Kluwer Academic Publishers", address = "Boston", year = "1995", pages = "134", URL = "http://kapis.www.wkap.nl/kapis/CGI-BIN/WORLD/book.htm?0-7923-9536-0", keywords = "Preface. 1. Introduction. 2. Principles of the Wavelet Transform 3. Multiresolution Analysis, Wavelets and Digital Filters. 4. Current Topics. References. Index.", abstract = "Since the study of wavelets is a relatively new area, much of the research coming from mathematicians, most of the literature uses terminology, concepts and proofs that may, at times, be difficult and intimidating for the engineer. Wavelet Basics has therefore been written as an introductory book for scientists and engineers. The mathematical presentation has been kept simple, the concepts being presented in elaborate detail in a terminology that engineers will find familiar. Difficult ideas are illustrated with examples which will also aid in the development of an intuitive insight. Chapter 1 reviews the basics of signal transformation and discusses the concepts of duals and frames. Chapter 2 introduces the wavelet transform, contrasts it with the short-time Fourier transform and clarifies the names of the different types of wavelet transforms. Chapter 3 links multiresolution analysis, orthonormal wavelets and the design of digital filters. Chapter 4 gives a tour d'horizon of topics of current interest: wavelet packets and discrete time wavelet transforms, and concludes with applications in signal processing.", } @Article{che-don-sau:atomic, title = "Atomic decomposition by basis pursuit", author = "Chen, S. S. B. and Donoho, D. L. and Saunders, M. A.", journal = SIAMJSC, volume = "20", number = "1", year = "1999", pages = "33--61", URL = "http://epubs.siam.org/sam-bin/getfile/SISC/articles/30401.ps", keywords = "overcomplete signal representation denoising time-frequency analysis time-scale analysis l 1 norm optimization matching pursuit wavelets wavelet packets cosine packetsinterior-point methods for linear programming total variation denoising multiscale edges", abstract = "The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries- stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an ``optimal'' superposition of dictionary elements, where optimal means having the smallest l(1) norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.", } @MastersThesis{che:msthesis, title = "Wavelet Analysis and Statistics of {CN} Tower Current Waveforms", author = "Ying Chen", year = "1997", pages = "93", school = "Department of Electrical and Computer Engineering, University of Western Ontario", URL = "http://www.ee.ryerson.ca:8080/~yichen/thesis", email = "yichen@ee.ryerson.ca", abstract = "", } @Article{chi-kol-mcc:abws, title = "Adaptive Bayesian Wavelet Shrinkage", author = "Hugh A. Chipman and Eric D. Kolaczyk and Robert E. McCulloch", journal = JASA, volume = "92", number = "440", pages = "1413--1421", year = "1997", URL = "http://galton.uchicago.edu/techreports/ABWS.ps.Z", } @Unpublished{chi-mor:estimation, title = "Estimation of Time Varying Linear Systems", author = "Chang Chiann and Pedro A. Morettin", year = "1999", note = "University of S\~{a}o Paulo, S\~{a}o Paulo, Brazil", URL = "ftp://ftp.ime.usp.br/pub/morettin/lsystem.ps.Z", abstract = "", } @Article{chi-mor:stationary, title = "A Wavelet Analysis for Time Series", author = "Chang Chiann and Pedro A. Morettin", journal = JNS, volume = "10", number = "1", year = "1999", pages = "1-46", email = "pam@ime.usp.br", URL = "ftp://ftp.ime.usp.br/pub/morettin/wavespec.ps.Z", keywords = "time series wavelets wavelet periodogram wavelet spectrum wavelet transform", abstract = "In this paper we develop a wavelet spectral analysis for a stationary discrete process. Some basic ideas on wavelets are given and the concept of wavelet spectrum is introduced. Asymptotic properties of the discrete wavelet transform of a sample of observed values from the process are derived and the wavelet periodogram is considered as an estimator of the wavelet spectrum. Applications to real and simulated series are given.", } @InProceedings{cho-bar:interpolation, title = "Interpolation and denoising of nonuniformly sampled data using wavelet-domain processing", author = "Hyeokho Choi and Richard Baraniuk", booktitle = "Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing", pages = "", year = "1999", URL = "http://www-dsp.rice.edu/publications/pub/choi99in.ps.Z", } @InProceedings{cho-bar:segmentation, title = "Image segmentation using wavelet-domain classification", author = "Hyeokho Choi and Richard Baraniuk", pages = "???--???", crossref = "uns-ald-lai:wavelet7", URL = "http://www-dsp.rice.edu/publications/pub/choi99im.ps.Z", } @InProceedings{cho-etal:complex, title = "Hidden {M}arkov tree modelling of complex wavelet transforms", author = "Hyeokho Choi and Justin Romberg and Richard Baraniuk and Nick Kingsbury", booktitle = "Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing", year = "2000", pages = "???--???", URL = "http://www-dsp.rice.edu/publications/pub/choi00icassp.ps.Z", } @Book{chu:introduction, title = "An Introduction to Wavelets", author = "C. K. Chui", volume = "1", series = "Wavelet Analysis and its Applications", year = "1992", publisher = "Academic Press, Inc.", ISBN = "0-12-174584-8", keywords = "An Overview, Fourier Analysis, Wavelet Transforms and Time-frequency Analysis, Cardinal Spline Analysis, scaling functions and Wavelets, Cardinal Spline Wavelets, Orthogonal Wavelets and Wavelet Packets", abstract = "This is the first volume in the series WAVELET ANALYSIS AND ITS APPLICATIONS. It is an introductory treatise on wavelet analysis, with an emphasis on spline wavelets and and time-frequency analysis. Among the basic topics covered are time frequency localization, intergral wavelet transforms, dyadic wavelets, frames, spine wavelets, orthonormal wavelet bases, and wavelet packets. Is is suitable as a textbook for a beginning course on wavelet analysis and is directed toward both mathematicians and engineers who wish to learn about the subject.", } @Book{chu:theory, title = "Wavelets: Theory, Algorithms, and Applications", editor = "Charles K. Chui and Laura Montefusco and Luigia Puccio", volume = "5", series = "Wavelet Analysis and its Applications", year = "1994", publisher = "Academic Press, Inc.", ISBN = "0-12-174575-9", keywords = "Multiresolution and Multilevel Analyses, Wavelet Transforms, Spline-Wavelets, Other Mathematical Tools for Time-Frequency Analysis, Wavelets and Fractals, Numerical Methods and Algorithms, Applications", abstract = "Wavelets: Theory, Algorithms, and Applications is the fifth volume in the highly respected series, WAVELET ANALYSIS AND ITS APPLICATIONS. This volume shows why wavelet analysis has become a tool of choice in fields ranging from image compression, to signal detection and analysis in electrical engineering and geophysics, to analysis of turbulent or intermittent processes. The 28 papers comprising this volume are organized into seven subject areas: multiresolution analysis, wavelet transforms, tools for time-frequency analysis, wavelets and fractals, numerical methods and algorithms, and applications. More than 135 figures supplement the text.", } @Book{chu:tool, title = "Wavelets: {A} Mathematical Tool for Signal Analysis", author = "Charles K. Chui", series = "SIAM Monographs on Mathematical Modeling and Computation", year = "1997", publisher = SIAM, address = "Philadelphia", ISBN = "0-89871-384-6", abstract = "Wavelets continue to be powerful mathematical tools that can be used to solve problems for which the Fourier (spectral) method does not perform well or cannot handle. This book is for engineers, applied mathematicians, and other scientists who want to learn about using wavelets to analyze, process, and synthesize images and signals. Applications are described in detail and there are step-by-step instructions about how to construct and apply wavelets. The only mathematically rigorous monograph written by a mathematician specifically for nonspecialists, it describes the basic concepts of these mathematical techniques, outlines the procedures for using them, compares the performance of various approaches, and provides information for problem solving, putting the reader at the forefront of current research.", } @Book{chu:tutorial, title = "Wavelets: {A} Tutorial in Theory and Applications", author = "C. K. Chui", volume = "2", series = "Wavelet Analysis and its Applications", year = "1992", publisher = "Academic Press, Inc.", ISBN = "0-12-174590-2", abstract = "Wavelets: A Tutorial in Theory and Applications is the second volume in the new series WAVELET ANALYSIS AND ITS APPLICATIONS. As a companion to the first volume in this series, this volume covers several of the most important areas in wavelets, ranging from the development of the basic theory such as construction and analysis of wavelet bases to an introduction of some of the key applictions, including Mallat's local wavelet maxima technique in second generagion image coding.", } @Proceedings{cia-cox-mon-pav:advanced, title = "Advanced Mathematical Tools in Metrology", booktitle = "Advanced Mathematical Tools in Metrology", editor = "P. Ciarlini and M. Cox and R. Monaco and F. Pavese", volume = "16", series = "Advances in Mathematics for Applied Sciences", publisher = "World Scientific", address = "Singapore", year = "1994", URL = "http://www.wspc.co.uk/wspc/Books/book_series.html", note = "Proceedings of the International Workshop", } @Article{cim-etal:time-series, title = "Time series analysis of geological data", author = "G. Cimino and G. Del Duce and L. K. Kadonaga and G. Rotundo and A. Sisani and G. Stabile and B. Tirozzi and M. Whiticar", journal = CG, volume = "161", number = "1-3", pages = "253--270", year = "1999", abstract = "", } @Article{cly-par-vid:multiple, title = "Multiple shrinkage and subset selection in wavelets", author = "Clyde, M. and Parmigiani, G. and Vidakovic, B.", journal = BKA, volume = "85", number = "2", pages = "391--401", year = "1998", keywords = "", abstract = "This paper discusses Bayesian methods for multiple shrinkage estimation in wavelets. Wavelets are used in applications for data denoising, via shrinkage of the coefficients towards zero, and for data compression, by shrinkage and setting small coefficients to zero. We approach wavelet shrinkage by using Bayesian hierarchical models, assigning a positive prior probability to the wavelet coefficients being zero. The resulting estimator for the wavelet coefficients is a multiple shrinkage estimator that exhibits a wide variety of nonlinear patterns. We discuss fast computational implementations, with a focus on easy-to-compute analytic approximations as well as importance sampling and Markov chain Monte Carlo methods. Multiple shrinkage estimators prove to have excellent mean squared error performance in reconstructing standard test functions. We demonstrate this in simulated test examples, comparing various implementations of multiple shrinkage to commonly-used shrinkage rules. Finally, we illustrate our approach with an application to the so-called 'glint' data.", } @TechReport{cly-geo:empirical, title = "Empirical Bayes Estimation in Wavelet Nonparametric Regression", author = "Merlise A. Clyde and Edward I. George", number = "99--06", institution = "Institute of Statistics and Decision Sciences, Duke University", year = "1999", } @Article{coh-dau-via:interval, title = "Wavelets on the interval and fast wavelet transforms", author = "A. Cohen and I. Daubechies and P. Vial", journal = ACHA, volume = "1", number = "1", year = "1993", pages = "54--81", keywords = "fast wavelet transform, orthonormal wavelet bases, interval", abstract = "The authors discuss several constructions of orthonormal wavelet bases on the interval, and they introduce a new construction that avoids some of the disadvantages of earlier constructions.", } @Book{coh-rya:multiscale, title = "Wavelets and Multiscale Signal Processing", author = "A. Cohen and R. D. Ryan", publisher = "Chapman \& Hall", year = "1995", pages = "248", keywords = "Introduction. Multiresolution analysis. Introduction. The continuous point of view. The discrete point of view. The multivariate case. Conclusions. Wavelets and conjugate quadrature filters. Introduction. The general case. The finite case. Wavelets with compact support. Action of the FWT on oscillating signals. The regularity of scaling functions and wavelets. Introduction. Regularity and oscillation. The subdivision algorithms. Spectral estimates of the regularity. Estimation of the Lp-Sobolev exponent. Applications. Biorthogonal wavelet bases. Introduction. General principles of Subband coding. Unconditional biorthogonal wavelet bases. Dual filters and biorthogonal Riesz bases. Examples and applications. Stochastic processes. Introduction. Linear approximation. Linear approximation of images. Approximation and compression of real images. Piecewise stationary processes. Nonlinear approximation. Quasi-analytic wavelet bases. Multivariate constructions. Multiscale unconditional bases. Notation. References.", abstract = "Since their appearance in the mid-1980s, wavelets and, more generally, multiscale methods have become powerful tools in mathematical analysis and in applications to numerical analysis and signal processing. This book is based on Ondelettes et Traitement Numerique du Signal by Albert Cohen. It has been translated from French by Robert D. Ryan and extensively updated by both Cohen and Ryan. It studies the existing relations between filter banks and wavelet decompositions and shows how these relations can be exploited in the context of digital signal processing. Throughout, the book concentrates on the fundamentals. It begins with a chapter on the concept of multiresolution analysis, which contains complete proofs of the basic results. The description of filter banks that are related to wavelet bases is elaborated in both the orthogonal case (Chapter 2), and in the biorthogonal case (Chapter 4). The regularity of wavelets, how this is related to the properties of the filters, and the importance of regularity for the algorithms are the subjects of Chapter 3. Chapter 5 looks at multiscale decomposition as it applies to stochastic processing, in particular to signal and image processing. Wavelets and Multiscale Signal Processing will be of particular interest to mathematicians working in analysis, academic and research electrical engineers, and researchers who need to analyse time series, in areas such as hydrodynamics, aeronautics, meteorology, geophysics, statistics and economics.", } @Article{coh-raz-mal:adaptive, title = "Adaptive suppression of {W}igner interference-terms using shift-invariant wavelet packet decompositions", author = "Cohen, Israel and Raz, Shalom and Malah, David", journal = SP, volume = "73", number = "3", year = "1999", pages = "203--223", URL = "http://www.elsevier.nl/cas/tree/store/sigpro/1999/73/3/1340.pdf", abstract = "The Wigner distribution (WD) possesses a number of desirable mathematical properties relevant to time-frequency analysis. However, the presence of interference terms renders the WD of multicomponent signals extremely difficult to interpret. In this work, we propose adaptive suppression of interference terms using the shift-invariant wavelet packet decomposition. A prescribed signal is expanded on its best basis and transformed into the Wigner domain. Subsequently, the interference terms are eliminated by adaptively thresholding the cross-WD of interactive basis functions, according to their amplitudes and distance in an idealized time--frequency plane. We define a distance measure that weighs the Euclidean distance with the local distribution of the signal. The amplitude and distance thresholds control the cross-term interference, the useful properties of the distribution, and the computational complexity. The properties of the resultant modified Wigner distribution (MWD) are investigated, and its performance in eliminating interference terms, while still retaining high-energy resolution, is compared with that of other existing approaches. It is shown that the proposed MWD is directly applicable to resolving multicomponent signals. Each component is determined as a partial sum of basis functions over a certain equivalence class in the time--frequency plane.", } @Article{coh-raz-mal:orthonormal, title = "Orthonormal shift-invariant wavelet packet decomposition and representation", author = "Cohen, Israel and Raz, Shalom and Malah, David", journal = SP, volume = "57", number = "3", year = "1997", pages = "251--270", URL = "http://www-sipl.technion.ac.il/Sipl/siltd.ps", keywords = "shift-invariant best basis time frequency wavelets wavelet packets algorithm translation", abstract = "In this work, a shifted wavelet packet (SWP) library, containing all the time shifted wavelet packet bases, is defined. A corresponding shift-invariant wavelet packet decomposition (SIWPD) search algorithm for a best basis' is introduced. The search algorithm is representable by a binary tree, in which a node symbolizes an appropriate subspace of the original signal. We prove that the resultant 'best basis' is orthonormal and the associated expansion, characterized by the lowest information cost, is shift- invariant. The shift invariance stems from an additional degree of freedom, generated at the decomposition stage and incorporated into the search algorithm. The added dimension is a relative shift between given parent node and its respective children nodes. We prove that for any subspace it suffices to consider one of two alternative decompositions, made feasible by the SWP library. These decompositions correspond to a zero shift and a 2(-l) relative shift where l denotes the resolution level. The optimal relative shifts, which minimize the information cost, are estimated using finite depth subtrees. By adjusting their depth, the quadratic computational complexity associated with SIWPD may be controlled at the expense of the attained information cost down to O(Nlog(2)N).", } @Article{coh-raz-mal:translation-invariant, title = "Translation-invariant denoising using the minimum description length criterion", author = "Cohen, Israel and Raz, Shalom and Malah, David", journal = SP, volume = "75", number = "3", year = "1999", pages = "201--223", URL = "http://www.elsevier.nl/cas/tree/store/sigpro/sub/1999/75/3/1380.pdf", keywords = "denoising signal estimation shift-invariant wavelet packet minimum description length best basis time-frequency representation Wigner distribution", abstract = "A translation-invariant denoising method based on the minimum description length (MDL) criterion and tree-structured best-basis algorithms is presented. A collection of signal models is generated using an extended library of orthonormal wavelet-packet bases, and an additive cost function, approximately representing the MDL principle, is derived. We show that the minimum description length of the noisy observed data is achieved by utilizing the shift-invarient wavelet packet decomposition (SIWPD) and thresholding the resulting coefficients. This approach is extendable to local trigonometric decompositions, and corresponding procedures to optimize either the library of bases or the filter banks used at each node of the expansion-tree are described. The signal estimator is efficiently combined with a modified Wigner distribution, yielding robust time-frequency representations, characterized by high resolution and suppressed interference-terms. The proposed method is compared to alternative existing methods, and its superiority is demonstrated by synthetic and real data examples.", } @Book{coh:time-frequency, title = "Time Frequency Analysis: Theory and Applications", author = "Leon Cohen", publisher = "Prentice Hall, Inc.", address = "New Jersey", year = "1994", pages = "320", keywords = "The Time and Frequency Description of Signals. Instantaneous Frequency and the Complex Signal. The Uncertainty Principle. Densities and Characteristic Functions. The Need for Time-Frequency Analysis. Time-Frequency Distributions: Fundamental Ideas. The Short-Time Fourier Transform. The Wigner Distribution. General Approach and the Kernel Method. Characteristic Function Operator Method. Kernel Design for Reduced Interference. Some Distributions. Further Developments. Positive Distributions Satisfying the Marginals. The Representation of Signals. Density of a Single Variable. Joint Representations for Arbitrary Variables. Scale. Joint Scale Representations", abstract = "Featuring traditional coverage as well as new research results that, until now, have been scattered throughout the professional literature, this book brings together --- in simple language --- the basic ideas and methods that have been developed to study natural and man-made signals whose frequency content changes with time; e.g., speech, sonar and radar, optical images, mechanical vibrations, acoustic signals, biological/biomedical and geophysical signals. Covers time analysis, frequency analysis, and scale analysis; time-bandwidth relations; instantaneous frequency; densities and local quantities; the short time Fourier Transform; time-frequency analysis; the Wigner representation; time-frequency representations; computation methods; the synthesis problem; spatial-spatial/frequency representations; time-scale representations; operators; general joint representations; stochastic signals; and higher order time-frequency distributions. Illustrates each concept with examples and shows how the methods have been extended to other variables, such as scale.", } @InProceedings{coi-don:spinning, title = "Time-Invariant Wavelet De-Noising", author = "Ronald R. Coifman and David Donoho", pages = "125--150", URL = "ftp://playfair.stanford.edu/pub/donoho/TIDeNoise.ps.Z", crossref = "ant-opp:wavelets", abstract = "", } @InCollection{coi-etal:packets, author = "Ronald R. Coifman and Yves Meyer and Stephen Quake and Mladen Victor Wickerhauser", title = "Signal Processing and Compression with Wavelet Packets", pages = "77--93", crossref = "mey-roq:progress", } @InCollection{coi-mey-wic:analysis, author = "Ronald R. Coifman and Yves Meyer and Mladen Victor Wickerhauser", title = "Wavelet analysis and signal processing", pages = "153--178", crossref = "rus-etal:wavelets", URL = "ftp://wuarchive.wustl.edu:/doc/techreports/wustl.edu/math/wasp.ps.Z.", abstract = "This describes the use of wavelet analysis for various tasks in signal processing.", } @InProceedings{coi-mey-wic:size, title = "Size properties of wavelet packets", author = "Ronald R. Coifman and Yves Meyer and Mladen Victor Wickerhauser", pages = "453--470", crossref = "rus-etal:wavelets", } @Article{coi-wic:entropy-based, title = "Entropy-Based Algorithms for Best Basis Selection", author = "Ronald R. Coifman and Mladen Victor Wickerhauser", journal = IEEETIT, volume = "38", number = "2", year = "1992", pages = "713--718", URL = "http://wuarchive.wustl.edu/doc/techreports/wustl.edu/math/papers/entbb.ps.Z", abstract = "Adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals. It permits efficient compression of a variety of signals, such as sound and images. The predefined libraries of modulated waveforms include orthogonal wavelet-packets and localized trigonometric functions, and have reasonably well-controlled time-frequency localization properties. The idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost. The method relies heavily on the remarkable orthogonality properties of the new libraries: all expansions in a given library conserve energy and are thus comparable. Several cost functionals are useful; one of the most attractive is Shannon entropy, which has a geometric interpretation in this context.", } @Article{cor-vet:time-varying, title = "Orthogonal time-varying filter banks and wavelet packets", author = "Herley, C. and Vetterli, M.", journal = IEEETSP, volume = "42", number = "10", year = "1994", pages = "2650--2663", keywords = "orthogonal time-varying filter banks wavelet packets construction time domain description two-channel orthogonal filter bank one-sided signals finite-length signals optimization subband tree structures filter sets iteration continuous-time bases half-line regions interval regions", abstract = "We consider the construction of orthogonal time-varying filter banks. By examining the time domain description of the two- channel orthogonal filter bank the authors find it possible to construct a set of orthogonal boundary filters, which allows to apply the filter bank to one-sided or finite-length signals, without redundancy or distortion. The method is constructive and complete. There is a whole space of orthogonal boundary solutions, and there is considerable freedom for optimization. This may be used to generate subband tree structures where the tree varies over time, and to change between different filter sets. The authors also show that the iteration of discrete- time time-varying filter banks gives continuous-time bases, just as in the stationary case. This gives rise to wavelet, or wavelet packet, bases for half-line and interval regions.", } @Article{cor:boundary, title = "Boundary filters for finite-length signals and time-varying filter banks", author = "Herley, C.", journal = IEEETCS2, volume = "42", number = "2", year = "1995", pages = "102--114", keywords = "finite-length signals time-varying filter banks M-channel nonorthogonal case boundary filters time-varying synthesis section nonorthogonal bases frequency localization multirate filter banks", abstract = "We examine the question of how to construct time-varying filter banks in the most general M-channel nonorthogonal case. We show that by associating with both analysis and synthesis operators a set of boundary filters, it is possible to make the analysis structure vary arbitrarily in time, and yet reconstruct the input with a similarly time-varying synthesis section. There is no redundancy or distortion introduced. This gives a solution to the problem of applying filter banks to finite length signals; it suffices to apply the boundary filters at the beginning and end of the signal segment. This also allows the construction of orthogonal and nonorthogonal bases with essentially any prescribed time and frequency localization, but which, nonetheless, are based on structures with efficient filter bank implementations.", } @Article{cou-cou:wavelet-HMMs, title = "Wavelet-based method for nonparametric estimation of {HMM}'s", author = "Couvreur, L. and Couvreur, C.", journal = IEEESPL, volume = "7", number = "2", year = "2000", pages = "25--27", keywords = "", abstract = "In this letter, we propose a new algorithm for nonparametric estimation of hidden Markov models (HMM's), The algorithm is based on a ``wavelet-shrinkage'' density estimator for the state-conditional probability density functions of the HMM's. It operates in an iterative fashion similar to that of the EM reestimation formulae used for maximum-likelihood estimation of parametric HMM's. We apply the resulting algorithm to simple examples and show its convergence. The proposed method is also compared to classical nonparametric HMM estimation based on quantization of observations (``histograms'') and discrete HMM's.", } @Article{cra:xy, title = "On the frequency function of $xy$", author = "Cecil C. Craig", journal = AofMS, volume = "7", pages = "1--15", year = "1936", } @InProceedings{cre-hew:neighbor, title = "A wavelet-based method of nearest neighbor pattern classification using scale sequential matching", author = "C. D. Creusere and G. Hewer", booktitle = "Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers", volume = "2", editor = "A. Singh", year = "1994", pages = "1123--1127", keywords = "nearest neighbor pattern classification, scale sequential matching. wavelet-based method. post-detection radar pulses, feature extraction. trained nearest neighbor codebook. pattern classifier, radar pulse fingerprinting. noise, Monte Carlo simulations, shift invariant classifier, undecimated wavelet transform, pulse edge, wavelet coefficients", abstract = "In this method of pattern classification a wavelet transform is used to extract features from the input signal which are then compared in a scale sequential manner (from coarse to fine) to a trained nearest neighbor codebook. At each scale, possible classification categories are eliminated until only one class is left. We apply this pattern classifier to the problem of fingerprinting post-detection radar pulses and analyze its performance in noise using Monte Carlo simulations. To make our classifier shift invariant, we process the input with an undecimated wavelet transform until the pulse edge is sensed and then start decimating the wavelet coefficients as appropriate to each scale.", } @Unpublished{cro-bar:fast-exact, title = "Fast, Exact Synthesis of {G}aussian and non{G}aussian Long-Range Dependent Processes", author = "Matthew S. Crouse and Richard G. Baraniuk", year = "1999", note = "submitted to {\em IEEE Trans. on Info. Theory}", URL = "http://www-dsp.rice.edu/publications/pub/fftMC.ps.Z", abstract = "Wavelet-based statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly Gaussian. These models are unrealistic for many real-world signals. In this paper, we develop a new framework for statistical signal processing based on wavelet-domain hidden Markov models (HMMs). The framework enables us to concisely model the statistical dependencies and nonGaussian statistics encountered with real-world signals. Wavelet-domain HMMs are designed with the intrinsic properties of the wavelet transform in mind and provide powerful yet tractable probabilistic signal models. Efficient Expectation Maximization algorithms are developed for fitting the HMMs to observational signal data. The new framework is suitable for a wide range of applications, including signal estimation, detection, classification, prediction, and even synthesis. To demonstrate the utility of wavelet-domain HMMs, we develop novel algorithms for signal denoising, classification, and detection.", } @Article{cro-now-bar:hidden, title = "Wavelet-Based Statistical Signal Processing Using Hidden Markov Models", author = "Matthew S. Crouse and Robert D. Nowak and Richard G. Baraniuk", journal = IEEETSP, volume = "46", number = "4", year = "1998", URL = "http://www-dsp.rice.edu/publications/pub/wmarkov.ps.Z", keywords = "", abstract = "Wavelet-based statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly Gaussian. These models are unrealistic for many real-world signals. In this paper, we develop a new framework for statistical signal processing based on wavelet-domain hidden Markov models (HMMs). The framework enables us to concisely model the statistical dependencies and nonGaussian statistics encountered with real-world signals. Wavelet-domain HMMs are designed with the intrinsic properties of the wavelet transform in mind and provide powerful yet tractable probabilistic signal models. Efficient Expectation Maximization algorithms are developed for fitting the HMMs to observational signal data. The new framework is suitable for a wide range of applications, including signal estimation, detection, classification, prediction, and even synthesis. To demonstrate the utility of wavelet-domain HMMs, we develop novel algorithms for signal denoising, classification, and detection.", } @Unpublished{dah-neu:locally, title = "Locally Adaptive...", author = "Rainer Dahlhaus and Michael H. Neumann", year = "1999", note = "StatLab Heidelberg, Institut f{\"u}r Angewandte Mathematik", URL = "ftp://statlab.uni-heidelberg.de/pub/reports/by.series/beitrag.60.ps", abstract = "We fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function \mu(.) and a p-dimensional function theta(.) = (theta^{(1)}(.), ..., theta^{(p)}(.))' that parametrizes the time-varying spectral density f_{theta(.)}(\lambda). Whereas the mean function is estimated by a usual kernel estimator, each component of theta(.) is estimated by a nonlinear wavelet method. According to a truncated wavelet series expansion of theta^{(i)}(.), we define empirical versions of the corresponding wavelet coefficients by minimizing an empirical version of the Kullback-Leibler distance. In the main smoothing step, we perform nonlinear thresholding on these coefficients, which finally provides a locally adaptive estimator of theta^{(i)}(.). This method is fully automatic and adapts to different smoothness classes. It is shown that usual rates of convergence in Besov smoothness classes are attained up to a logarithmic factor.", } @Article{dah-new-von:nonlinear, title = "Nonlinear wavelet estimation of time-varying autoregressive processes", author = "Rainer Dahlhaus and Michael H. Neumann and Rainer {von Sachs}", journal = "Bernoulli", volume = "5", number = "5", year = "1999", URL = "ftp://statlab.uni-heidelberg.de/pub/reports/by.series/report.09.ps", abstract = "We consider nonparametric estimation of the parameter functions a(i)(.), i = 1, ..., p, of a time-varying autoregressive process. Choosing an orthonormal wavelet basis representation of the Functions a(i), the empirical wavelet coefficients are derived from the time series data as the solution of a least-squares minimization problem. In order to allow the a(i) to be functions of inhomogeneous regularity, we apply nonlinear thresholding to the empirical coefficients and obtain locally smoothed estimates of the a(i). We show that the resulting estimators attain the usual minimax L-2 rates up to a logarithmic factor, simultaneously in a large scale of Besov classes. The finite-sample behaviour of our procedure is demonstrated by application to two typical simulated examples." } @Article{dao-fra-wil:MAR, title = "Multiscale autoregressive models and wavelets", author = "Khalid Daoudi and Austin B. Frakt and Alan S. Willsky", journal = IEEETIT, volume = "45", number = "3", year = "1999", pages = "828--845", } @Article{dat-hir:wavelet-based, title = "Wavelet-based estimations in fractional {B}rownian motion", author = "{D'Attellis}, C. E. and Hirchoren, G. A.", journal = "Latin American Applied Research", volume = "29", number = "3-4", year = "1999", pages = "221--225", abstract = "Fractional Brownian motion (fBm) is used as a model of 1/f-type stochastic processes, an important class of processes that characterize a large number of physical phenomena. Since the Hurst parameter H characterizes these processes, its estimation from a signal is an important issue in many applications. This paper points out some applications of recent results on estimations of fBm, and shows a new method for estimating the Hurst parameter. Since variation in H indicates changes in the physical system that produces the measured signal, the determination of these variations is relevant for failure detection. In this paper an algorithm that allows the estimation of changes in the Hurst parameter is introduced. The method is based on multiresolution analysis. The results obtained processing an acoustic emission signal from a coating breakdown are presented.", } @Article{dau-lag:difference1, title = "Two-scale difference equations, {I}", author = "Ingrid Daubechies and J. Lagarias", journal = SIAMJMA, volume = "22", year = "1991", pages = "1388--1410", } @Article{dau-lag:difference2, title = "Two-scale difference equations. {II}. {L}ocal regularity, infinite products of matrices and fractals", author = "Ingrid Daubechies and J. Lagarias", journal = SIAMJMA, volume = "23", year = "1992", pages = "1031--1079", abstract = "We study solutions of the functional equation $f(x)=\sum\sp N\sb {n=0}c\sb nf(kx-n)$, where $k\geq 2$ is an integer, and $\sum\sp N\sb {n=0}c\sb n=k$. Part I showed [SIAM J. Math. Anal. 22 (1991), no. 5, 1388--1410; MR 92d:39001] that equations of this type have at most one $L\sp 1$-solution up to a multiplicative constant, which necessarily has compact support in $[0,N/k-1]$. This paper gives a time-domain representation for such a function $f(x)$ (if it exists) in terms of infinite products of matrices (that vary as $x$ varies). Sufficient conditions are given on $\{c\sb n\}$ for a continuous nonzero $L\sp 1$-solution to exist. Additional conditions sufficient to guarantee $f\in C\sp r$ are also given. The infinite matrix product representations are used to bound from below the degree of regularity of such an $L\sp 1$-solution and to estimate the Holder exponent of continuity of the highest-order well-defined derivative of $f(x)$. Such solutions $f(x)$ are often smoother at some points than others. For certain $f(x)$ a hierarchy of fractal sets in $\bold R$ corresponding to different Holder exponents of continuity for $f(x)$ is described.", } @TechReport{dau-swe:factor, author = "I. Daubechies and W. Sweldens", title = "Factoring Wavelet Transforms into Lifting Steps", institution = "Bell Laboratories, Lucent Technologies", year = "1996", URL = "http://cm.bell-labs.com/who/wim/papers/factor.ps.gz", abstract = "The lifting scheme is a new flexible tool for constructing wavelets and wavelet transforms. In this paper, we use the Euclidean algorithm to show how any discrete wavelet transform or two band subband transform with finite filters can be obtained with a finite number of lifting steps starting from the Lazy wavelet (or polyphase transform). We show a bound on the number of lifting steps which is proportional to the length of the filters. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal (non-unitary) case. The lifting factorization asymptotically reduces the computational complexity of the transform by a factor of two and allows for wavelet transforms that map integers to integers.", } @InCollection{dau:connection, title = "Orthonormal Bases of Wavelets with Finite Support -- Connection with Discrete Filters", author = "Ingrid Daubechies", pages = "38--66", crossref = "com-gro-tch:wavelets", abstract = "", } @Article{dau:orthonormal, title = "Orthonormal bases of compactly supported wavelets", author = "Ingrid Daubechies", journal = "Communications in Pure and Applied Mathematics", volume = "41", pages = "909--996", year = "1988", } @Book{dau:ten, title = "Ten Lectures on Wavelets", author = "Ingrid Daubechies", series = "CBMS-NSF Regional Conference Series in Applied Mathematics", volume = "61", publisher = SIAM, address = "Philadelphia", year = "1992", URL = "http://www.siam.org/catalog/mcc02/daubechi.htm", keywords = "Introduction; Preliminaries and Notation; The What, Why, and How of Wavelets; The Continuous Wavelet Transform; Discrete Wavelet Transforms: Frames; Time-Frequency Density and Orthonormal Bases; Orthonormal Bases of Wavelets and Multiresolutional Analysis; Orthonormal Bases of Compactly Supported Wavelets; More About the Regularity of Compactly Supported Wavelets; Symmetry for Compactly Supported Wavelet Bases; Characterization of Functional Spaces by Means of Wavelets; Generalizations and Tricks for Orthonormal Wavelet Bases; References; Indexes.", abstract = "Wavelets are a mathematical development that may revolutionize the world of information storage and retrieval according to many experts. They are a fairly simple mathematical tool now being applied to the compression of data--such as fingerprints, weather satellite photographs, and medical x-rays--that were previously thought to be impossible to condense without losing crucial details. This monograph contains 10 lectures presented by Dr. Daubechies as the principal speaker at the 1990 CBMS-NSF Conference on Wavelets and Applications. The author has worked on several aspects of the wavelet transform and has developed a collection of wavelets that are remarkably efficient. The opening chapter provides an overview of the main problems presented in the book. Following chapters discuss the theoretical and practical aspects of wavelet theory, including wavelet transforms, orthonormal bases of wavelets, and characterization of functional spaces by means of wavelets. The last chapter presents several topics under active research, as multidimensional wavelets, wavelet packet bases, and a construction of wavelets tailored to decompose functions defined in a finite interval. Because of their interdisciplinary origins, wavelets appeal to scientists and engineers of many different backgrounds.", } @Article{dau:time-frequency, title = "The Wavelet Transform, Time-Frequency Localization and Signal Analysis", author = "I. Daubechies", journal = IEEETIT, volume = "36", number = "5", year = "1990", pages = "961--1005", abstract = "Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems.", } @InCollection{dau:wavelet-transform, title = "The Wavelet Transform: {A} Method for Time-Frequency Localization", author = "Ingrid Daubechies", pages = "366--417", crossref = "hay:advances", keywords = "", abstract = "", } @Article{dav-lab-les:commodity, title = "Wavelet Analysis of Commodity Price Behavior", author = "Russel Davidson and Walter C. Labys and Jean-Baptiste Lesourd", journal = CE, volume = "11", number = "", year = "1998", pages = "103--128", } @InCollection{dav-mar-wis:multifractal, title = "Wavelet-based multifractal analysis of non-stationary and/or intermittent geophysical signals", author = "Anthony Davis and Alexander Marshak and Warren Wiscombe", pages = "249--298", crossref = "fou-kum:geophysics", URL = "ftp://climate.gsfc.nasa.gov/pub/davis/Wavelets/Wavelets.text.PS.Z", abstract = "", } @Article{dav:distribution, title = "The Distribution of a Linear Combination of $\chi^2$ Random Variables", author = "Robert B. Davies", journal = AS, volume = "29", number = "", year = "1980", pages = "323--333", keywords = "Quadratic form", } @InCollection{dav:tables, title = "Tables of the Correlation Coefficient", author = "F. N. David", booktitle = "Biometrika Tables for Statisticians", editor = "E. S. Pearson and H. O. Hartley", publisher = "Cambridge University Press", address = "Cambridge", volume = "1", edition = "3", year = "1966", } @Article{del-gam-sal:interannual, title = "Interannual signals in length of day and atmospheric angular momentum", author = "{del Rio}, R. A and Gambis, D. and Salstein, D. A.", journal = AG, volume = "18", number = "3", year = "2000", pages = "347--364", keywords = "Meteorology and atmospheric dynamics (general circulation) - Solar physics, astrophysics and astronomy (celestial mechanics)", abstract = "Atmospheric angular momentum (AAM) and length of day (LOD) series are investigated for their characteristics on interannual time scales during the half-century period 1949 to 1998. During this epoch, the interannual variability in LOD can be separated naturally into three bands: a quasi-biennial, a triennial-quadrennial and one at six-seven years. The atmosphere appears to excite the first two bands, while it does not contribute to the last. Considering the quasi-biennial (QB) band alone, the atmosphere appears to excite most of its signal in LOD, but it arises from separate fluctuations with stratospheric and tropospheric origin. Thus, although close in frequency, stratospheric and tropospheric processes differ in their amplitude and phase variability. The time shift can be noted especially during the strong El Nino events of 1982-83 and 1997-98 when both processes have positive phase and thus combine to help produce particularly strong peak in AAM and LOD. In addition, we have reconfirmed the downward propagation in the stratosphere and upward propagation in the troposphere of AAM observed in earlier studies for other variables. In the triennial-quadrennial (TQ) band, time-variable spectral analyses reveal that LOD and AAM contain strong variability, with periods shorter than four years before 1975 and longer thereafter. This signal originates mainly within the troposphere and propagates upwards from the lower to the higher layers of the troposphere. According to a zonal analysis, an equatorial poleward mode, strongly linked to the SOI, explains more than 60% of the total variability at these ranges. In addition, this study also indicates that an equatorward mode, originating within polar latitudes, explains, on average, more than 15% of the triennial-quadrennial oscillation (TQO) variability in AAM, and up to 30% at certain epochs. Finally, a six year period in LOD noted in earlier studies, as well as in lengthier series covering much of the century, is found to be absent in atmospheric excitations, and it is thus likely to arise from mantle/core interactions.", } @Article{del-jud:computation, journal = JAT, volume = "88", number = "1", year = "1997", pages = "47--79", title = "{O}n the computation of wavelet coefficients", author = "B. Delyon and A. Juditsky", URL = "http://www.irisa.fr/EXTERNE/bibli/pi/pi856.html", abstract = "We consider fast algorithms of wavelet decomposition of a function f when discrete observations of f (supp f subset of or equal to[0,1](d)) are available. The properties of the algorithms are studied for three types of observation design which for $d=1$ can be described as follows: the regular design, when the observations $f(xi)$ are taken on the regular grid $x(i)=i/N$, $i=1, ..., N$; the case of a jittered regular grid, when it is only known that for all 1 less than or equal to i less than or equal to $N$, $i/N$ less than or equal to $x(i)<i+1)/N$; and the random design case; in which $x(i), i=1, ..., N$, are independent and identically distributed random variables on [0,1]. We show that these algorithms are in a certain sense efficient when the accuracy of the approximation is concerned. The proposed algorithms are computationally straightforward; the whole effort to compute the decomposition is order $N$ for the sample size $N$.", } @InCollection{del-jud:estimation, title = "Estimating Wavelet Coefficients", author = "Bernard Delyon and Anatoli Juditsky", pages = "151--168", crossref = "ant-opp:wavelets", URL = "", email = "iouditsk@irisa.fr", abstract = "", } @Article{del-wei:M-band, title = "{\em M}-band wavepacket-based transient signal detector using a translation-invariant wavelet transform", author = "Stephen del Marco and John Weiss", journal = OE, volume = "33", number = "7", year = "1994", pages = "2175--2182", abstract = "This paper develops a two-dimensional M-band translation-invariant wavelet transform (2-D MTI). Use of the MTI overcomes the shift-variance of the wavelet transform by applying a cost function over M shifts of the input signal. The new transform is proven to be translation-invariant. Use of M-band wavelets enables a finer frequency partitioning and greater energy compaction in the transform representation. Examples are presented which show that the translation-invariant transforms provide superior energy concentration compared to the corresponding nominal wavelet transforms. Examples are also presented comparing the energy concentration capability of M-band wavelets and the modulated lapped transform (MLT). We explored the MTI as a tool for image processing by using it to represent several different images.", } @Article{del-wei:improved, title = "Improved transient signal detection using a wavepacket-based detector with an extended translation-invariant wavelet transform", author = "Stephen del Marco and John Weiss", journal = IEEETSP, volume = "45", number = "4", year = "1997", pages = "841--850", URL = "http://www.tiac.net/users/nurit/ETI/ETI.html", abstract = "This paper presents the theory of M-band, extended translation-invariant (ETI) wavelet transforms. The ETI generalizes the translation-invariant wavelet transform of Weiss. It is shown that iteration of the ETI, in a tree structure, provides a signal decomposition into an orthonormal wavepacket basis, Other properties such as translation invariance and invertibility of the transform are proven, The theory is then applied to transient signal detection through development of a family of translation-invariant wavepacket-based detectors. This family of detectors provides improved performance over previously defined wavepacket-based detectors, A performance analysis is conducted. ROC curves generated by Monte-Carlo simulation are presented, indicating detector performance, Detector performance is demonstrated to be independent of the signal translation.", keywords = "compactly supported wavelets, representation, algorithms", } @TechReport{der-lou-war:arbitrary, title = "Multiresolution Analysis for Sufaces of Arbitrary Topological Type", author = "Tony D. DeRose and Michael Lounsbery and Joe Warren", institution = "Department of Computer Science and Engineering, University of Washington", number = "93-10-05", year = "1993", } @Article{dij-maz-bag:reciprocal, title = "Reciprocal processes on a tree-modeling and estimation issues", author = "R. W. Dijkerman and R. R. Mazumdar and A. Bagchi", journal = IEEETAC, volume = "40", number = "2", year = "1995", pages = "330--335", keywords = "reciprocal processes, estimation issues, multiresolution decomposition methods, discrete wavelet transformation, truncated N-ary trees, nearest neighbor models, recursive description, autoregressive processes, zero-valued boundary values, smoothing equations", abstract = "Motivated by multiresolution decomposition methods such as the discrete wavelet transformation, the authors introduce reciprocal processes on truncated N-ary trees. The authors discuss the relationship between such processes and nearest neighbor models. The authors show that they can derive a recursive description of the process, and that all reciprocal processes on N-ary trees reduce to autoregressive processes in the case of zero-valued boundary values at the bottom of the tree, corresponding to truncation of the tree. The authors then study the smoothing equations associated with such models.", } @Article{dij-maz:correlation, title = "On the correlation structure of the wavelet coefficients of fractional {B}rownian motion", author = "R. W. Dijkerman and R. R. Mazumdar", journal = IEEETIT, volume = "40", number = "5", year = "1994", pages = "1609--1612", keywords = "correlation structure, wavelet coefficients, fractional Brownian motion, normalized correlation, decay", abstract = "Shows that the interdependence of the discrete wavelet coefficients of fractional Brownian motion, defined by normalized correlation, decays exponentially fast across scales and hyperbolically fast along time.", } @Article{dij-maz:representation, title = "Wavelet representations of stochastic processes and multiresolution stochastic models", author = "R. W. Dijkerman and R. R. Mazumdar", journal = IEEETSP, volume = "42", number = "7", year = "1994", pages = "1640--1652", keywords = "stochastic processes, multiresolution stochastic models, deterministic signal analysis, wavelet representations, wavelet bases, time domain, correlation structure, discrete wavelet coefficients, correlation decay, trees, time process, approximations, wavelet transform, decorrelation", abstract = "Deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years. In the context of stochastic processes, the use of wavelet bases has not yet been fully investigated. We use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in L/sup 2/ defined in the time domain. We derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales. We study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by Basseville et al. (see IEEE Trans. Inform. Theory, vol.38, p.766-784, Mar. 1992). We propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process. These models are simple due to the strong decorrelation of the wavelet transform. Experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated.", } @Article{don-joh-ker-pic:asymptopia, title = "Wavelet Shrinkage: Asymptopia? (with discussion)", author = "David L. Donoho and Iain M. Johnstone and G{\'e}rard Kerkyacharian and Dominique Picard", journal = JRSSB, volume = "57", number = "2", year = "1995", pages = "301--369", URL = "http://playfair.Stanford.EDU/reports/donoho/asymp.ps.Z", abstract = "Much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects - curves, densities, spectral densities, images - from noisy data. A now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems. Unfortunately, the results have rarely moved into practice, for a variety of reasons - among them being similarity to known methods, computational intractability and lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data: translate the empirical wavelet coefficients towards the origin by an amount [sq.root](2 log n) [sigma]/[sq.root]n. The proposal differs from those in current use, is computationally practical and is spatially adaptive; it thus avoids several of the previous objections. Further, the method is nearly minimax both for a wide variety of loss functions - pointwise error, global error measured in Lp-norms, pointwise and global errors in estimation of derivatives - and for a wide range of smoothness classes, including standard Hölder and Sobolev classes, and bounded variation. This is a much broader near optimality than anything previously proposed: we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity.", keywords = "Adaptive estimation Besov spaces Density estimation Minimax estimation Nonparametric regression Optimal recovery Spatial adaptation Wavelet orthonormal bases", } @Article{don-joh:adapting, title = "Adapting to unknown smoothness by wavelet shrinkage", author = "David L. Donoho and Iain M. Johnstone", journal = JASA, volume = "90", number = "", year = "1995", pages = "1200--1224", URL = "http://playfair.Stanford.EDU/reports/donoho/isaws.ps.Z", } @Article{don-joh:asymptotic, title = "Asymptotic minimaxity of wavelet estimators with sampled data", author = "David L. Donoho and Iain M. Johnstone", journal = SSin, volume = "9", number = "1", year = "1999", pages = "1--32", keywords = "Besov spaces bounded operators between Besov spaces Minimax estimation thresholding wavelet transforms of sampled data wavelets white noise equivalence", abstract = "Donoho and Johnstone (1998) studied a setting where data were obtained in the continuum white noise model and shop;ed that scalar nonlinearities applied to wavelet coefficients gave estimators which were asymptotically minimax over Besov balls. They claimed that this implied similar asymptotic minimaxity results in the sampled-data model. In this paper we carefully develop and fully prove this implication. Our results are based on a careful definition of an empirical wavelet transform and precise bounds on the discrepancy between empirical wavelet coefficients and the theoretical wavelet coefficients.", } @Article{don-joh:ideal, title = "Ideal spatial adaptation by wavelet shrinkage", author = "David L. Donoho and Iain M. Johnstone", journal = BKA, volume = "81", number = "3", year = "1994", pages = "425--455", URL = "http://playfair.Stanford.EDU/reports/donoho/ausws.ps.Z", } @Article{don-joh:minimax, title = "Minimax estimation via wavelet shrinkage", author = "David L. Donoho and Iain M. Johnstone", journal = AofS, volume = "26", number = "3", year = "1998", pages = "879--921", URL = "http://playfair.Stanford.EDU/reports/donoho/mews.ps.Z", keywords = "minimax decision theory minimax Bayes estimation Besov Holder Sobolev Triebel spaces nonlinear estimation white noise model nonparametric regression orthonormal bases of compactly supported wavelets renormalization white noise approximation", abstract = "We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets, we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficient. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel- and Besov-type smoothness constraints and asymptotically minimax over Besov bodies with p less than or equal to q. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with p < 2, so the method can significantly outperform every linear method (e.g., kernel, smoothing spline, sieve) in a minimax sense. Variants of our method based on simple threshold nonlinear estimators are nearly minimax. Our method possesses the interpretation of spatial adaptivity; it reconstructs using a kernel which may vary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper, which was first drafted in November 1990, discuss practical implementation, spatial adaptation properties, universal near minimaxity and applications to inverse problems.", } @Article{don-joh:neo-classical, title = "Neo-classical minimax problems, thresholding and adaptive function estimation", author = "David L. Donoho and Iain M. Johnstone", journal = Ber, volume = "2", number = "1", year = "1996", pages = "39--62", URL = "", abstract = "We study the problem of estimating θ from data Y&126;N(θ, σ2) under squared-error loss. We define three new scalar minimax problems in which the risk is weighted by the size of θ. Simple thresholding gives asymptotically minimax estimates in all three problems. We indicate the relationships of the new problems to each other and to two other neo-classical problems: the problems of the bounded normal mean and of the risk-constrained normal mean. Via the wavelet transform, these results have implications for adaptive function estimation in two settings: estimating functions of unknown type and degree of smoothness in a global l2 norm; and estimating a function of unknown degree of local Hölder smoothness at a fixed point. In the latter setting, the scalar minimax results imply: Lepskii's results that it is not possible fully toadapt the unknown degree of smoothness without incurring a performance cost; and that simple thresholding of the empirical wavelet transform gives an estimate of a function at a fixed point which is, to within constants, optimally adaptive to unknown degree of smoothness.", keywords = "adaptive estimation; lp balls; minimax estimation; weak lp balls", } @Unpublished{don-mal-von:covariances, title = "Estimating Covariances of Locally Stationary Processes: Rates of Convergence of Best Basis Methods", author = "David L. Donoho and Steph{\'a}ne Mallat and Rainer {von Sachs}", year = "1997", note = "Unpublished", } @TechReport{don:interpolating, title = "Interpolating Wavelet Transforms", author = "David L. Donoho", year = "1992", institution = "Technical Report 408, Department of Statistics, Stanford University", URL = "http://playfair.Stanford.EDU/reports/donoho/interpol.ps.Z", } @InProceedings{don:nonlinear, title = "Nonlinear Wavelet Methods for Recovery of Signals, Densities, and Spectra from Indirect and Noisy Data", author = "David L. Donoho", booktitle = "Proceedings of Symposia in Applied Mathematics", organization = AMS, volume = "47", year = "1993", pages = "173--205", URL = "http://playfair.Stanford.EDU/reports/donoho/ShortCourseFigs.epsf.shar.Z", abstract = "Wavelet methods for the recovery of objects from noisy and incomplete data are described. The common themes: (a) the new methods use nonlinear operations in the wavelet domain; (b) they accomplish tasks which are not possible by traditional linear/Fourier approaches to such problems. An attempt is made to indicate the heuristic principles, theoretical foundations and possible application areas for these methods, i.e. wavelet de-noising, wavelet approaches to linear inverse problems, wavelet packet de-noising, segmented multiresolutions, and nonlinear multi-resolutions.", } @InCollection{don:smooth, author = "David L. Donoho", title = "Smooth Wavelet Decompositions with Blocky Coefficient Kernels", pages = "1--43", URL = "http://playfair.Stanford.EDU/reports/donoho/blocky.ps.Z", crossref = "sch-web:recent", abstract = "", } @Article{don:soft, title = "De-noising by Soft-Thresholding", author = "David L. Donoho", journal = IEEETIT, volume = "41", number = "3", year = "1995", pages = "613--627", URL = "http://playfair.Stanford.EDU/reports/donoho/denoiserelease3.ps.Z", keywords = "soft-thresholding, de-noising, reconstruction, unknown function, noisy data, standard Gaussian random variables, wavelet domain, empirical wavelet coefficients, estimator, probability, smoothness measures, statistical inference, optimal recovery model", abstract = "Donoho and Johnstone (1994) proposed a method for reconstructing an unknown function f on (0,1) from noisy data d/sub i/=f(t/sub i/)+ sigma z/sub i/, i=0, ..., n-1,t/sub i/=i/n, where the z/sub i/ are independent and identically distributed standard Gaussian random variables. The reconstruction f*/sub n/ is defined in the wavelet domain by translating all the empirical wavelet coefficients of d toward 0 by an amount sigma . square root (2log (n)/n). The authors prove two results about this type of estimator. (Smooth): with high probability f*/sub n/ is at least as smooth as f, in any of a wide variety of smoothness measures. (Adapt): the estimator comes nearly as close in mean square to f as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. The present proof of these results develops new facts about abstract statistical inference and its connection with an optimal recovery model.", } @Article{dor:least-asymmetric, title = "On the least asymmetric wavelets", author = "Doroslova\v{c}ki, Milo\v{s} L.", journal = IEEETSP, volume = "46", number = "4", year = "1998", pages = "1125-1130", keywords = "least asymmetric wavelets scaling functions second moment wavelet-generating discrete-time filter uncertainty relation discrete-time signals asymmetry measures signal processing transfer function phase nonlinearities minimisation", abstract = "The asymmetry of Daubechies' (1988, 1992) scaling functions and wavelets can be diminished by minimizing a special second moment in time for the wavelet-generating discrete-time filter. The moment is involved in an uncertainty relation for discrete-time signals. Other measures of asymmetry are addressed as well, and corresponding results are compared.", } @Article{dow-sil:multiple, title = "The discrete multiple wavelet transform and thresholding methods", author = "T. R. Downie and B. W. Silverman", journal = IEEETSP, volume = "46", number = "9", year = "1998", pages = "2558--2562", URL = "http://www.stats.bris.ac.uk/~bernard/threshmwave3.ps.gz", keywords = "", abstract = "", } @Article{dro-kat:new, title = "New filter banks and more regular wavelets", author = "Karim Drouiche and Djalil Kateb", journal = IEEETSP, volume = "47", number = "8", year = "1999", pages = "2220--2227", URL = "", keywords = "", abstract = "", } @Article{duf-mil:prior, title = "Statistical signal restoration with $1/f$ wavelet domain prior models", author = "R. M. Dufour and E. L. Miller", journal = SP, volume = "78", number = "3", year = "1999", pages = "289--307", email = "rdufour@cdsp.neu.edu,elmiller@ece.neu.edu", } @InProceedings{dut:algorithm, title = "An Implementation of the ``algorithme {\`a} trous'' to Compute the Wavelet Transform", author = "P. Dutilleux", pages = "298--304", crossref = "com-gro-tch:wavelets", abstract = "", } @Unpublished{edw:discrete, title = "Discrete Wavelet Transforms: Theory and Implementation", author = "Tim Edwards", note = "Deptartment of Statistics, Stanford University", year = "1991", email = "tim@sinh.stanford.edu", abstract = "This includes a brief introduction to wavelets in general and the discrete wavelet transform in particular, covering a number of implementation issues that are often missed in the literature. A hardware implementation on a commercially available DSP system is described along with a program listing to show how such an implementation can be simulated.", } @Article{efr-mor:stein, title = "Data Analysis Using {S}tein's Estimator and Its Generalizations", author = "Bradley Efron and Carl Morris", journal = JASA, volume = "70", number = "350", year = "1975", pages = "311--319", } @Article{efr:overcome, title = "How to overcome the curse of long-memory errors", author = "Efromovich, S.", journal = IEEETIT, volume = "45", number = "5", year = "1999", pages = "1735--1741", author = "Long-memory errors dramatically slow down the convergence of minimax risks in a fixed design nonparametric regression. The problem becomes even more complicated for the case of adaptive estimation. This defines the curse of long-memory errors. I show that using a random design, instead of a fixed one, allows one to overcome this curse and make familiar data-driven estimators robust. Moreover, the result holds for a wide class of nonstationary errors with bounded moments (including bounded deterministic errors). Possible extensions are discussed.", } @Article{efr:quasi-linear, title = "Quasi-linear wavelet estimation", author = "Efromovich, S.", journal = JASA, volume = "94", number = "445", year = "1999", pages = "189--204", keywords = "adaptation asymptotic besov space data compression filtering monotone function Monte Carlo nonparametric regression rate optimality sharp optimality small sample", abstract = "The main paradigm of the modern wavelet theory of spatial adaptation formulated by Donoho and Johnstone is that there is a divergence between the linear minimax adaptation theory and the heuristic guiding algorithm development that leads to the necessity of using strongly nonlinear adaptive thresholded methods. On the other hand, it is well known that linear adaptive estimates are the best whenever an estimated function is smooth. Is it possible to suggest a quasi-linear wavelet estimate, by adding to a linear adaptive estimate a minimal number of nonlinear terms on finest scales, that offers advantages of linear adaptive estimates and at the same time matches asymptotic properties of strongly nonlinear procedures like the benchmark SureShrink? The answer is ``yes,'' and we discuss quasi-linear estimation both theoretically and via a Monte Carlo study. In particular, I show that, asymptotically, a quasi-linear procedure not only matches properties of SureShrink over the Besov scale, but also allows us to relax familiar assumptions and solve a long standing problem of rate and sharp optimal estimation of monotone functions. For the case of small sample sizes and functions that contain spiky/jumps parts and smooth parts, a quasi-linear estimate performs exceptionally well in terms of visual aesthetic appeal, approximation, and data compression.", } @InProceedings{erd-bao-che:interpolation, title = "Wavelet Interpolation: {F}rom Orthonomal To The Oversampled Wavelet Transform", author = "Nurgun Erdol and Feng Bao and Zajing Chen", booktitle = "International Conference on Acoustics, Speech, and Signal Processing", volume = "2", pages = "1093--1096", year = "1995", note = "9-12 May 1995, Detroit, MI, USA", keywords = "oversampled wavelet transform. wavelet interpolation. orthonormal wavelet transform. signal representation. aliasing. decimation stage. signal resolution. signal processing. wideband correlation processing. time alignment sensitivity. nonorthogonal wavelet transform. redundancy. wavelet transform coefficients. feature extraction. filter banks.", abstract = "The orthonormal wavelet transform is an efficient way for signal representation since there is no redundancy in its expression, but due to aliasing in the decimation stage it lacks the often desired property of shift invariance. On the other hand, the oversampled or nonorthogonal wavelet offers a finer resolution in translation; thus reducing the effect of shift of origin, it becomes more robust to changes in the initial phase of the signal. In some areas of signal processing, such as wideband correlation processing, sensitivity to time alignment necessitates the use of the nonorthogonal wavelet transform. The price paid for the advantage of robustness to shifting is the introduction of redundancy in the expression. In many applications, both of these two properties are needed in different stages of signal processing. Thus there is a need to know the conditions under which the redundant and nonorthonormal wavelet transform coefficients can be derived from the orthonormal wavelet transform coefficients. The answer provides us with a convenient way to switch between these two forms: the orthonormal wavelet for efficient expression, and the nonorthogonal one whenever it is necessary for feature extraction.", } @InProceedings{erd-bao:use, title = "{U}se of shift variance of the wavelet transform for signal detection", booktitle = "Sixth IEEE Digital Signal Processing Workshop", year = "1994", author = "N. Erdol and Bao. Feng", note = "2-5 Oct. 1994, Yosemite National Park, CA, USA", abstract = "Characterizes signals according to the degree with which a time shift affects their wavelet series coefficients and develops a measure called the `shift index' to quantify that effect. The authors argue that the shift index can be used to locate, separate and cluster and/or detect pulse like signals with random arrival times. Examples are given to verify the established theory.", keywords = "wavelet transform. signal detection. shift variance. time shift. wavelet series coefficients. shift index. pulse like signals. random arrival times.", } @Article{esk-etal:seperating, title = "Separating different scales of motion in time series of meteorological variables", author = "Robert E. Eskridge and Jia-Yeong Ku and S. Trivikrama Rao and P. Steven Porter and Igor G. Zurbenko", journal = BAMetS, volume = "78", number = "7", year = "1997", pages = "1473--1484", abstract = "The removal of synoptic and seasonal signals from time series of meteorological variables leaves datasets amenable to the study of trends, climate change, and the reasons for such trends and changes. In this paper, four techniques for separating different scales of motion are examined and their effectiveness compared. These techniques are PEST, anomalies, wavelet transform, and the Kolmogorov-Zurbenko (KZ) filter. It is shown that PEST and anomalies do not cleanly separate the synoptic and seasonal signals from the data as well as the other two methods. The KZ filter method is shown to have the same level of accuracy as the wavelet transform method. However, the KZ filter method can be applied to datasets with missing observations and is much easier to use than the wavelet transform method.", } @Article{fan-hal-mar-pat:adaptation, title = "Adaptation to high spatial inhomogeneity using wavelet methods", author = "J. Q. Fan and P. Hall and M. A. Martin and P. Patil", journal = SSin, volume = "9", number = "1", year = "1999", pages = "85--102", keywords = "convergence rate. fine-scale. local adaptivity. resolution. wavelet", abstract = "Many of the signals to which waselet methods are applied, including those encountered in simulation experiments, are essentially smooth but contain a small number of high-frequency episodes such as spikes. In principle it is possible to employ a different amount of smoothing at different spatial locations, but in the context of wavelets this is so awkward to implement that it is not really practicable. Instead, it is attractive to select the primary resolution level (or smoothing parameter) so as to give good performance for smooth parts of the signal. While this is readily accomplished using a cross-validation argument., it is unclear whether it has a deleterious impact on performance at high-frequency episodes. In this paper we show that it does not. We derive upper and lower bounds to pointwise rates of convergence for functions whose `spikiness' increases with sample size. (This allows us to model contexts where wavelet methods have to work hard to recover high-frequency events.) We show that, in order to achieve optimal rates of convergence, it is necessary for the primary resolution level of the empirical wavelet transform to vary with location, sometimes extensively. nevertheless, the convergence rate penalty incurred through using a non-varying resolution level, chosen to provide good performance for coarse-scale features, equals a factor that is less than the logarithm of sample size.", } @Article{fan-hal-mar-pat:local, journal = JASA, volume = "91", number = "433", year = "1996", pages = "258--266", title = "{O}n local smoothing of nonparametric curve estimators", author = "J. Q. Fan and P. Hall and M. A. Martin and P. Patil", abstract = "We develop new local versions of familiar smoothing methods; such as cross-validation and smoothed cross-validation, in the contexts of density estimation and regression. These new methods are locally adaptive in the sense that they capture smooth local fluctuations in the curve by using smoothly varying bandwidths that change as the character of the curve changes. Moreover, the new methods are accurate, easy to apply, and computationally expedient.", keywords = "density-estimation. bandwidth choice. cross-validation.", } @Article{fan-lin:curves, title = "Test of significance when data are curves", author = "Fan, J. Q. and Lin, S. K.", journal = JASA, volume = "93", number = "443", year = "1998", pages = "1007--1021", keywords = "adaptive analysis of variance adaptive Neyman rest functional data repeated measurements thresholding wavelets", abstract = "With modern technology, massive data can easily be collected in a form of multiple sets of curves. New statistical challenge includes testing whether there is any statistically significant difference among these sets of curves. In this article we propose some new tests for comparing two groups of curves based on the adaptive Neyman test and the wavelet thresholding techniques introduced earlier by Fan. We demonstrate that these tests inherit the properties outlined by Fan and that they are simple and powerful for detecting differences between two sets of curves. We then further generalize the idea to compare multiple sets of curves, resulting in an adaptive high-dimensional analysis of variance, called HANOVA. These newly developed techniques are illustrated by using a dataset on pizza commercials where observations are curves and an analysis of cornea topography in ophthalmology where images of individuals are observed. A simulation example is also presented to illustrate the power of the adaptive Neyman test.", } @Article{fan:test, journal = JASA, volume = "91", number = "434", year = "1996", pages = "674--688", title = "{T}est of significance based on wavelet thresholding and {N}eyman's truncation", author = "J. Q. Fan", abstract = "Traditional nonparametric tests, such as the Kolomogorov-Smirnov test and the Cramer-Von Mises test, are based on the empirical distribution functions. Although these tests possess root-n consistency, they effectively use only information contained in the low frequencies. This leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high-frequency alternations. The drawback can be repaired via smoothing-based test statistics. In this article we propose two such kind of test statistics based on the wavelet thresholding and the Neyman truncation. We provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations, while maintaining the same capability in detecting smooth alternative densities as the traditional tests. Similar conclusions can be made for two-sample nonparametric tests of distribution functions. In that case, the traditional linear rank tests such as the Wilcoxon test and the Fisher-Yates test have low power in detecting two nearby densities where one has local features or contains high-frequency components, because these procedures are essentially testing the uniform distribution based on the sample mean of rank statistics. In contrast, the proposed tests use more fully the sampling information and have better ability in detecting subtle features.", keywords = "adaptive Neyman test. goodness-of-fit. hard-thresholding parameter. soft-thresholding parameter. two-sample test. wavelet thresholding.", } @InProceedings{far-bro:applications, title = "Applications of Time-Frequency and Time-Scale Transforms to Ultra-Wideband Radar Transient Signal Detection", author = "Monique P. Fargues and William A. Brooks", booktitle = "Advanced Signal Processing Algorithms, Architectures, and Implementations IV", editor = "Franklin T. Luk", volume = "2027", organization = SPIE, year = "1993", pages = "180--193", address = "San Diego, California", abstract = "Compared to conventional radars, ultra-wideband (UWB) radars are characterized by very large bandwidth and fine range resolution. Potential applications of this type of radar include terrain mapping, and target identification/classification. In this paper we use a non- stationary approach and analyze UWB radar data using time- frequency and time-scale transformations. The time-frequency transformations considered are the Short-Time Fourier Transform (STFT), the Wigner-Ville Distribution (WD), the Instantaneous Power Spectrum (IPS), and the ZAM transform. Two discrete implementations of the Wavelet Transform (DWT) are also investigated: the decimated A-trous algorithm proposed by Holschneider et al, which uses non-orthogonal wavelets; and the Mallat algorithm, which employs orthogonal wavelets. The transients under study are UWB radar returns from a boat (with and without corner reflector) in the presence of sea clutter, multipath, and radio frequency interferences (RFI). Results show that all time-frequency and time-scale transforms clearly detect the transient radar returns corresponding to the boat with a corner reflector. However, as the radar cross section of the target decreases (boat without a corner reflector), results change drastically as the RFI component dominates the signal. Simulations show that the Instantaneous Power Spectrum may be better adapted for localizing the transient among the time-frequency techniques studied. The decimated A-trous algorithm has the best time resolution of the techniques studied as the return appears better localized in the scalogram.", } @Proceedings{far-hun-vas:wavelets, title = "Wavelets, fractals, and Fourier transforms", booktitle = "Wavelets, fractals, and Fourier transforms", editor = "M. Farge and Julian C. R. Hunt and J. C Vassilicos", volume = "43", series = "Institute of Mathematics and Its Applications conference series", year = "1993", publisher = "Clarendon Press", address = NY, keywords = "Wavelets, Mathematics, Fractals, Fourier transformations", loc = "QA403.3 .W39 1993", note = "Based on the proceedings of a conference on wavelets, fractals, and Fourier transforms held at Newnham College, Cambridge in December 1990", } @Article{far-kev-per-goi:turbulence, journal = PIEEE, volume = "84", number = "4", year = "1996", pages = "639--669", title = "{W}avelets and turbulence", author = "M. Farge and N. Kevlahan and V. Perrier and U. Goirand", abstract = "We have used wavelet transform techniques to analyze, model, and compute turbulent flows. The theory and open questions encountered in turbulence are presented The wavelet-based techniques that we have applied to turbulence problems are explained and the main results obtained are summarized.", keywords = "partial-differential equations. two-dimensional turbulence. statistical equilibrium states. negative-temperature states. fully-developed turbulence. perfect fluid-dynamics. reynolds-number. isotropic turbulence. numerical-solution. kinetic-equations", } @Article{far:AS256, title = "The Distribution of a Quadratic Form in Normal Variables", author = "R. W. Farebrother", journal = AS, volume = "39", number = "2", pages = "294--309", year = "1990", keywords = "Imhof procedure; Koerts-Ambrose algorithm; Linear combination of chi-squared variables", } @Article{far:eigenvalue, title = "Eigenvalue-{F}ree Methods for Computing the Distribution of a Quadratic Form in Normal Variables", author = "R. W. Farebrother", journal = "Statistische Hefte", volume = "26", number = "", pages = "287--302", year = "1985", keywords = "", } @Article{far:wavelets, title = "Wavelet transforms and their applications to turbulence", author = "M. Farge", journal = "Annual Review of Fluid Mechanics", volume = "24", year = "1992", pages = "395--457", } @InProceedings{fer-per-swe:spie96, title = "{LIFTPACK}: {A} software package for wavelet transforms using lifting", author = "G. Fern\'{a}ndez and S. Periaswamy and Wim Sweldens", crossref = "uns-ald-lai:wavelet4", URL = "http://cm.bell-labs.com/who/wim/papers/spi96.ps", abstract = "We present LIFTPACK: A software package written in C for fast calculation of 2D biorthogonal wavelet transforms using the lifting scheme. The lifting scheme is a new approach for the construction of biorthogonal wavelets entirely in the spatial domain, i.e., independent of the Fourier Transform. Constructing wavelets using lifting consists of three simple phases: the first step or Lazy wavelet splits the data into two subsets, even and odd, the second step calculates the wavelet coefficients (high pass) as the failure to predict the odd set based on the even, and finally the third step updates the even set using the wavelet coefficients to compute the scaling function coefficients (low pass). The predict phase ensures polynomial cancelation in the high pass (vanishing moments of the dual wavelet) and the update phase ensures preservation of moments in the low pass (vanishing moments of the primal wavelet). By varying the order, an entire family of transforms can be built. The lifting scheme ensures fast calculation of the forward and inverse wavelet transforms that only involve FIR filters. The transform works for images of arbitrary size with correct treatment of the boundaries. Also, all computations can be done in-place.", } @Article{fis:frequency, title = "Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population", author = "R. A. Fisher", journal = BKA, volume = "10", year = "1915", pages = "507--521", } @Article{fis:tests, title = "Tests of significance in harmonic analysis", author = "R. A. Fisher", journal = PRSLA, volume = "125", year = "1929", pages = "54--59", } @InCollection{fla-gon:from, title = "From Wavelets to Time-Scale Energy Distributions", author = "P. Flandrin and P. Gon\c{c}lav{\`e}s", pages = "309--334", crossref = "sch-web:recent", } @Article{fla:brownian, title = "Wavelet Analysis and Synthesis of Fractional {B}rownian Motion", author = "Patrick Flandrin", journal = IEEETIT, volume = "38", number = "2", year = "1992", pages = "910--917", abstract = "Fractional Brownian motion (FBM) offers a convenient modeling for nonstationary stochastic processes with long-term dependencies and 1/f-type spectral behavior over wide ranges of frequencies. Statistical self-similarity is an essential feature of FBM and makes natural the use of wavelets for both its analysis and its synthesis. A detailed second-order analysis is carried out for wavelet coefficients of FBM. It reveals a stationary structure at each scale and a power-law behavior of the coefficients' variance from which the fractal dimension of FBM can be estimated. Conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed, consequences of discretization are considered, and some connections between the wavelet point of view and previous approaches based on length measurements (analysis) or dyadic interpolation (synthesis) are briefly pointed out.", } @Article{fos:unevenly, title = "Wavelets for Period Analysis of Unevenly Sampled Time Series", author = "Grant Foster", journal = AnJ, volume = "112", number = "4", year = "1996", pages = "1709--1729", email = "gfoster@aavso.org", } @InProceedings{fou:blocking, title = "Wavelet analysis of observed geopotential and wind: {B}locking and local energy coupling across scales", author = "Aim\'{e} Fournier", crossref = "uns-ald-lai:wavelet4", email = "fournier@cloudy.geology.yale.edu", abstract = "Atmospheric blocking during three unusual winter months is studied by multiresolution analysis and a wavelet based adaptation of traditional Fourier series based energetics. We demonstrate that blocking, in part a large and localized meteorological phenomenon, is largely described by just the largest scale part of the multiresolution analysis. New forms of the transfer functions of kinetic energy with the mean and eddy parts of the atmospheric circulation are introduced. These quantify the spatially localized conversion of energy between scales. A new accounting method for wavelet indexed transfers permits the introduction of a physically meaningful localized scale flux function. These techniques are applied to the data, and some support is found for the hypothesis that blocking is partially maintained by an inverse cascade.", } @Unpublished{fou:introduction, title = "An introduction to orthonormal wavelet analysis with shift invariance: {A}pplication to observed atmospheric-blocking spatial structure", author = "Aim\'{e} Fournier", year = "1999", note = "Submitted to the {\em Journal of the Atmospheric Sciences}", abstract = "Orthonormal wavelet analysis (OWA) is a special form of wavelet analysis which is especially suitable for analyzing spatial structures, such as atmospheric fields. For this purpose, OWA is more efficient and accurate than the nonorthogonal wavelet analysis (NWA) which has been introduced to the meteorological community in recent years, and which is more suitable for ``making time series sing.'' Since NWA is strictly correct only for infinite domains, OWA must be derived from NWA by use of appropriate boundary conditions, in this case, by periodization of OWA on an infinite domain. OWA is not translation invariant, unlike NWA. Two remedies are to use all possible translations, known as the Stationary Wavelet Transform, or else to use a `best' translation, known as the Best Shift OWA. OWA is a generalization of Fourier series, and the associated multiresolution analysis (MRA) is a generalization of Reynolds averaging. Like these, OWA/MRA on discrete and continuous domains satisfy analogous identities arithmetically exactly, unlike NWA. OWA/MRA is more efficient than Fourier series for analyzing nearly-singular synthetic functions, and also sequences of observed geopotential height maps which contain atmospheric blocking. Close to or more than 90% of the spatial variance of latitude-band averaged geopotential height, averaged over five different blocking events, is reconstructed by only two wavelets using best shift. This is comparable to EOF analysis, but is much faster and less data-dependent. All of the basic OWA programs are freely available for MATLAB (R), in the package WaveLab (c).", } @InProceedings{fou:lower, title = "Wavelet representation of lower-atmospheric long nonlinear wave dynamics, governed by the {B}enjamin-{D}avis-{O}no-{B}urgers equation", author = "Aim\'{e} Fournier", crossref = "szu:wavelet2", pages = "672--681", } @InProceedings{fou:simulated, title = "Wavelet multiresolution analysis of numerically simulated 3{D} radiative convection", author = "Aim\'{e} Fournier", pages = "642--653", crossref = "szu:wavelet3", email = "fournier@cloudy.geology.yale.edu", abstract = "A wavelet multiresolution analysis is performed on atmospheric fields simulated by a multilevel 3-dimensional atmospheric boundary layer model. Wavelet cospectra of the vertical wind and potential temperature are calculated and compared with radial Fourier cospectra. The former indicate most of the field variance to have horizontal scales roughly equal to the vertical scale, as should be the case for convectively driven turbulence. Fourier spectra exhibit a -3 power law, suggesting that the statistics may depend only on a quantity with units of time. Observations of time-and scale-dependent structures suggest certain physical mechanisms at work. The multiresolution analysis analogue of turbulent energy equations are formulated. This framework supports the proposed physical mechanisms.", } @Article{fri-gro-tch:gaps, title = "Wavelet analysis of signals with gaps", author = "Frick, P. and Grossmann, A. and Tchamitchian, P.", journal = JMP, volume = "39", number = "8", year = "1998", pages = "4091--4107", abstract = "A recently introduced algorithm [Frick et al., Astrophys. J. 483, 426 (1997)] of spectral analysis of data with gaps via a modified continuous wavelet transform is developed and studied. This algorithm is based on a family of functions called ``gapped wavelets'' which fulfill the admissibility condition on the gapped support. The wavelet family is characterized by an additional parameter which should be calculated for every scale and position. Three theorems concerning the properties of gapped wavelet transform are formulated and proved. They affirm the global stability of the algorithm as well as its stability in both limits of large and small scales. These results are illustrated by some numerical examples, which show that the algorithm really attenuates the artifacts coming from gaps (and/or boundaries), and is particularly efficient at small and large scales.", } @InProceedings{gag:numerical-results, title = "Wavelet filtering of speckle noise -- some numerical results", author = "Langis Gagnon", booktitle = "Proceedings of the Conference Vision Interface", year = "1999", pages = "???--???", email = "langis.gagnon@crim.ca", URL = "http://www.crm.umontreal.ca/~lgagnon/articles/vi99.pdf", } @Article{gal-hut:terrain, title = "Scale dependence in terrain analysis", author = "J. C. Gallant and M. F. Hutchinson", journal = "Mathematics and Computers in Simulation", volume = "43", number = "3--6", month = mar, year = "1997", pages = "313--321", keywords = "elevation", abstract = "Topographic attributes computed from digital elevation models are dependent on the resolution of the elevation data from which they are computed. A regular rectangular grid is not an ideal representation of topographic surfaces for the study of scale effects. Spectral and wavelet techniques are obvious alternatives but have several deficiencies, particularly in their use of oscillatory basis functions. The positive wavelet representation has very attractive properties of localisation and feature representation. Preliminary application to one-dimensional topographic data (profiles) yields useful results, including the identification of changes in topographic structure with scale. Extension to two-dimensional analysis will allow quantification of characteristic shapes, scales and orientations in the landscape.", } @Article{gam:wavelet, title = "Wavelet transform analysis of the length of the day and {E}l-{N}i{\~n}o/{S}outhern {O}scillation variations at intraseasonal and interannual time scales", author = "Gambis, D.", journal = AG, volume = "10", number = "", year = "1992", pages = "429--437", } @Article{gao-bru:firm, title = "{W}ave{S}hrink with Firm Shrinkage", author = "Hong-Ye Gao and Andrew Bruce", journal = SSin, volume = "7", number = "4", year = "1997", pages = "855--874", URL = "ftp://ftp.statsci.com/pub/gao/firm.ps.Z", keywords = "Bias Estimation; Confidence Interval; Firm Shrinkage; Minimax Thresholds; Non-parametric Regression; Signal De-noising; Trend Estimation; Variance Estimation; Wavelet Transform; WaveShrink", abstract = "Donoho and Johnstone's WaveShrink procedure has proven valuable for signal de-noising and non-parametric regression. WaveShrink has very broad asymptotic near-optimality properties. In this paper, we introduce a new shrinkage scheme, {\em firm}, which generalizes the hard and soft shrinkage proposed by Donoho and Johnstone. We derive minimax thresholds and provide formulas for computing the pointwise variance, bias, and risk for WaveShrink with firm shrinkage. We study the properties of the shrinkage functions, and demonstrate that firm shrinkage offers advantages over both hard shrinkage (uniformly smaller risk and less sensitivity to small perturbations in the data) and soft shrinkage (smaller bias and overall $L_2$ risk). Software is provided to reproduce all results in this paper.", } @Article{gao-li:coherent, journal = JAM, volume = "32", number = "11", year = "1993", pages = "1717--1725", title = "{W}avelet analysis of coherent structures at the atmosphere-forest interface", author = "W. Gao and BL. Li", abstract = "Wavelet studies were used for the turbulent data obtained inside and over a deciduous forest to investigate spatial and scale properties of a coherent structure in the area. Discrete warm and cool centers are linked to organized updrafts and downdrafts. Their patterns are alike, but the magnitudes vary at various heights. Temperature structures over the canopy possess a shorter duration, but the rate of reduction in the time scale with increasing height seems proportional to the rise in mean wind speed.", } @Article{gao:choice, title = "Choice of Thresholds for Wavelet Shrinkage Estimate of the Spectrum", author = "Hong-Ye Gao", journal = JTSA, volume = "18", number = "3", year = "1997", pages = "231--251", URL = "ftp://ftp.mathsoft.com/pub/wavelets/wavethresh.ps", keywords = "Log Spectrum Estimation; Orthogonal Wavelet Transformation; Shrinkage Estimator", abstract = "We study the problem of estimating the log spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds $t_{j,n}$ depending on sample size n, wavelet basis and resolution level j. At fine resolution levels (j=1, 2,...), we propose \[ t_{j,n} = A_j\log n, \] where $A_j$ are level-dependent constants and at coarse levels (j>>1), \[ t_{j,n} = \frac{\pi}{\sqrt{3}}\sqrt{\log n}. \] The purpose of this thresholding level is to make the reconstructed log-spectrum as nearly noise-free as possible. In addition to being pleasant from a visual point of view, the noise-free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties.", } @Article{gao:garrote, title = "Wavelet Shrinkage DeNoising Using Non-Negative Garrote", author = "Hong-Ye Gao", journal = JCGS, volume = "7", number = "4", year = "1998", pages = "469--488", URL = "http://www.amstat.org/publications/jcgs/pdf_98/Gao.pdf", keywords = "Cycle-Spinning; Minimax Threshold; Non-negative Garrote; Nonparametric Regression; Shrinkage Functions; Stein's Unbiased Risk Estimate (SURE); Wavelet Transform", abstract = "In this paper, we combine Donoho and Johnstone's Wavelet Shrinkage denoising technique (known as WaveShrink) with Breiman's non-negative garrote. We show that the non-negative garrote shrinkage estimate enjoys the same asymptotic convergence rate as the hard and the soft shrinkage estimates. For finite sample simulations, non-negative garrote performs better (smaller mean-square-error) than both hard and soft, and comparable to the firm shrinkage. We derive the minimax thresholds for the non-negative garrote. We study the threshold selection procedure based on Stein's Unbiased Risk Estimate (SURE) for both non-negative garrote and soft shrinkages. We propose a new threshold selection procedure based on combining Coifman and Donoho's cycle-spinning and SURE. We call our new procedure SPINSURE. We use examples to show that SPINSURE is more stable than SURE: smaller standard deviation and smaller range.", } @Unpublished{gao:heteroscedastic, title = "Wavelet Shrinkage Estimates For Heteroscedastic Regression Models", author = "Hong-Ye Gao", note = "Statistical Sciences Division, MathSoft, Inc", year = "1997", URL = "ftp://ftp.statsci.com/pub/gao/wsshd.ps.Z", keywords = "Heteroscedasticity; Nonparametric Regression; Running MAD; Wavelet Transform; Variance Estimation.", abstract = "We extend Donoho and Johnstone's wavelet shrinkage smoothing technique (known as WaveShrink) to handle data with heteroscedastic noise. We first show that if the noise variances are known, WaveShrink estimate achieves the same near-optimal convergence rate as in the white noise case. We then propose a procedure for estimating the noise variances. Our procedure is based on applying running MAD (Median Absolute Deviation from the median) to the non-decimated finest level wavelet coefficients. We apply our technique to various numerical examples.", } @Unpublished{gao:isontonic, title = "Wavelets and Isotonic Regression", author = "Hong-Ye Gao", note = "Statistical Sciences Division, MathSoft, Inc", year = "199?", URL = "ftp://ftp.statsci.com/pub/gao/isotonic.ps.Z", keywords = "Grenander Estimator; Isotonic Regression; Monotone Curve; Orthogonal Wavelet Transformation; Shrinkage Estimator", abstract = "Consider the following isotonic regression model: \[ y_i = f(t_i) + z_i\] where $f$ is only known to be a decreasing function and $\{z_i\}$ are iid Gaussian with mean zero and variance $\sigma^2$. We propose a simple thresholding procedure based on the fact that the wavelet coefficients for $f$, under Haar basis, are non-negative. We show that our estimator is competitive with the Grenander estimator both theoretically and numerically (in the sense of mean-square-error).", } @Unpublished{gao:selection, title = "Threshold Selection in {W}ave{S}hrink", author = "Hong-Ye Gao", note = "Statistical Sciences Division, MathSoft, Inc", year = "1997", URL = "ftp://ftp.statsci.com/pub/gao/threshold.ps.Z", keywords = "Cycle-Spinning; Minimax Threshold; Non-negative Garrote; Nonparametric Regression; Shrinkage Functions; Stein's Unbiased Risk Estimate (SURE); Wavelet Transform; WaveShrink", abstract = "Donoho and Johnstone's wavelet shrinkage denoising technique (known as WaveShrink) consists three steps: (1) transform data into wavelet domain; (2) shrink the wavelet coefficients; and (3) transform the shrunk coefficients back. The choice of shrinkage function and thresholds in step (2) plays an important role for WaveShrink both theoretically and in practice. In this paper, we discuss the issue of threshold selection in WaveShrink. We first review the threshold selection procedure based minimax thresholds and Stein's Unbiased Risk Estimate (SURE). We then propose a new threshold selection procedure based on combining Coifman and Donoho's cycle-spinning and SURE. We call our new procedure SPINSURE. We use examples to show that SPINSURE is numerically more stable than SURE: smaller standard deviation and smaller range. Various comparisons with the ideal and minimax thresholds are also presented.", } @Unpublished{gao:spectral, title = "Spectral Density Estimation via Wavelet Shrinkage", author = "Hong-Ye Gao", note = "Statistical Sciences Division, MathSoft, Inc", year = "1996", URL = "ftp://ftp.statsci.com/pub/gao/spec.ps.Z", keywords = "Non-Gaussian Model; Periodized Meyer Wavelets; Shrinkage Estimator; Spectral Density Estimation; Wavelet Transform", abstract = "We study the problem of estimating the spectral density of a stationary Gaussian time series. We use an orthogonal wavelet system whose members are periodic functions and have a finite number of non-zero Fourier coefficients -- periodized Meyer wavelets. We apply shrinkage rules to the empirical wavelet coefficients. We show that estimates based on thresholds $t_{j,n} = \lm_j\log n$ for certain $\lm_j$, with $n$ the sample size, have near-optimal $L_2$ convergence rates, over any Besov class in a wide range. In some cases, which includes the Bump Algebra, wavelet shrinkage procedures significantly outperform classical linear procedures, such as window methods and AR approximation methods.", } @PhdThesis{gao:thesis, title = "Wavelet Estimation of Spectral Densities in Time Series Analysis", author = "Hong-Ye Gao", school = "University of California, Berkeley", year = "1993", } @Unpublished{gen-sel-whi:differentiating, title = "Differentiating intraday seasonalities through wavelet multi-scaling", author = "R. Gen\c{c}ay and F. Sel\c{c}uk and B. Whitcher", year = "1999", note = "submitted", } @Unpublished{gen-sel-whi:high-frequency, title = "Wavelet Scale Analysis of High-Frequency Foreign Exchange Rates", author = "R. Gen\c{c}ay and F. Sel\c{c}uk and B. Whitcher", year = "2000", note = "submitted", } @Article{ger-har-mas:fractal, title = "Fractal functions and wavelet expansions based on several scaling functions", author = "Jeffrey S. Geronimo and Douglas P. Hardin and Peter R. Massopust", journal = JAT, volume = "78", number = "3", year = "1994", pages = "373--401", } @InProceedings{gil-wil-fel:visualizing, title = "Visualizing multifractal scaling behavior: {A} simple coloring heuristic", author = "A. C. Gilbert and W. Willinger and A. Feldman", booktitle = "Conference Record of The Thirty-Second Asilomar Conference on Signals, Systems and Computers", volume = "1", year = "1998", pages = "715--722", email = "agilbert@research.att.com, walter@research.att.com, anja@research.att.com", URL = "http://www.research.att.com/~agilbert/ps.files/asilomar98.ps.Z", } @Article{gil:trend, title = "Testing for the onset of trend, using wavelets", author = "Scott D. Gilbert", journal = JTSA, volume = "20", number = "5", year = "1999", pages = "513--526", } @InCollection{gof:wavelets, title = "Wavelets in macroeconomics: {A}n introduction", author = "W. L. Goffe", booktitle = "Computational Techniques for Econometrics and Economic Analysis", editor = "D. A. Belsley", publisher = "Kluwar Academic Publishers", address = "Dordrecht", series = "Advances in Computational Economics", volume = "3", year = "1993", pages = "???--???", } @InProceedings{gon-abr:multiple-window, title = "Multiple-window wavelet transform and local scaling exponent estimation", author = "P. Gon\c{c}alv{\`e}s and P. Abry", booktitle = "Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing", volume = "4", year = "1997", pages = "3433--3436", email = "paulo.goncalves@inria.fr, pabry@physique.ens-lyon.fr", abstract = "We propose here a multiple-window wavelet transform for the purpose of identifying non-stationary self-similar structures in random processes and estimating the time-varying scaling exponent H(t) that controls the local regularity and correlation of the process. More specifically, our final aim is to be able to track even rapidly varying trajectories (t,H(t)). The solution described here combines analysis obtained from scalograms computed with a set of multi-windows designed so as to satysfy to a decorrelation condition. We derive here the statistics for the estimate of H(t), compare it against numerical simulations and show that we obtain a substantial reduction of variance in estimation, without introducing bias.", } @InCollection{gon-fla:bilinear, title = "Bilinear Time-Scale Analysis Applied to Local Scaling Exponents Estimation", author = "P. Gon\c{c}alv{\`e}s and P. Flandrin", pages = "271--276", crossref = "mey-roq:progress", } @InProceedings{gon-fla:scaling, title = "Scaling exponents estimation from time-scale energy distributions", author = "P. Gon\c{c}alv{\`e}s and P. Flandrin", booktitle = "Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing", volume = "5", year = "1992", pages = "157--160", } @Unpublished{gon-etal:financial, title = "The Wavelet Transform for Filtering Financial Data Streams", author = "Zheng Gonghui and Jean-Luc Stark and Jonathan Campbell and Fionn Murtagh", year = "1998", note = "Submitted to {\em Journal of Computational Intelligence in Finance}", } @InProceedings{gon-rie-bar:simple, title = "Simple Statistical Analysis of Wavelet-based Multifractal Spectrum Estimation", author = "P. Gon\c{c}alv{\`e}s and R. H. Riedi and R. G. Baraniuk", booktitle = "Conference Record of The Thirty-Second Asilomar Conference on Signals, Systems and Computers", volume = "?", year = "1998", pages = "???--???", URL = "http://www-dsp.rice.edu/~riedi/cv_frame.html", abstract = "The multifractal spectrum characterizes the scaling and singularity structures of signals and proves useful in numerous applications, from network traffic analysis to turbulence. Of great concern is the estimation of the spectrum from a finite data record. In this paper, we derive asymptotic expressions for the bias and variance of a wavelet-based estimator for a fractional Brownian motion (fBm) process. Numerous numerical simulations demonstrate the accuracy and utility of our results.", } @Article{gou-auv-bag:dwarfs, title = "Wavelet Analysis of Pulsating White Dwarfs", author = "M. J. Goupil and M. Auvergne and A. Baglin", journal = AandA, volume = "250", number = "1", year = "1991", pages = "89--98", abstract = "Parts of light curves of two variable white dwarfs, Giclas 191-16 (BR Cam) and PG 1351+489 (EM UMa), are investigated by means of a wavelet analysis. This time-frequency analysis decomposes the light curves into their different oscillating components whose temporal behaviors are then individually studied. In addition to an oscillation of large amplitude, small amplitude oscillations are thereby clearly emphasized for both stars. Amplitude variations are found for most detected oscillations with periods of modulation as long or greater than the time intervals of the corresponding runs. A wavelet analysis of a comparison star gives the quality of the night in localizing perturbative atmospheric events.", } @Article{gra:introduction, title = "An introduction to wavelets", author = "Amara Graps", journal = IEEECSE, volume = "2", number = "2", year = "1995", pages = "50--61", URL = "ftp://ftp.best.com/pub/agraps/papers/IEEEwavelet.ps.gz", keywords = "wavelets, scale-based analysis, mathematical requirements, data represention, packet techniques, applications, data analysis, discretely sampled time-series data", abstract = "Wavelets were developed independently by mathematicians, quantum physicists, electrical engineers and geologists, but collaborations among these fields during the last decade have led to new and varied applications. What are wavelets, and why might they be useful to you? The fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers feel that using wavelets means adopting a whole new mind-set or perspective in processing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. Most of the basic wavelet theory has now been done. The mathematics have been worked out in excruciating detail, and wavelet theory is now in the refinement stage. This involves generalizing and extending wavelets, such as in extending wavelet packet techniques. The future of wavelets lies in the as-yet uncharted territory of applications. Wavelet techniques have not been thoroughly worked out in such applications as practical data analysis, where, for example, discretely sampled time-series data might need to be analyzed. Such applications offer exciting avenues for exploration.", } @Article{gre-gie-lip:invariance, title = "Translational invariance in turbulent cascade models", author = "M. Greiner and J. Giesemann and P. Lipa", journal = PRE, volume = "56", number = "4", year = "1997", pages = "4263--4274", URL = "http://ojps.aip.org/journal_cgi/getabs?KEY=PLEEE8&cvips=PLEEE8000056000004004263000001&gifs=No", keywords = "fully-developed turbulence hydrodynamic turbulence exact resummations wavelet correlations", abstract = "Due to the underlying hierarchical structure, spatial correlation functions calculated from multiplicative cascade models are not translationally invariant. A scheme is presented that restores translational invariance by averaging over the experimentally unknown spatial location of cascade realizations with respect to the observation window. The impact of this scheme on multiplier distributions for the energy dissipation field in fully developed turbulence is analyzed; only the experimental multiplier distribution is found to be invariant under a wide range of scales.", } @Article{gre-ros:fast, title = "A fast wavelet-based Karhunen-Loeve transform", author = "Greenshields, I. R. and Rosiene, J. A.", journal = PR, volume = "31", number = "7", year = "1998", pages = "839--845", abstract = "The paper describes the role of the standard wavelet decomposition in computing a fast Karhunen-Loeve transform. The standard wavelet decomposition (which we show is different from the conventional wavelet transform) leads to a highly sparse and simply structured transformed version of the correlation matrix which can be easily subsetted (with little loss of Frobenius norm). The eigenstructure of this smaller matrix can be efficiently computed using standard algorithms such as QL. Finally, we provide an example of the use of the efficient transform by classifying a 219-channel AVIRIS image with respect to its eigensystem.", } @InCollection{gre:cross-validation, title = "Wavelet basis selection for regression by cross-validation", author = "Seth A. Greenblatt", booktitle = "Computational Approaches to Economic Problems", editor = "H. M. Amman and B. Rustem and A. B. Whinston", publisher = "Kluwar Academic Publishers", address = "Dordrecht", series = "Advances in Computational Economics", volume = "6", year = "1997", pages = "39--55", } @InCollection{gre:outlier, title = "Wavelets in econometrics: {A}n application to outlier testing", author = "Seth A. Greenblatt", booktitle = "Computational Economic Systems: Models, Methods \& Econometrics", editor = "M. Gilli", publisher = "Kluwar Academic Publishers", address = "Dordrecht", series = "Advances in Computational Economics", volume = "5", year = "1996", pages = "139--160", } @InProceedings{gro-kro-mar:reading, title = "Reading and understanding continuous wavelet transforms", author = "A. Grossmann and R. Kronland-Martinet and J. Morlet", pages = "2--20", crossref = "com-gro-tch:wavelets", keywords = "wavelets", abstract = "An introduction to continuous wavelet transforms and a description of the representation methods that have evolved. Also discusses the influence of the choice of the wavelet in the interpretation of wavelet transforms.", } @Article{gro-mor:hardy, title = "Decomposition of {H}ardy Functions into Square Integrable Wavelets of Constant Shape", author = "A. Grossmann and J. Morlet", journal = SIAMJMA, volume = "15", number = "4", year = "1984", pages = "723--736", abstract = "", } @Article{gru-wal:seismic, title = "{C}haracterizing seismic time series using the discrete wavelet transform", author = "H. J. Grubb and A. T. Walden", journal = GP, volume = "45", number = "2", year = "1997", pages = "183--205", abstract = "The discrete wavelet transform (DWT) has potential as a tool for supplying discriminatory attributes with which to characterize or cluster groups of seismic traces in reservoir studies. The wavelet transform has the great advantage over the Fourier transform in being able to better localize changes. The multiscale nature and structure of the DWT leads to a method of display which highlights this and allows comparison of changes in the transform with changing data. Many different sorts of wavelet exist and it is found that the quality of reconstruction of a seismic trace segment, using some of the coefficients, is dependent on the choice of wavelet, which leads us to consider choosing a wavelet under a 'best reconstruction' criterion. Location shifts, time zero uncertainties, are also shown to affect the transform, as do truncations, resampling, etc. Using real data, examples of utilizing the DWT coefficients as attributes for whole trace segments or fractional trace segments are given. Provided the DWT is applied consistently, for example with a fixed wavelet, and non-truncated data, the transform produces useful results. Care must be exercised if it is applied to data of different lengths. However, as the algorithm is refined and improved in the future, the DWT should prove increasingly useful.", keywords = "projection pursuit, sampling theory, propagation, signal", } @MastersThesis{guo:msthesis, title = "Theory and Applications of the Shift-Invariant, Time-Varying and Undercimated Wavelet Transforms", author = "Haitao Guo", year = "1995", pages = "144", school = "Electrical and Computer Engineering Department, Rice University", URL = "http://www-dsp.rice.edu/~harry/paper/ms.ps.Z", email = "harry@ece.rice.edu", abstract = "In this thesis, we generalize the classical discrete wavelet transform, and construct wavelet transforms that are shift-invariant, time-varying, undecimated, and signal dependent. The result is a set of powerful and efficient algorithms suitable for a wide variety of signal processing tasks, e.g., data compression, signal analysis, noise reduction, statistical estimation, and detection. These algorithms are comparable and often superior to traditional methods. In this sense, we put wavelets in action.", } @Article{haa:haar, title = "Zur {T}heorie der orthogonalen {F}unktionensysteme", author = "Alfred Haar", journal = "Mathematische Annalen", volume = "69", pages = "331--371", year = "1910", note = "In German", } @Article{hal-ker-pic:block, title = "Block threshold rules for curve estimation using kernel and wavelet methods", author = "Hall, P. and Kerkyacharian, G. and Picard, D.", journal = AofS, volume = "26", number = "3", year = "1998", pages = "922--942", keywords = "adaptivity bias convergence rate density estimation minimax nonparametric regression smoothing parameter variance", abstract = "Motivated by recently developed threshold rules for wavelet estimators, we suggest threshold methods for general kernel density estimators, including those of classical Rosenblatt-Parzen type. Thresholding makes kernel methods competitive in terms of their adaptivity to a wide variety of aberrations in complex signals. It is argued that term-by-term thresholding does not always produce optimal performance, since individual coefficients cannot be estimated sufficiently accurately for reliable decisions to be made. Therefore, we suggest grouping coefficients into blocks and making simultaneous threshold decisions about all coefficients within a given block. It is argued that block thresholding has a number of advantages, including that it produces adaptive estimators which achieve minimax-optimal convergence rates without the logarithmic penalty that is sometimes associated with term-by-term thresholding. More than this, the convergence rates are achieved over large classes of functions with discontinuities, indeed with a number of discontinuities that diverges polynomially fast with sample size. These results are also established for block thresholded wavelet estimators, which, although they can be interpreted within the kernel framework, are often most conveniently constructed in a slightly different way.", } @Article{hal-ker-pic:minimax, title = "On the minimax optimality of block thresholded wavelet estimators", author = "Hall, P. and Kerkyacharian, G. and Picard, D.", journal = SSin, volume = "9", number = "1", year = "1999", pages = "33--49", keywords = "Besov space. chirp function. convergence rate. Doppler function. mean squared error. nonparametric regression. smoothing parameter", abstract = "Block thresholding methods have been proposed by Hall, Kerkyacharian and Picard (1995) as a means of obtaining increased adaptivity when estimating a function using wavelet methods. For example, it has been shown that block thresholding reduces mean squared error by rendering the estimator more adaptive to relatively subtle, local changes in curvature, of the type that local bandwidth choice is designed to accommodate in traditional kernel methods. In this paper we show that block thresholding also provides extensive adaptivity to many varieties of aberration, including those of chirp and Doppler type. Indeed, in a wide variety of function classes, block thresholding methods possess minimax-optimal convergence rates, and in particular enjoy those rates without the extraneous logarithmic penalties that are usually suffered by term-by-term thresholding methods.", } @Article{hal-mck-tur:performance, title = "Performance of wavelet methods for functions with many discontinuities", author = "Peter Hall and Ian McKay and Berwin Turlach", journal = AofS, volume = "24", number = "6", year = "1996", pages = "???--???", abstract = "", } @TechReport{hal-nas:non-integer, title = "On Choosing a Non-integer Resolution Level when Using Wavelet Methods", author = "Peter Hall and Guy P. Nason", year = "1996", institution = "Centre for Mathematics and its Applications, Australian National University", } @Article{hal-pat:effect, title = "Effect of Thresholding Rules on Performance of Wavelet-Based Curve Estimators", author = "Peter Hall and Prakash Patil", journal = SSin, volume = "6", year = "1996", pages = "331--345", abstract = "", } @Article{hal-pat:methods, title = "On wavelet methods for estimating smooth functions", author = "Peter Hall and Prakash Patil", journal = Ber, volume = "1", number = "1", year = "1995", pages = "41--58", abstract = "Without assuming any prior knowledge of wavelet methods, we develop theory describing their performance as estimators of smooth functions. The linear part of the wavelet estimator is discussed by analogy with classical kernel methods. Concise formulae are developed for its bias, variance and mean square error. These quantities oscillate somewhat erratically on a wavelength that is equivalent to the bandwidth, reflecting the irregular numerical fluctuations that are observed in practice. Nevertheless, the contributions of these oscillations to mean integrated square error tend to dampen one another out, even over very small intervals, with the result that mean integrated square error properties of linear wavelet methods are much closer to those of kernel methods than is perhaps reasonable, given the local behaviour. We illustrate the adaptive qualities of the nonlinear component of a wavelet estimator by describing its performance when the target function is smooth but has high-frequency oscillations. It is shown that the nonlinear component automatically adapts to changing local conditions, to the extent of achieving (except for a logarithmic factor) the same convergence rate as the optimal linear estimator, but without a need to adjust the underlying bandwidth. This makes explicitly clear the way in which the linear part of the estimator takes care of the ‘average’ characteristics of the unknown curve, while the nonlinear part corrects for more erratic fluctuations, in a manner which is virtually independent of the construction of the linear part.", keywords = "convergence rate density estimation differentiability dilation equation kernel method nonparametric curve estimation orthogonal series regression scaling function smoothness wavelet", } @Article{hal-pat:smoothing, journal = JRSSB, volume = "58", number = "2", year = "1996", pages = "361--377", title = "{O}n the choice of smoothing parameter, threshold and truncation in nonparametric regression by non-linear wavelet methods", author = "P. Hall and P. Patil", abstract = "Concise asymptotic theory is developed for non-linear wavelet estimators of regression means, in the context of general error distributions, general designs, general normalizations in the case of stochastic design, and non-structural assumptions about the mean. The influence of the tail weight of the error distribution is addressed in the setting of choosing threshold and truncation parameters. Mainly, the tail weight is described in an extremely simple way, by a moment condition; previous work on this topic has generally imposed the much more stringent assumption that the error distribution be normal. Different approaches to correction for stochastic design are suggested. These include conventional kernel estimation of the design density, in which case the interaction between the smoothing parameters of the non-linear wavelet estimator and the linear kernel method is described.", } @Article{hal-pen-ker-pic:block, journal = SC, volume = "7", number = "2", year = "1997", pages = "115--124", title = "Numerical performance of block thresholded wavelet estimators", author = "Peter Hall and Spiridon Penev and G{\'e}rard Kerkyacharian and Dominique Picard", keywords = "Adaptivity bias density estimation mean squared error non-parametric regression smoothing parameter variance", abstract = "Usually, methods for thresholding wavelet estimators are implemented term by term, with empirical coefficients included or excluded depending on whether their absolute values exceed a level that reflects plausible moderate deviations of the noise. We argue that performance may be improved by pooling coefficients into groups and thresholding them together. This procedure exploits the information that coefficients convey about the sizes of their neighbours. In the present paper we show that in the context of moderate to low signal-to-noise ratios, this `block thresholding' approach does indeed improve performance, by allowing greater adaptivity and reducing mean squared error. Block thresholded estimators are less biased than term-by-term thresholded ones, and so react more rapidly to sudden changes in the frequency of the underlying signal. They also suffer less from spurious aberrations of Gibbs type, produced by excessive bias. On the other hand, they are more susceptible to spurious features produced by noise, and are more sensitive to selection of the truncation parameter.", } @TechReport{hal-tur:competitors, title = "Convolution and Interpolation: {C}ompetitors with Local Polynomial Smoothing", author = "Peter Hall and Berwin A. Turlach", institution = "Centre for Mathematics and its Applications, Australian National University", number = "SRR95-037", year = "1995", email = "berwin@alphasun.anu.edu.au", abstract = "Local polynomial smoothing enjoys a variety of very attractive features. It is often viewed as superior to convolution and interpolation methods, which offer greater numerical stability but inferior theoretical performance. In this paper we show that modifications to convolution and interpolation techniques produce effective competitors with local polynomial smoothing, enjoying similar bias, variance and mean squared error properties but without the downside of numerical instability. The methods suggested here may be employed as the basis for empirical wavelet transforms of ungridded data.", } @Article{hal-tur:enhancing, title = "Enhancing Convolution and Interpolation Methods for Nonparametric Regression", author = "Peter Hall and Berwin A. Turlach", journal = BKA, volume = "84", number = "4", year = "1997", pages = "779--790", email = "halstat@fac.anu.edu.au,berwin.turlach@anu.edu.au", } @Book{har-etal:wavelets, title = "Wavelets, Approximation, and Statisticsl Applications", author = "Wolfgang H{\"a}rdle and Gerard Kerkyacharian and Dominique Picard and Alexander Tsybokov", booktitle = "Wavelets, Approximation, and Statisticsl Applications", series = "Lecture Notes in Statistics", volume = "129", publisher = "Springer-Verlag", address = "New York", pages = "265", year = "1998", } @Article{hen-etal:invest, title = "Investigating the Nonlinear Dynamics of Cellular Motion in the Inner Ear Using the Short-Time {F}ourier and Continuous Wavelet Transforms", author = "Conor Heneghan and Shyam Khanna and {\AA}ke Flock and Mats Ulfendahl and Lou Brundin and Malvin C. Teich", journal = IEEETSP, volume = "42", number = "12", year = "1994", pages = "3335--3352", abstract = "The short-time Fourier transform (STFT) and the continuous wavelet transform (CWT) are used to analyze the time course of cellular motion in the inner ear. The velocity responses of individual outer hair cells and Hensen's cells to sinusoidal and amplitude modulated (AM) acoustical signals applied at the ear canal display characteristics typical of nonlinear systems, including the generation of harmonic and half-harmonic components. The STFT proves to be valuable for following the time course of the frequency components generated using sinusoidal and ARM input signals. The CWT is also useful for analyzing these signals; however, it is generally not as effective as the STFT when octave-band-based CWT's are used. For the transient response, the spectrogram (which is the squared magnitude of the STFT) and the octave-band-based scalogram (which is the squared magnitude of the CWT) prove equally valuable, and the authors have used both to study the responses of these cells to step-onset tones of different frequencies. Such analyses reveal information about the preferred vibration frequencies of cells in the inner ear and are useful for deciding among alternative mathematical models of nonlinear cellular dynamics. A modified Duffing oscillator model yields results that bear some similarity to the data.", } @Article{her-vet:time-varying, title = "Orthogonal Time-Varying Filter Banks and Wavelet Packets", author = "Cormac Herley and Martin Vetterli", journal = IEEETSP, volume = "42", number = "10", year = "1994", pages = "2650--2663", } @Book{her-wei:first, title = "A First Course on Wavelets", author = "Eugenio Hern\'{a}ndez and Guido Weiss", publisher = "CRC Press Inc.", address = "Boca Raton", pages = "512", year = "1996", URL = "http://www.crcpress.com/prods/8274.htm", abstract = "Wavelet theory had its origin in quantum field theory, signal analysis, and function space theory. In these areas wavelet-like algorithms replace the classical Fourier-type expansion of a function. This unique new book is an excellent introduction to the basic properties of wavelets, from background math to powerful applications. The authors provide elementary methods for constructing wavelets, and illustrate several new classes of wavelets. The text begins with a description of local sine and cosine bases that have been shown to be very effective in applications. Very little mathematical background is needed to follow this material. A complete treatment of band-limited wavelets follows. These are characterized by some elementary equations, allowing the authors to introduce many new wavelets. Next, the idea of multiresolution analysis (MRA) is developed, and the authors include simplified presentations of previous studies, particularly for compactly supported wavelets. Some of the topics treated include: Several bases generated by a single function via translations and dilations; Multiresolution analysis, compactly supported wavelets, and spline wavelets; Band-limited wavelets; Unconditionality of wavelet bases; Characterizations of many of the principal objects in the theory of wavelets, such as low-pass filters and scaling functions. The authors also present the basic philosophy that all orthonormal wavelets are completely characterized by two simple equations, and that most properties and constructions of wavelets can be developed using these two equations. Material related to applications is provided, and constructions of splines wavelets are presented. Mathematicians, engineers, physicists, and anyone with a mathematical background will find this to be an important text for furthering their studies on wavelets.", } @Article{her:boundary, title = "Boundary Filters for Finite-Length Signals and Time-Varying Fitler Banks", author = "Cormac Herley", journal = IEEETCS2, volume = "42", number = "2", year = "1995", pages = "102--114", } @Article{hes-wic:wavelets, title = "Wavelets and time-frequency analysis", author = "Nikolaj Hess-Nielsen and Mladen Victor Wickerhauser", journal = PIEEE, volume = "84", number = "4", year = "1996", pages = "523--540", keywords = "time-frequency analysis, wavelet packet analysis, wavelet analysis, numerical algorithms, continuous models, frequency spreading control, high frequencies, nonstationary filtering, periodic wavelet packets, time localization, linear phase filters, signal decomposition", abstract = "We present a selective overview of time-frequency analysis and some of its key problems. In particular we motivate the introduction of wavelet and wavelet packet analysis. Different types of decompositions of an idealized time-frequency plane provide the basis for understanding the performance of the numerical algorithms and their corresponding interpretations within the continuous models. As examples we show how to control the frequency spreading of wavelet packets at high frequencies using nonstationary filtering and study some properties of periodic wavelet packets. Furthermore we derive a formula to compute the time localization of a wavelet packet from its indexes which is exact for linear phase filters, and show how this estimate deteriorates with deviation from linear phase.", } @Article{hic-dat:fractal, title = "Estimation of fractal signals using wavelets and filter banks", author = "Hirchoren, G. A. and DAttellis, C. E.", journal = IEEETSP, volume = "46", number = "6", year = "1998", pages = "1624--1630", keywords = "filter banks fractal signals 1/f processes wavelets", abstract = "A filter bank design based on orthonormal wavelets and equipped with a multiscale Wiener filter mas recently proposed for signal restoration and for signal smoothing of 1/f family of fractal signals corrupted by external noise. The conclusions obtained in these papers are based on the following simplificative hypotheses: 1) The wavelet transformation is a whitening filter, and 2) the approximation term of the wavelet expansion can be avoided when the number of octaves in the multiresolution analysis is large enough. In this paper, we shelf that the estimation of 1/f processes in noise can be improved avoiding these two hypotheses. Explicit expressions of the mean-square error are given, and numerical comparisons with previous results are shown.", } @InCollection{hil-etal:mri, title = "Wavelet denoising of functional {MRI} Data", author = "M. Hilton and Ogden T. and D. Hattery and G. Jawerth and B. Eden", pages = "93--114", crossref = "ald-uns:medicine", URL = "", abstract = "", } @Article{hir-dat:estimation, title = "Estimation of Fractal Signals Using Wavelets and Filter Banks", author = "Hirchoren, G. A. and {D'Attellis}, C. E.", journal = IEEETSP, volume = "46", number = "6", year = "1998", pages = "1624--1630", } @InCollection{hol-etal:real-time, title = "A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform", author = "M. Holschneider and R. Kronland-Martinet and J. Morlet and Ph. Tchamitchian", pages = "286--297", crossref = "com-gro-tch:wavelets", abstract = "", } @InCollection{hon:bridges, title = "Wavelets, probability and statistics: some bridges", author = "C. Hondr{\'e}", pages = "", crossref = "ben-fra:wavelets", abstract = "", note = "", } @Article{how-per:wavelet, title = "Wavelet variance, {A}llan variance, and leakage", author = "David A. Howe and Donald B. Percival", journal = IEEETIM, volume = "44", number = "2", year = "1995", pages = "94--97", URL = "http://weber.u.washington.edu/~dbp/PSFILES/wvavleak.ps.Z", keywords = "Analysis of variance; Atomic clocks; Frequency stability; Power-law processes", abstract = "Wavelets have recently been a subject of great interest in geophysics, mathematics and signal processing. The discrete wavelet transform can be used to decompose a time series with respect to a set of basis functions, each one of which is associated with a particular scale. The properties of a time series at different scales can then be summarized by the wavelet variance, which decomposes the variance of a time series on a scale by scale basis. The wavelet variance corresponding to some of the recently discovered wavelets can provide a more accurate conversion between the time and frequency domains than can be accomplished using the Allan variance. This increase in accuracy is due to the fat that these wavelet variances give better protection against leakage than does the Allan variance.", } @Article{hoy-sch:sunspot, title = "Group Sunspot Numbers: {A} New Solar Activity Reconstruction", author = "Douglas V. Hoyt and Kenneth H. Schatten", journal = "Solar Physics", volume = "181", number = "", year = "1998", pages = "491--512", keywords = "", abstract = "", } @Unpublished{hua-cre:deterministic, title = "Deterministic/Stochastic Wavelet Decomposition for Recovery of Signal from Noise Data", author = "Hsin-Cheng Huang and Noel Cressie", year = "1999", note = "Submitted to {\em Technometrics}", } @InCollection{hua-cre:empirical, title = "Empirical {B}ayesian spatial prediction using wavelets", authors = "Hsin-Cheng Huang and Noel Cressie", crossref = "mul-vid:biwbm", pages = "203--222", URL = "http://wagner.stat.sinica.edu.tw/~hchuang/decomp.ps", } @Article{hua:density, title = "Density estimation by wavelet-based reproducing kernels", author = "S. Y. Huang", journal = SSin, volume = "9", number = "1", year = "1999", pages = "137--151", keywords = "asymptotics Bernoulli numbers Bernoulli polynomials density estimation efficiency multiresolution approximation projection kernel reproducing kernel reproducing kernel Hilbert space wavelets", abstract = "Density estimation by wavelet-based reproducing kernels is studied. Asymptotic bias and variance are derived. Estimators using spline- wavelets and Daubechies wavelets are presented as examples. Kernel order and kernel efficiency are also discussed. By an integral property of the bias and an idea from Scott's averaged shifted histograms, a bias reduction technique based on a grid point average is proposed. This bias reduction technique is shown to be variance stable.", } @Article{hua:projection, title = "Projection estimation in multiple regression with application to functional ANOVA models", author = "Huang, J.-H. Z.", journal = AofS, volume = "26", number = "1", year = "1998", pages = "242--272", keywords = "ANOVA curse of dimensionality finite elements interaction least squares polynomials rate of convergence regression splines tensor product trigonometric polynomials wavelets", abstract = "A general theory on rates of convergence of the least-squares projection estimate in multiple regression is developed. The theory is applied to the functional ANOVA model, where the multivariate regression function is modeled as a specified sum of a constant term, main effects (functions of one variable) and selected interaction terms (functions of two or more variables). The least-squares projection is onto an approximating space constructed from arbitrary linear spaces of functions and their tensor products respecting the assumed ANOVA structure of the regression function. The linear spaces that serve as building blocks can be any of the ones commonly used in practice: polynomials, trigonometric polynomials, splines, wavelets and finite elements. The rate of convergence result that is obtained reinforces the intuition that low-order ANOVA modeling can achieve dimension reduction and thus overcome the curse of dimensionality. Moreover, the components of the projection estimate in an appropriately defined ANOVA decomposition provide consistent estimates of the corresponding components of the regression function. When the regression function does not satisfy the assumed ANOVA form, the projection estimate converges to its best approximation of that form.", } @Book{hub:world, title = "The World According to Wavelets: The Story of a Mathematical Technique in the Making", author = "Barbara Burke Hubbard", year = "1996", publisher = "A K Peters", address = "Wellesley, Massachusetts", ISBN = "1-56881-047-4", abstract = "This book, lovingly written and highly accessible, embraces the often unheralded notion that mathematics contains ideas that can, and deserve, to be communicated to a wider public ­p; even if what is communicated is at the level of appreciation rather than practical knowledge. Put simply, it is a book about the wavelet transform, that strange and scientifically intriguing new method of encoding information with an abundance of practical applications. This book is a wonderfully successful attempt to entice the non-mathematical reader into formerly uncharted territory without sacrificing precision. The material is masterfully organized so mathematical details can be assimilated at one's own pace; the main text is devoid of formulas and relates a story of people and ideas, while separate boxes and appendices contain intricate discussions for the more mathematically adventurous. This book is a rarity in mathematics books in that it recognizes that both mathematicians and readers interested in mathematics have a human side.", } @Article{hud-fri-may:atmospheric, title = "Wavelet Transforms and Atmospheric Turbulence", author = "Lonnie Hudgins and Carl. A. Friehe and Meinhard E. Mayer", journal = PRL, volume = "71", number = "20", year = "1993", pages = "3279--3282", abstract = "Wavelet cross spectra and cross scalograms are used to analyze the time-scale structure of bivariate turbulence data from the boundary layer over the ocean. The cross scalogram for the streamwise and vertical turbulent velocity components shows a highly intermittent pattern with significant contributions of opposite signs appearing at two specific scales, approximately 60 m and approximately 2 km, believed to be related to small- scale turbulent mixing and large-scale secondary flow in the boundary layer.", } @InCollection{hud-fri-may:fourier, title = "Fourier and wavelet analysis of atmospheric turbulence", author = "Lonnie Hudgins and Carl. A. Friehe and Meinhard E. Mayer", pages = "491--498", crossref = "mey-roq:progress", } @Article{hud-hua:bivariate, title = "Bivariate wavelet analysis of {A}sia monsoon and {ENSO}", author = "Lonnie Hudgins and Jianping Huang", journal = "Advances in Atmospheric Sciences", volume = "13", number = "", year = "1996", pages = "", URL = "http://www.eurandom.tue.nl/whitcher/other_papers/hud0hua96.pdf", abstract = "", } @PhdThesis{hud:thesis, title = "Wavelet Analysis of Atmospheric Turbulence", author = "Lonnie H. Hudgins", school = "University of California, Irvine", year = "1992", email = "lonnie@lynx.ps.uci.edu", } @Article{hwa:estimation, title = "Estimation of fractional Brownian motion embedded in a noisy environment using nonorthogonal wavelets", author = "Hwang, W. L.", journal = IEEETSP, volume = "47", number = "8", year = "1999", pages = "2211--2219", abstract = "We show that nonorthogonal wavelets can characterize the fractional Brownian motion (fBm) that is in white noise. We demonstrate the point that discriminating the parameter of an fBm from that of noise is equivalent to discriminating the composite singularity formed by superimposing a peak singularity on a Dirac singularity. We characterize the composite singularity by formalizing this problem as a nonlinear optimization problem, This yields our parameter estimation algorithm. For fractal signal estimation, Wiener filtering is explicitly formulated as a function of the signal and noise parameters and the wavelets. We show that the estimated signal is a 1/f process, Comparative studies through numerical simulations of our methods with those of Wornell and Oppenheim are presented.", } @Article{imh:computing, title = "Computing the distribution of a quadratic form in normal variables", author = "J. P. Imhof", journal = BKA, volume = "48", pages = "419--426", year = "1961", } @Article{iss:product-moment, title = "On a Formula for the Product-Moment Coefficient of Any Order of a Normal Frequency Distribution in Any Number of Variables", author = "L. Isserlis", journal = BKA, volume = "12", pages = "134--139", year = "1918", } @Article{ist:coefficients, title = "Wavelet Coefficients of a {G}aussian Process and Applications", author = "Jacques Istas", journal = "Annales de l'Institut Henri Poincare, Section B, Calcul des Probabilities et Statistique", volume = "28", pages = "537--556", year = "1992", note = "In French", keywords = "covariance functions, spectral densities, Gaussian stationary process, square error, wavelet transform", abstract = "The author gives the relations between the covariance functions and the spectral densities of the approximation and the details of a Gaussian stationary process at different resolutions. He studies the rate of convergence of the square error between the process and its wavelet transform. Then he shows the convergence in distribution of the projection of the process to the original process. Finally, proposes a choice of the regularity of the wavelet in order to minimize the correlation between the approximation and the details.", } @Article{iya-kuw:earthquake, title = "Application of wavelets to analysis and simulation of earthquake motions", author = "Iyama, J. Kuwamura, H.", journal = "Earthquake Engineering \& Structural Dynamics", volume = "28", number = "3", pages = "255--272", year = "1999", keywords = "wavelet transform simulated earthquake motions energy input Fourier transform time-frequency characteristics energy input rate", abstract = "A method of applying wavelet transform to earthquake motion analysis is developed from the viewpoint of energy input structures, in which relationships between wavelet coefficients and energy input, namely energy principles in wavelet analysis are derived. By using the principles, time-frequency characteristics of the 1995 Hyogoken-Nanbu earthquake ground motions are analysed and time histories of energy input for various ranges of frequencies and epicentral distances are identified. Furthermore, a technique to simulate earthquake ground accelerations by wavelet inverse transform is developed on the condition that target time-frequency characteristics are specified. Structural responses to the simulated accelerations are compared with the target time-frequency characteristics, which shows satisfactory correlations between wavelet coefficients and energy responses in both time and frequency domains.", } @Article{jaw-swe:overview, author = "B. Jawerth and Wim Sweldens", title = "An overview of wavelet based multiresolution analyses", journal = "SIAM Rev.", volume = "36", number = "3", pages = "377--412", year = "1994", URL = "http://cm.bell-labs.com/who/wim/papers/overview.ps", abstract = "Wavelet-based multiresolution analysis helps in data compression, operator analysis and developing a periodic fast wavelet transform algorithm. The analysis requires definition of a multiresolution analysis and investigation of the method in which wavelets fit into the multiresolution analysis. The fitting process requires a consideration of the semiorthogonal, orthogonal and biorthogonal wavelets. The application process requires an understanding of the wavelets on an interval, wavelet packets, multidimensional waves and fast wavelet transforms.", } @Unpublished{jen-whi:local, title = "A Semiparametric Wavelet-Based Estimator of a Locally Stationary Long-Memory Model", author = "Mark J. Jensen and Brandon Whitcher", year = "1999", note = "Under preparation", } @Unpublished{jen:alternative, title = "An Alternative Maximum Likelihood Estimator of Long-Memeory Processes Using Compactly Supported Wavelets", author = "Mark J. Jensen", journal = JEDC, volume = "24", number = "3", year = "1997", pages = "361--387", URL = "http://econwpa.wustl.edu/eprints/em/papers/9709/9709002.abs", keywords = "ARFIMA, Fractional Integration, Long-memory, MLE, Wavelets", abstract = "In this paper we apply compactly supported wavelets to the ARFIMA(p,d,q) long-memory process to develop an alternative maximum likelihood estimator of the differencing parameter, d, that is invariant to unknown means, model specification, and contamination. We show that this class of time series have wavelet transforms whose covariance matrix is sparse when the wavelet is compactly supported. It is shown that the sparse covariance matrix can be approximated to a high level of precision by a matrix equal to the covariance matrix except with the off-diagonal elements set equal to zero. This diagonal matrix is shown to reduce the order of calculating the likelihood function to an order smaller than those associated with the exact MLE method. We test the robustness of the wavelet MLE of the fractional differencing parameter to a variety of compactly supported wavelets, series length, and contamination levels by generating ARFIMA(p,d,q) processes for different values of p, d, and q, and calculating the wavelet MLE using only the main diagonal elements of its covariance matrix. In our simulations we find the wavelet MLE to be superior to the approximate frequency MLE when estimating contaminated ARFIMA(0,d,0), and uncontaminated ARFIMA(1,d,0) and ARFIMA(0,d,1) processes except when the MA parameter is close to one. We also find the wavelet MLE to be robust to model specification and as such is an attractive alternative semiparameter estimator to the Geweke, Porter--Hudak estimator.", } @Article{jen:approximate, title = "An approximate wavelet {MLE} of short and long memory parameters", author = "Mark J. Jensen", journal = SNDE, volume = "3", number = "4", pages = "239--253", year = "1999", URL = "http://econwpa.wustl.edu/eprints/em/papers/9802/9802003.abs", keywords = "Long Memory, Fractional Integration, ARFIMA, Wavelets", abstract = "By design a wavelet's strength rests in its ability to localize a process simultaneously in time-scale space. The wavelet's ability to localize a time series in time-scale space directly leads to the computational efficiency of the wavelet representation of a N x N matrix operator by allowing the N largest elements of the wavelet represented operator to represent the matrix operator [Devore, et al. (1992a) and (1992b)]. This property allows many dense matrices to have sparse representation when transformed by wavelets.In this paper we generalize the long-memory parameter estimator of McCoy and Walden (1996) to estimate simultaneously the short and long-memory parameters. Using the sparse wavelet representation of a matrix operator, we are able to approximate an ARFIMA model's likelihood function with the series' wavelet coefficients and their variances. Maximization of this approximate likelihood function over the short and long-memory parameter space results in the approximate wavelet maximum likelihood estimates of the ARFIMA model. By simultaneously maximizing the likelihood function over both the short and long-memory parameters and using only the wavelet coefficient's variances, the approximate wavelet MLE provides a fast alternative to the frequency-domain MLE. Furthermore, the simulation studies found herein reveal the approximate wavelet MLE to be robust over the invertible parameter region of the ARFIMA model's moving average parameter, whereas the frequency-domain MLE dramatically deteriorates as the moving average parameter approaches the boundaries of invertibility.", } @Article{jen:ols, title = "Using wavelets to obtain a consistent ordinary least squares estimator of the long-memory parameter", author = "Mark J. Jensen", journal = JF, volume = "18", number = "1", year = "1999", pages = "17--32", email = "jensen@haar.econ.siu.edu", URL = "http://econwpa.wustl.edu/eprints/em/papers/9710/9710002.abs", keywords = "fractionally integrated processes long-memory wavelets", abstract = "We develop an ordinary least squares estimator of the long-memory parameter from a fractionally integrated process that is an alternative to the Geweke and Porter-Hudak (1983) estimator. Using the wavelet transform from a fractionally integrated process, we establish a log-linear relationship between the wavelet coefficients' variance and the scaling parameter equal to the log-memory parameter. This log-linear relationship yields a consistent ordinary least squares estimator of the long-memory parameter when the wavelet coefficients' population variance is replaced by their sample variance. We derive the small sample bias and variance of the ordinary least squares estimator and test it against the GPH estimator and the McCoy-Walden maximum likelihood wavelet estimator by conducting a number of Monte Carlo experiments. Based upon the criterion of choosing the estimator which minimizes the mean squared error, the wavelet OLS approach was superior to the GPH estimator, but inferior to the McCoy-Walden wavelet estimator for the processes simulated. However, given the simplicity of programming and running the wavelet OLS estimator and its statistical inference of the long-memory parameter we feel the general practitioner will be attracted to the wavelet OLS estimator.", } @Unpublished{jen:wafip, title = "Wavelet Analysis of Fractionally Integrated Processes", author = "Mark J. Jensen", year = "1994", note = "Department of Economics, Washington University", email = "jensen@wuecona.wustl.edu", URL = "http://econwpa.wustl.edu/eprints/em/papers/9405/9405001.abs", keywords = "Long-Memory, Wavelets, Spectral Analysis, 1/f Processes", abstract = "In this paper we apply wavelet analysis to the class of fractionally integrated processes to show that this class is a member of the $1/f$ family of processes as defined by Wornell (1993) and to produce an alternative method of estimating the fractional differencing parameter. Currently the method by Geweke and Porter-Hudak (1983) is used most often to estimate and test the fractional differencing parameter. The GPH approach, however, has been shown to have poor statistical properties and suffers from subjective decisions that the users must make. The wavelet analysis estimate of the fractional differencing parameter is shown to be more straightforward and to provide results that are more robust than the GPH method.", } @Book{joh-kot:book70, title = "Continuous Univariate Distributions", booktitle = "Continuous Univariate Distributions", author = "Norman L. Johnson and Samuel Kotz", volume = "", publisher = "Houghton Mifflin", address = NY, year = "1970", } @Book{joh-kot:book72, title = "Continuous Multivariate Distributions", booktitle = "Continuous Multivariate Distributions", author = "Norman L. Johnson and Samuel Kotz", volume = "", publisher = "John Wiley \& Sons, Inc.", address = NY, year = "1972", } @Article{joh-sil:correlated, journal = JRSSB, volume = "59", number = "2", year = "1997", pages = "319--351", title = "{W}avelet threshold estimators for data with correlated noise", author = "I. M. Johnstone and B. W. Silverman", URL = "http://playfair.Stanford.EDU/reports/johnstone/correl.ps.gz", abstract = "Wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level-dependent soft threshold to the coefficients in the wavelet transform. A variety of threshold choices is proposed, including one based on an unbiased estimate of mean-squared error. The practical performance of the method is demonstrated on examples, including data from a neurophysiological context. The theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable 'bench-mark', that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity, and more generally is obtained by the use of an 'oracle' that provides information that is not actually available in the data. It is shown that the level-dependent threshold estimator performs well relative to the bench-mark risk, and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator. The wavelet domain structure of both short-and long-range dependent noise is considered, and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes, adapting automatically to the regularity properties of the underlying model. The proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem.", keywords = "decomposition. regression. shrinkage.", } @Book{joh-wic:multivariate, title = "Applied Multivariate Statistical Analysis", booktitle = "Applied Multivariate Statistical Analysis", author = "Richard A. Johnson and Dean W. Wichern", edition = "4", publisher = "Prentice-Hall, Inc.", address = "Englewood Cliffs, NJ", year = "1998", pages = "799", } @Article{joh:adaptivity, title = "Wavelet shrinkage for correlated data and inverse problems: {A}daptivity results", author = "Johnstone, I. M.", journal = SSin, volume = "9", number = "1", year = "1999", pages = "51--83", keywords = "adaptation correlated data fractional Brownian motion linear inverse problems long range dependence mixing conditions oracle inequalities rates of convergence unbiased risk estimate wavelet Vaguelette decomposition wavelet shrinkage wavelet thresholding", abstract = "Johnstone and Silverman (1997) described a level-dependent thresholding method for extracting signals from correlated noise. The thresholds were chosen to minimize a data based unbiased risk criterion. Here we show that in certain asymptotic models encompassing short and long range dependence, these methods are simultaneously asymptotically minimax up to constants over a broad range of Besov classes. We indicate the extension of the methods and results to a class of linear inverse problems possessing a wavelet vaguelette decomposition.", } @Article{jon-lon-mai:hurst, title = "Wavelet packet computation of the {H}urst exponent", author = "C. L. Jones and G. T. Lonergan and D. E. Mainwaring", journal = JPA, volume = "29", number = "10", year = "1996", pages = "2509--2527", keywords = "signals representations diffusion fractals", abstract = "Wavelet packet analysis was used to measure the global scaling behaviour of homogeneous fractal signals from the slope of decay for discrete wavelet coefficients belonging to the adapted wavelet best basis. A new scaling function for the size distribution correlation between wavelet coefficient energy magnitude and position in a sorted vector listing is described in terms of a power law to estimate the Hurst exponent. Profile irregularity and long-range correlations in self-affine systems can be identified and indexed with the Hurst exponent, and synthetic one-dimensional fractional Brownian motion (fBm) type profiles are used to illustrate and test the proposed wavelet packet expansion. We also demonstrate an initial application to a biological problem concerning the spatial distribution of local enzyme concentration in fungal colonies which can be modelled as a self-affine trace or an `enzyme walk'. The robustness of the wavelet approach applied to this stochastic system is presented, and comparison is made between the wavelet packet method and the root-mean-square roughness and second-moment approaches for both examples. The wavelet packet method to estimate the global Hurst exponent appears to have similar accuracy compared with other methods, but its main advantage is the extensive choice of available analysing wavelet filter functions for characterizing periodic and oscillatory signals.", } @InProceedings{kad:choice, title = "On the Choice of a Wavelet, and the Energy and the Phase Distributions of the Wavelet Transform", author = "Shubha Kadambe", booktitle = "Time-Frequency and Time-Scale Analysis", organization = IEEESPS, year = "1992", pages = "379--382", address = "Victoria, B.C., Canada", } @InProceedings{kai:filtering, title = "Wavelet filtering in the scale domain", author = "Gerald Kaiser", pages = "51--54", crossref = "szu:wavelet3", abstract = "It is shown that any convolution operator in the time domain can be represented exactly as a multiplication operator in the time-scale (wavelet) domain. The Mellin transform establishes a one-to-one correspondence between frequency filters (system or transfer functions) and scale filters, which are defined as multiplication operators in the scale domain, subject to the convergence of the defining integrals. Applications to the denoising of random signals are proposed. We argue that the present method is more suitable for removing the effects of atmospheric turbulence than the conventional procedures based on Fourier analysis because it is ideally suited for resolving spectral power laws.", } @Book{kai:friendly, title = "A Friendly Guide to Wavelets", author = "Gerald Kaiser", publisher = "Springer-Verlag", address = NY, year = "1994", pages = "300", ISBN = "0-8176-3711-7", keywords = "Suggestions to the Reader. List of Symbols, Conventions and Transforms. Preliminaries: Background and Notation. Windowed Fourier Transforms. Continuous Wavelet Transforms. Generalized Frames: Key to Analysis and Synthesis. Discrete Time-Frequency Analysis and Sampling. Discrete Time-Scale Analysis. Multiresolution Analysis. Daubechies' Orthonormal Wavelet Bases. Introduction to Wavelet Electromagnetics. Applications to Radar and Scattering. Wavelet Acoustics", abstract = "This volume consists of two parts. The first part, forming chapters 1-8, is designed as a textbook for an introductory one-semester course on wavelet analysis and time-frequency analysis aimed at graduate students or advanced undergraduates in science and engineering. Each of the first eight chapters ends with a set of straightforward exercises designed to drive home the concepts just covered, and the graphics should further facilitate absorption. The second part, form-ing chapters 9-11, represents original research and is written in a more advanced style. This section can be used as a textbook for a second-semester course or, when combined with chapters 1 \& 3, as a reference for an advanced research-level seminar.", } @Article{kai:mellin, title = "Wavelet filtering with the {M}ellin transform", author = "Gerald Kaiser", journal = AML, volume = "9", number = "5", year = "1996", pages = "69--74", abstract = "It is shown that any convolution operator in the time domain can be represented exactly as a multiplication operator in the time-scale (wavelet) domain. The Mellin transform gives a one-to-one correspondence between frequency filters (system functions) and scale filters (multiplication operators in the scale domain), subject to the convergence of the defining integrals. Applications to the denoising of random signals are proposed. It is argued that the present method is more suitable for removing the effects of atmospheric turbulence than the conventional procedures because it is ideally suited for resolving spectral power laws.", } @Article{kai:physical, title = "Physical wavelets and radar: {A} variational approach to remote sensing", author = "Gerald Kaiser", journal = IEEEAPM, volume = "38", number = "1", year = "1996", pages = "15--24", abstract = "Physical wavelets are acoustic or electromagnetic waves, resulting from the emission of a time signal by a localized acoustic or electromagnetic source moving along an arbitrary trajectory in space. Thus, they are localized solutions of the wave equation or Maxwell`s equations. Under suitable conditions, such wavelets can be used as ``basis'' functions, to construct general acoustic or electromagnetic waves. This gives a local alternative to the construction of such waves in terms of (nonlocal) plane waves, via Fourier transforms. We give a brief, self-contained introduction to physical wavelets, and apply them to remote sensing. We define the ambiguity functional, generalization of the radar and sonar ambiguity functions, which applies not only to wideband signals, but also to targets and radar platforms executing arbitrary nonlinear motions.", } @Article{kap-kuo:fractal, title = "Fractal Estimation from Noisy Data via Discrete Fractional Gaussian Noise ({DFGN}) and the Haar Basis", author = "Lance M. Kaplan and C.-C. Jay Kuo", journal = IEEETSP, volume = "41", number = "12", year = "1993", pages = "3554--3562", abstract = "The authors show that the application of the discrete wavelet transform (DWT) using the Haar basis to the increments of fractional Brownian motion (fBm), also known as discrete fractional Gaussian noise (DFGN), yields coefficients which are weakly correlated and have a variance that is exponentially related to scale. Similar results were derived by Flandrin (1989), Tewfik, and Kim for a continuous-time fBm going through a continuous wavelet transform (CWT). The new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by Wornell and Oppenheim. The performance of the new algorithm is compared with that of Wornell and Oppenheim's (see IEEE Trans. Signal Processing, vol. 40, p. 611-623, Mar. 1992) algorithm in numerical simulation.", } @Article{kar-jon-kni:bias, title = "Testing for Bias in the Climate Record", author = "Thomas R. Karl and Philip D. Jones and Richard W. Knight", journal = "Science", volume = "271", number = "5257", year = "1996", pages = "1879--1883", abstract = "The method climatologists use to calculate trends on monthly and annual time series do not introduce significant bias as has been suggested. Perihelion calender shifts were used to test for bias because they have no impact on annual mean temperature trends. Monthly differences were insignificant.", } @TechReport{kat-vid-alb:global-local, author = {Gabriel Katul and Brani Vidakovic and John Albertson}, title = {Estimating Global and Local Scaling Exponents in Turbulent Flows using Wavelet Transformations}, institution = {Institute for Statistics and Decision Sciences, Duke University}, year = {1999}, number = {99-24}, } @Article{kat-mas:fBm, title = "On the spectral density of the wavelet transform of fractional {B}rownian motion", author = "Takeshi Kato and Elias Masry", journal = JTSA, volume = "20", number = "5", year = "1999", pages = "559--563", } @Article{kaw-ari:matching, title = "Signal matching using wavelet correlation", author = "Kouzou Kawata and Suguru Arimoto", journal = ECJ3, volume = "79", number = "9", year = "1996", pages = "23--34", note = "Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. 78-A, No. 12, December 1995, pp. 1655--1664", keywords = "wavelet correlation, local correlation, bandpass cross correlation, corresponding problem, complex filter", abstract = "The problem of detecting corresponding points is studied in the case in which local deformations exist and a new method named ``wavelet correlation'' is proposed. There is a difficulty in that a reasonable window width cannot be designed in local correlation, which is one of the methods for a corresponding problem. The wavelet correlation is derived by extending the notion of local correlation and is considered to overcome difficulty. The fundamental concept is derived by the belief that a signal can be decomposed to several (sinusoidal) components and the window width can be varied according to each component. It is claimed that any algorithm using local correlation can be replaced by the one using wavelet correlation. In this paper, the wavelet correlation derived from local correlation is compared with the Laplacian distance and the local correlation itself by experiments. Further, a matching method that uses a narrow-band property of a wavelet correlation function is proposed and the matching error is evaluated through experiments using one-dimensional signals. Finally, an absolute measure of matching by using normalized wavelet correlation is introduced and applied for detecting discontinuities of local deformations.", } @Article{ken-woo:fractal, title = "Estimating the fractal dimension of a locally self-similar {G}aussian process by using increments", author = "J. T. Kent and A. T. A. Wood", journal = JRSSB, volume = "59", number = "3", year = "1997", pages = "679--700", } @Article{ker-pic-tri:lp, title = "Lp adaptive density estimation", author = "G{\'e}rard Kerkyacharian and Dominique Picard and Karine Tribouley", journal = Ber, volume = "2", number = "3", year = "1996", pages = "229--247", URL = "", abstract = "We provide global adaptive wavelet-type density estimates. Our procedures illustrate the refinement which can be obtained by replacing the Fourier basis by the wavelet basis in estimation methods. The main argument consists in observing that the estimated total energy of the details of a specified level j will be smaller or greater than some known threshold if precisely j is above or below the theoretical optimal level calculated with the a priori knowledge of the regularity of the density. This balancing effect leads directly to an adaptation procedure, and some natural extensions. We investigate the minimax properties of these procedures and explain their evolution for different global error measures.", keywords = "adaptive estimation Besov spaces density estimation minimax estimation U-estimate wavelet orthonormal bases", } @Article{kes-mou:matching, title = "Matching wavelet packets to Gaussian random processes", author = "Keshava, N. and Moura, J. M. F.", journal = IEEETSP, volume = "47", number = "6", year = "1999", pages = "1604--1614", keywords = "basis functions best basis search Bhattacharyya coefficient binary detection classification nonadditive cost function random processes wavelet packet", abstract = "In this paper, we consider the problem of approximating a set of arbitrary, discrete-time, Gaussian random processes by a single, representative wavelet-based, Gaussian process. We measure the similarity between the original processes and the wavelet-based process with the Bhattacharyya coefficient, By manipulating the Bhattacharyya coefficient, we reduce the task of defining the representative process to finding an optimal unitary matrix of wavelet-based eigenvectors, an associated diagonal matrix of eigenvalues, and a mean vector. The matching algorithm we derive maximizes the nonadditive Bhattacharyya coefficient in three steps: a migration algorithm that determines the best basis by searching through a wavelet packet tree for the optimal unitary matrix of wavelet-based eigenvectors; and two separate fixed-point algorithms that derive an appropriate set of eigenvalues and a mean vector. We illustrate the method with two different classes of processes: first-order Markov and bandlimited, The technique is also applied to the problem of robust terrain classification in polarimetric SAR images.", } @Article{kha-duc:detection, title = "Detection and classification of multiple events in piecewise stationary signals: {C}omparison between autoregressive and multiscale approaches", author = "Khalil, M. and Duch{\^e}ne, J.", journal = SP, volume = "75", number = "3", year = "1999", pages = "239--251", email = "mohamad.khalil@univ-troyes.fr", URL = "http://www.elsevier.nl/cas/tree/store/sigpro/sub/1999/75/3/1382.pdf", keywords = "detection classification rejection wavelet AR modelling multiscale decomposition", abstract = "In this paper, we present methods of detection and classification of events in nonstationary signals which are well adapted to uterine EMG processing. Two sequential methods of detection are presented: the first one is monodimensional and based on AR modelling, the second is multidimensional and achieved by decomposing the signal onto scales using wavelet transform. Hypothesis rejection is achieved using either AR coefficients or a variance covariance matrix computed from the scales. Both methods are adaptive and allow event detection without necessarily returning to the null hypothesis H-o. They have been applied to simulated data and uterine EMG. Their performances have been compared together.", } @InCollection{kol:application, title = "An application of wavelet shrinkage to tomography", author = "Eric D. Kolaczyk", pages = "77--92", crossref = "ald-uns:medicine", URL = "", abstract = "", } @Article{kol:burst, title = "Non-Parametric Estimation of Gamma-Ray Burst Intensities Using Haar Wavelets", author = "Eric D. Kolaczyk", journal = ApJ, volume = "483", number = "1", year = "1997", pages = "340--349", URL = "ftp://galton.uchicago.edu/pub/kolaczyk/TIPSH_Appl.ps.Z", abstract = "In this article, I present a method for the non-parametric (model-free) estimation of intensity profiles underlying gamma-ray bursts. The method, TIPSH, is based on applying specially calibrated thresholds to the Haar wavelet coefficients of binned counts gathered from such bursts. As functions well-localized with respect to both time and scale, wavelets are an ideal tool for working with the often sharp, abrupt nature of gamma-ray burst signals. When applied to an idealized signal in a small simulation study and a selection of actual gamma-ray bursts, the TIPSH algorithm is found to be well capable of simultaneously estimating the smooth, uniform background and the pulse-like structure of gamma-ray burst signals.", } @Article{kol:method, title = "Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds", author = "Eric D. Kolaczyk", journal = SSin, volume = "9", number = "1", year = "1999", pages = "119--135", URL = "ftp://galton.uchicago.edu/pub/kolaczyk/PoisTholds.ps.Z", keywords = "gamma-ray bursts. large deviations. Poisson processes. wavelets", abstract = "Wavelet shrinkage estimation has been found to be a powerful tool for the non-parametric estimation of spatially variable phenomena. Most work in this area to date has concentrated primarily on the use of wavelet shrinkage techniques in contexts where the data are modeled as observations of a signal plus additive, Gaussian noise. In this paper, I introduce an approach to estimating intensity functions for a certain class of ``burst-like'' Poisson processes using wavelet shrinkage. The proposed method is based on the shrinkage of wavelet coefficients of the original count data, as opposed to the current approach of pre-processing the data using Anscombe's square root transform and working with the resulting data in a Gaussian framework. ``Corrected'' versions of the usual Gaussian-based shrinkage thresholds are used. The corrections explicitly account for effects of the first few cumulants of the Poisson distribution on the tails of the coefficient distributions. A large deviations argument is used to justify these corrections. The performance of the new method is examined, and compared to that of the pre-processing approach, in the context of an application to an astronomical gamma-ray burst signal.", } @Unpublished{kol:poisson, title = "Estimation of Intensities of Burst-Like Poisson Processes Using Haar Wavelets", author = "Eric D. Kolaczyk", note = "Submitted to the {\em Journal of the Royal Statistical Society, Series B}", year = "1997", URL = "ftp://galton.uchicago.edu/pub/kolaczyk/TIPSH_Mod.ps.Z", abstract = "I present a method for producing estimates of the intensity function of certain `burst-like' inhomogeneous Poisson processes, based on the shrinkage of Haar wavelet coefficients of the observed counts. The Haar basis is a natural wavelet basis in which to work in this context, and I derive thresholds for shrinkage estimation based on the distribution of the coefficients. The translation-invariant de-noising approach of Donoho and Coifman (1995) is used in conjunction with these level-dependent thresholds to yield smooth estimates, without the usual `staircase' structure associated with Haar wavelets. This work is motivated by recent efforts in astronomy to model the intensity functions underlying gamma-ray bursts. It is demonstrated that the method proposed herein (TIPSH) yields sharper estimates of the detail structure in these signals than those obtained through an analogous version of the standard adaptation of wavelet shrinkage for Poisson counts, based on the square-root transformation.", } @Article{kol:shrinkage, journal = JASA, volume = "91", number = "435", year = "1996", pages = "1079--1090", title = "{A} wavelet shrinkage approach to tomographic image reconstruction", author = "E. D. Kolaczyk", abstract = "A method is proposed for reconstructing images from tomographic data with respect to a two-dimensional wavelet basis. The Wavelet-vaguelette decomposition (WVD) is used as a framework within which expressions for the necessary wavelet coefficients may be derived. These coefficients are calculated using a version of the filtered back-projection algorithm as a computational tool, in a multiresolution fashion. The necessary filters are defined in terms of the underlying wavelets. Denoising is accomplished through an adaptation of the wavelet shrinkage (WS) approach of Donoho et al. and amounts to a form of regularization. Combining these two steps yields the proposed WVD/WS reconstruction algorithm, which is compared to the traditional filtered backprojection method in a small study.", keywords = "decomposition, backprojection. tomography. wavelet-vaguelette", } @Article{kom-etal:helioseismic, title = "Multitaper spectral analysis and wavelet denoising applied to helioseismic data", author = "Komm, R. W. and Gu, Y. and Hill, F. and Stark, P. B. and Fodor, I. K.", journal = ApJ, volume = "519", number = "1", year = "1999", pages = "407--421", abstract = "Estimates of solar normal mode frequencies from helioseismic observations can be improved by using multitaper spectral analysis (MTSA) to estimate spectra from the time series, then using wavelet denoising of the log spectra. MTSA leads to a power spectrum estimate with reduced variance and better leakage properties than the conventional periodogram. Under the assumption of stationarity and mild regularity conditions, the log multitaper spectrum has a statistical distribution that is approximately Gaussian, so wavelet denoising is asymptotically an optimal method to reduce the noise in the estimated spectra. We find that a single m-v spectrum benefits greatly from MTSA followed by wavelet denoising and that wavelet denoising by itself can be used to improve m-averaged spectra. We compare estimates using two different five-taper estimates (Slepian and sine tapers) and the periodogram estimate for Global Oscillation Network Group (GONG) time series at selected angular degrees l. We compare those three spectra with and without wavelet denoising, both visually and in terms of the mode parameters estimated from the preprocessed spectra using the GONG peak-fitting algorithm. The two multitaper estimates give equivalent results. The number of modes fitted well by the GONG algorithm is 20%-60% larger (depending on l and the temporal frequency) when applied to the multitaper estimates than when applied to the periodogram. The estimated mode parameters (frequency, amplitude, and width) are comparable for the three power spectrum estimates, except for modes with very small mode widths (a few frequency bins), where the multitaper spectra broaden the modes compared with the periodogram. At frequencies below 3 mHz, wavelet denoising of the log multitaper power spectra tends to increase the number of modes for which the GONG peak-fitting algorithm converges well. Close to 3 mHz, where all modes are resolved, wavelet denoising makes little difference. At higher frequencies close to the acoustic cutoff frequency, where modes are blended into ridges, wavelet denoising the multitaper spectra reduces the number of good fits. We tested the influence of the number of tapers used and found that narrow modes at low n-values are broadened to the extent that they can no longer be fitted if the number of tapers is too large. For helioseismic time series of this length and temporal resolution, the optimal number of tapers is less than 10.", } @Book{koo:wavelets, title = "Wavelets: {A}n Elementary Treatment of Theory and Applications", editor = "Tom H. Koornwinder", series = "Approximations and Decompositions", volume = "1", publisher = "World Scientific", address = "Singapore", pages = "225", year = "1993", keywords = "", abstract = "", } @Book{kot-joh-rea:encyclopedia, title = "Encyclopedia of Statistical Sciences", booktitle = "Encyclopedia of Statistical Sciences", editor = "Samuel Kotz and Norman L. Johnson and Campbell B. Read", publisher = "Wiley", address = NY, year = "1982", } @Article{kov-sil:extending, title = "Extending the scope of wavelet regression methods by coefficient-dependent thresholding", author = "Arne Kovac and Bernard W. Silverman", journal = JASA, volume = "95", number = "449", year = "2000", pages = "172--183", } @InProceedings{kri-dro-pes:multiscale, title = "{M}ultiscale detection of nonstationary signals", booktitle = "Time-Frequency and Time-Scale Analysis", organization = IEEESPS, year = "1992", pages = "105--108", address = "Victoria, B.C., Canada", author = "H. Krim and K. Drouiche and J. C. Pesquet", abstract = "A statistical method for detecting and/or localizing nonstationarities in a process observed over a time interval T is presented. Stationarity is induced by taking a wavelet transform of the process. A parametric model is fitted to the result. The error incurred in fitting the model is shown to preserve the singularity manifested in the transform. The error is then used to establish a statistical detection test that is free of any prior knowledge about the class of signals being analyzed, and of any user input.", keywords = "multiscale detection. nonstationary signals. statistical method. time interval. wavelet transform. parametric model. singularity. statistical detection test.", } @InProceedings{kri-pes-wil:robust, title = "{R}obust multiscale representation of processes and optimal signal reconstruction", booktitle = "Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis", pages = "1--4", year = "1994", author = "H. Krim and J. C. Pesquet and A. S. Willsky", note = "25-28 Oct. 1994, Philadelphia, PA, USA", abstract = "We propose a statistical approach to obtain a ``best basis'' representation of an observed random process. We derive statistical properties of a criterion first proposed to determine the best wavelet packet basis, and, proceed to use it in constructing a statistically sound algorithm. For signal enhancement, this best basis algorithm is followed by a nonlinear filter based on the minimum description length (MDL) criterion. We show that it is equivalent to a min-max based algorithm proposed by Donoho and Johnstone (1992).", keywords = "optimal signal reconstruction. statistical approach. robust multiscale representation. random process. wavelet packet basis. statistically sound algorithm. signal enhancement. best basis algorithm. nonlinear filter. minimum description length criterion. min-max based algorithm. white noise. Gaussian noise.", } @Article{kri-pes:nonstationary, title = "Multiresolution analysis of a class of nonstationary processes", author = "H. Krim and J. C. Pesquet", journal = IEEETIT, volume = "41", number = "4", year = "1995", pages = "1010--1020", keywords = "multiresolution analysis, nonstationary processes, signal processing, nonstationary signals, multiscale framework, discrete-time analysis, parametric model, wide-sense stationarity, analysis wavelet, wavelet packet analysis, nonstationarities", abstract = "Processing nonstationary signals is an important and challenging problem. We focus on the class of nonstationary processes with stationary increments of an arbitrary order, and place them in a multiscale framework. Unlike other related studies, we concentrate on the discrete-time analysis and derive a number of new results in addition to placing the related existing ones in the same framework. We extend the study to various parametric models for which we derive the resulting multiresolution description. We show that wide-sense stationarity may be achieved by adequately selecting the analysis wavelet. After generalizing the study to wavelet packet analysis, we show that the latter possesses additional properties which are useful in the presence of other types of nonstationarities.", } @Article{kri-etal:denoising, title = "On denoising and best signal representation", author = "Krim, H. and Tucker, D. and Mallat, S. and Donoho, D.", journal = IEEETIT, volume = "45", number = "7", year = "1999", pages = "2225--2238", abstract = "We propose a best basis algorithm for signal enhancement in white Gaussian noise. The best basis search is performed in families of orthonormal bases constructed with wavelet packets or local cosine bases. We base our search for the ``best'' basis on a criterion of minimal reconstruction error of the underlying signal. This approach is intuitively appealing because the enhanced or estimated signal has an associated measure of performance, namely, the resulting mean-square error. Previous approaches in this framework have focused on obtaining the most ``compact'' signal representations, which consequently contribute to effective denoising, These approaches, however, do not possess the inherent measure of performance which our algorithm provides. We first propose an estimator of the mean-square error, based on a heuristic argument and subsequently compare the reconstruction performance based upon it to that based on the Stein unbiased risk estimator. We compare the two proposed estimators by providing both qualitative and quantitative analyses of the bias term. Having two estimators of the mean-square error, we incorporate these cost functions into the search for the ``best'' basis, and subsequently provide a substantiating example to demonstrate their performance.", } @Article{kro-ram-jon:frequency-shift-invariant, title = "Frequency-shift-invariant orthonormal wavelet packet representations", author = "Krongold, B. S. and Ramchandran, K. and Jones, D. L.", journal = IEEETSP, volume = "47", number = "9", pages = "2579--2582", abstract = "It is commonly known that the dyadic structure of wavelet expansions results in both time- and frequency-translation sensitivity of an input signal. We develop the first efficient method to reduce frequency-alignment sensitivity by introducing a wavelet packet decomposition that is invariant to frequency shifts of a signal. A frequency-shifted wavelet packet library is presented, and an efficient best-basis algorithm is developed to determine the best signal representation among all frequency shifts of a signal. The algorithm computes all frequency-shifted coefficients in O(N-2) operations, followed by an efficient depth-first tree search of the same complexity." } @Article{kul-sad-mur:trough, title = "Wavelet analysis of intermittent turbulent transport in the atmospheric surface layer over a monsoon trough region", author = "Kulkarni, J. R. and Sadani, L. K. and Murthy, B. S.", journal = BLM, volume = "90", number = "2", year = "1999", pages = "217--239", keywords = "atmospheric surface layer quadrant analysis turbulence intermittency turbulent fluxes wavelet analysis", abstract = "The structure of the turbulence in the atmospheric surface layer over a monsoon trough region has been studied using structural analysis based on wavelet transform. The observational site is located at the eastern (wet) end of the monsoon trough region, characterized by high moisture in the atmospheric surface layer. On the average relative humidity varied from 70% to 100% during the experiment. The wind and temperature data, collected at Kharagpur (22 degrees 25' N, 87 degrees 18' E) at six observational hours of a day in June 1990 during the Monsoon Trough Boundary Layer Experiment (MONTBLEX), have been utilized in the study. The wind and instantaneous momentum flux time series were decomposed into 12 scales using the Haar wavelet transform. The eddies exhibited a large temporal variability generating intermittency in the energy and Bur distributions. A criterion based on the isotropy has been suggested for separating the large eddies from the small eddies. At the separation scale the isotropy coefficient drops sharply. It is shown that the intermittency in the small eddies resulted from the spatial variation of energy, and deviation of velocity statistics from the Gaussian distribution known as flatness. The deviation from the -5/3 power law has been attributed to the increased mean values of, (i) the coefficient of variation of energy, and (ii) the flatness factor, in the inertial subrange. The decomposition of the instantaneous momentum Bur time series reveals that the major contribution to the total flux arises from the large eddies. The quadrant analysis of the momentum flux shows that ejections and sweeps account for a substantial part of the total flux. and quantifies the relative importance of the various spatial scales that contribute to the transport of momentum.", } @Article{kul:monsoon, title = "Wavelet analysis of the association between the {S}outhern {O}scillation and the {I}ndian {S}ummer {M}onsoon", author = "Kulkarni, J. R.", journal = IJC, volume = "20", number = "1", year = "2000", pages = "89--104", abstract = "A new aspect of the monsoon-Southern Oscillation (SO) link has been investigated. All India Summer Monsoon Rainfall (AISMR) and Southern Oscillation Index (SOI) data (for August-September-October months) for the period 1871-1998 have been processed for wavelet analysis. Using the Haar wavelet function, the data are decomposed into seven dyadic scales corresponding to periods of 2, 4, 8, 16, 32, 64 and 128 years. The time frequency localization in the wavelet analysis was used to study the temporal variability of modes in AISMR and SOI. The 2 and 8 year modes in both are found to exhibit low frequency modulation. The 4 year mode in both showed large intermittency. The periods of high/low activities of 2, 4 and 8 year modes were associated with a large/low number of deficient AISMR years. The SOI derived from 2, 4 and 8 year modes in the ENSO years, is found to be related to AISMR variability, at 1% level of significance. The 2, 4 and 8 year modes in AISMR and SOI are found to be correlated at a 5% level of significance. There is a large temporal variability in the correlations of these modes. The occurrences of maxima and minima in these correlations followed a sequence, first in the 8 year mode, then in the 4 year mode and in the end, in the 2 year mode. The reasons for de-association between AISMR activity and SOI in the last 8 years of the present decade have been attributed to (i) the negative contributions by 128, 64, 32 and 16 year modes, (ii) the low activity of 4 and 8 year modes and (iii) the weak correlation between AISMR and SOI in 4 and 8 year modes during this period.", } @InCollection{kum-fou:introduction, title = "Wavelet analysis in geophysics: {A}n introduction", author = "Praveen Kumar and Efi Foufoula-Georgiou", pages = "1--43", crossref = "fou-kum:geophysics", keywords = "", abstract = "", } @Article{kum-fou:rainfall, title = "A New Look at Rainfall Fluctuations and Scaling Properties of Spatail Rainfall Using Orthogonal Wavelets", author = "Praveen Kumar and Efi Foufoula-Georgiou", journal = JAM, volume = "32", year = "1993", pages = "209--222", abstract = "Orthogonal wavelet transforms of the rainfall fields are analyzed. Results show that wavelet multiresolution analysis provides methods for the study of nonhomogeneous anisotropic processes and for defining fluctuations in two dimensions. Moreover, orthogonal wavelet transforms segregate large-scale features from small-scale features by providing convenient orthogonal decompositions with directionality. Lastly, orthogonal wavelet analysis is applied to a squall-line storm.", } @Article{kum-fou:review, title = "Wavelet analysis for geophysical applications", author = "Praveen Kumar and Efi Foufoula-Georgiou", journal = "Review of Geophysics", volume = "35", number = "4", year = "1997", pages = "385--412", keywords = "fractional brownian motion 2-scale difference equations turbulent coherent motions spatial rainfall fields signal analysis forest canopy multicomponent decomposition sampling theory transforms representation", abstract = "Wavelet transforms originated in geophysics in the early 1980s for the analysis of seismic signals. Since then, significant mathematical advances in wavelet theory have enabled a suite of applications in diverse fields. In geophysics the power of wavelets for analysis of nonstationary processes that contain multiscale features, detection of singularities, analysis of transient phenomena, fractal and multifractal processes, and signal compression is now being exploited for the study of several processes including space-time precipitation, remotely sensed hydrologic fluxes, atmospheric turbulence, canopy cover, land surface topography, seafloor bathymetry, and ocean wind waves. It is anticipated that in the near future, significant further advances in understanding and modeling geophysical processes will result from the use of wavelet analysis. In this paper we review the basic properties of wavelets that make them such an attractive and powerful tool for geophysical applications, We discuss continuous, discrete, orthogonal wavelets and wavelet packets and present applications to geophysical processes.", } @Article{kum:role, title = "Role of Coherent Structures in the Stochastic-Dynamic Variability of Precipitation", author = "Praveen Kumar", journal = JGRA, volume = "101", number = "D21", year = "1996", pages = "26,393--26,404", keywords = "spatial rainfall fields, multicomponent decomposition, wave-propagation, sampling theory, signal", abstract = "Using time-frequency-scale elements obtained from wavelet packets as a basis, we describe a broad framework of analysis which can be used to reveal the essential dynamics, identified as coherent structures, of precipitation. We show that the matching pursuits algorithm with nearly symmetric orthogonal wavelets provides an optimal representation of the inner structure of rainfall time series and can describe features that range from scales of isolated singularity to synoptically forced large-scale features. We describe the analysis of time series of several storms and show that there exist distinct scales of variation identifiable with rain cell and synoptic-scale activity, which is in contradistinction to the scale invariance hypothesis.", } @Article{kum:scale-space, title = "A Wavelet Based Methodology for Scale-Space Anisotropic Analysis", author = "Praveen Kumar", journal = "Geophysical Research Letters", volume = "22", number = "20", year = "1995", pages = "2777--2780", abstract = "It is well known that several geophysical fields exhibit characteristic features at different scales. For some such fields scale-space anisotropy is also present, that is, features contributing a significant fraction of energy are oriented in different directions at different scales. Examples of such fields include clouds, rainfall, hurricanes etc. A technique based on wavelet transforms (with two-dimensional directionally oriented Morlet wavelet) is developed to analyze such random fields. This methodology has significant advantage over Fourier transform based techniques and is demonstrated using the analysis of a spatial rainfall field.", } @Article{lan-etal:noise, title = "Noise reduction using an undecimated discrete wavelet transform", author = "M. Lang and H. Guo and J. E. Odegard and C. S. Burrus and R. O. Wells", journal = IEEESPL, volume = "3", number = "1", year = "1996", pages = "10--12", abstract = "A new nonlinear noise reduction method is presented that uses the discrete wavelet transform. Similar to Donoho (1995) and Donoho and Johnstone (1994, 1995), the authors employ thresholding in the wavelet transform domain but, following a suggestion by Coifman, they use an undecimated, shift-invariant, nonorthogonal wavelet transform instead of the usual orthogonal one. This new approach can be interpreted as a repeated application of the original Donoho and Johnstone method for different shifts. The main feature of the new algorithm is a significantly improved noise reduction compared to the original wavelet based approach. This holds for a large class of signals, both visually and in the l/sub 2/ sense, and is shown theoretically as well as by experimental results.", } @InProceedings{lan-etal:nonlinear, title = "Nonlinear processing of a shift invariant {DWT} for noise reduction", author = "M. Lang and H. Guo and J. E. Odegard and C. S. Burrus and R. O. Wells", pages = "640--651", crossref = "szu:wavelet2", keywords = "nonlinear processing, shift invariant DWT, noise reduction, thresholding, redundant wavelet transform, nondecimated redundant wavelet transform, synthetic aperture radar, SAR images", abstract = "A novel approach for noise reduction is presented. Similar to Donoho, we employ thresholding in some wavelet transform domain but use a nondecimated and consequently redundant wavelet transform instead of the usual orthogonal one. Another difference is the shift invariance as opposed to the traditional orthogonal wavelet transform. We show that this new approach can be interpreted as a repeated application of Donoho`s original method. The main feature is, however, a dramatically improved noise reduction compared to Donoho`s approach, both in terms of the l/sub 2/ error and visually, for a large class of signals. This is shown by theoretical and experimental results, including synthetic aperture radar (SAR) images.", } @InProceedings{lar-zak:automatic, title = "Automatic Classification of Active Sonar Data Using Time-Frequency Transforms", author = "Francesco Lari and Avideh Zakhor", booktitle = "Time-Frequency and Time-Scale Analysis", organization = IEEESPS, year = "1992", pages = "21--24", address = "Victoria, B.C., Canada", abstract = "Automatic classification of active sonar signals using the Wigner-Ville transform (WVT), the wavelet transform (WT) and the scalogram is addressed. Features are extracted by integrating over regions in the time-frequency (TF) distribution, and are classified by a decision tree. Experimental results show classification and detection rates of up to 92% at -4 dB of SNR. The WT outperforms the WVT and the scalogram, particularly at high noise levels. This can be partially attributed to the absence of cross terms in the WT.", } @Article{lau-wen:climate, title = "Climate signal detection using wavelet transform: {H}ow to make a time series sing", author = "K. M. Lau and Hengyi Weng", journal = BAMetS, volume = "76", number = "12", pages = "23--41", year = "1995", keywords = "", abstract = "In this paper, the application of the wavelet transform (WT) to climate time series analyses is introduced. A tutorial description of the basic concept of WT, compared with similar concepts used in music, is also provided. Using an analogy between WT representation of a time series and a music score, the authors illustrate the importance of local versus global information in the time-frequency localization of climate signals. Examples of WT applied to climate data analysis are demonstrated using analytic signals as well as real climate time series. Results of WT applied to two climate time series - that is, a proxy paleoclimate time series with a 2.5-Myr deep-sea sediment record of [[Delta].sup.18]O and a 140-yr monthly record of Northern Hemisphere surface temperature - are presented. The former shows the presence of a 40-kyr and a 100-kyr oscillation and an abrupt transition in the oscillation regime at 0.7 Myr before the present, consistent with previous studies. The latter possesses a myriad of oscillatory modes from interannual (2-5 yr), interdecadal (10-12 yr, 20-25 yr, and 40-60 yr), and century ([approximately]180 yr) scales at different periods of the data record. In spite of the large difference in timescales, common features in time-frequency characteristics of these two time series have been identified. These features suggest that the variations of the earth's climate are consistent with those exhibited by a nonlinear dynamical system under external forcings.", } @Article{leb-vet:balanced, title = "Balanced multiwavelets theory and design", author = "J{\'e}r{\^o}me Lebrun and Martin Vetterli", journal = IEEETSP, volume = "46", number = "4", pages = "1119--1125", year = "1998", keywords = "balanced multiwavelets time-varying filter banks signal compression multichannel signal processing multifilter banks multiwavelets filter bank design multiresolution analysis", abstract = "This article deals with multiwavelets, which are a generalization of wavelets in the context of time-varying filter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are fit for processing multichannel signals. This is the main issue in which we are interested. First, we review material on multiwavelets and their links with multifilter banks and, especially, time-varying filter banks. Then, we have a close look at the problems encountered when using multiwavelets in applications, and we propose new solutions for the design of multiwavelets filter banks by introducing the so-called balanced multiwavelets", } @Unpublished{led-muj-mur-smi:motion, title = "Spatio-Temporal Wavelet Transforms for Motion Tracking", author = "Jean-Pierre Leduc and Fernando Mujica and Romain Murenzi and Mark Smith", year = "1997", email = "leduc@ee.gatech.edu", note = "Georgia Institute of Technology", } @Article{led:spatio-temporal, title = "Spatio-Temporal Wavelet Transforms for Digital Signal Analysis", author = "Jean-Pierre Leduc", journal = SP, volume = "60", number = "1", pages = "23--41", year = "1997", email = "leduc@ee.gatech.edu, leduc@irisa.fr", keywords = "Wavelet transforms; Spatio-temporal signals; Image seqeunces; Motion analysis; Group of transfromations; Regions of interest; Selective feature extraction", abstract = "The goal of this paper is to investigate spatio-temporal continuous wavelet transforms. A new wavelet family called the Galilean wavelet has been designed to tune to four main parameters namely the scale, the spatio-temporal position, the spatial orientation, and the velocity. The paper starts with the theory of motion-compensated wavelet filtering in the discrete realm of image processing. As a major difference to multi-dimensional homogeneous spaces, the spatio-temporal signals involve motions that warp the signal along the temporal dimension. Modeling motion with 2-D affine transformations leads to spatio-temporal generalizations. Decomposition in to elementary operators lead to developing transformation groups and exploiting the related representation theory. The construction of continuous spatio-temporal wavelets in $R^n \times R$ spaces is then handled with classical techniques of calculation. Close connections may then be established among all the spatio-temp oral wavelet transforms through different sets of transformations. This approachgenerates a general framework for the study of future tools. Frames of wavelets are thereafter investigated to revisit discrete wavelet transforms in a more general way. Eventually illustrations demonstrate the ability of the Galilean wavelet transforms to analyze spatio-temporal contents.", } @TechReport{lee-har-spe:tides, title = "Automated Smoothing of Tides Data Using Wavelets", author = "GeungHee Lee and Jeffrey D. Hart and F. Michael Speed", institution = "Department of Statistics, Texas A\&M University", number = "268", year = "1996", keywords = "automatic trend, jump, outlier, shrinkage estimator, wavelet packet", } @Article{lee:wavelets, title = "Wavelets and wavelet estimation: {A} review", author = "Lee, G. H.", journal = JETE, volume = "4", number = "1", year = "1998", pages = "123--157", abstract = "Wavelets have received a lot of attention in statistics since Donoho and Johnstone (1994) introduced wavelet shrinkage estimators, which include some important ideas--wavelets as a new local basis, multiresolution analysis and thresholding. From these ideas, we can reilluminate the methods and data analyses based on Fourier series and truncated estimators. In this paper, we review wavelets, wavelet series estimators, their model selection methods and their applications in econometrics and economics.", } @Book{leh:tpe, title = "Theory of Point Estimation", author = "E. L. Lehmann", publisher = "Wiley", address = NY, pages = "506", year = "1983", } @Book{leh:tsh, title = "Testing Statistical Hypotheses", author = "E. L. Lehmann", publisher = "Wiley", address = NY, edition = "2", pages = "600", year = "1986", } @TechReport{len:denoising, title = "Denoising not equispaced data with wavelets", author = "L. Lenarduzzi", year = "1997", number = "IAMI 97.1", institution = "Instituto per le Applicazioni della Matematica e dell'Informatica", } @InProceedings{lew-bur:approximate, title = "Approximate continuous wavelet transform with an application to noise reduction", author = "James M. Lewis and C. Sidney Burrus", booktitle = "Proceedings of the International Conference on Acoustics, Speech, and Signal Processing", pages = "???--???", year = "1998", URL = "http://www-dsp.rice.edu/publications/pub/lewi-icassp98.ps.Z", abstract = "We describe a generalized scale-redundant wavelet transform which approximates a dense sampling of the continuous wavelet transform (CWT) in both time and scale. The dyadic scaling requirement of the usual wavelet transform is relaxed in favor of an approximate scaling relationship which in the case of a Gaussian scaling function is known to be asymptotically exact and irrational. This scheme yields an arbitrarily dense sampling of the scale axis in the limit. Similar behavior is observed for other scaling functions with no explicit analytic form. We investigate characteristics of the family of Lagrange interpolating filters (related to the Daubechies family of compactly-supported orthonormal wavelets), and finally present applications of the transform to denoising and edge detection.", } @Article{li-noz:cross-correlation, journal = "Japanese Society of Mechanical Engineers International Journal, Series B", volume = "40", number = "1", year = "1997", pages = "58--66", title = "{A}pplication of wavelet cross-correlation analysis to a plane turbulent jet", author = "Hui Li and Tsutomu Nozaki", abstract = "A new cross-correlation method, which is called wavelet cross-correlation analysis and is used to express the statistical cross-correlation of two arbitrary signals in terms of scale and time delay, is proposed and its main properties are presented, analyzing two test signals, it is shown that wavelet cross-correlation does not have the limitations of classical cross-correlation. As a practical application to fluid mechanics, wavelet cross-correlation is employed to determine the cross-correlation relationships between the x-components of the fluctuation velocities at two points on opposite sides of the centerline and along the centerline of a plane turbulent jet in terms of period and time delay. In the distributions of the wavelet cross-correlation coefficients, similar structures with various scales are observed instantaneously, and the period of eddy and apparent flapping motions can be determined easily in terms of period and time delay. It is found that the apparent flapping behavior appears first in region with an intermediate period. It is also revealed that a similar structure with a high period consists of similar structures with a low period, i.e., a large eddy contains small eddies.", keywords = "turbulent flow, jet, eddy, coherent structure, branching structure, mixing layer, wavelet analysis, wavelet cross-correlation function, wavelet cross-correlation coefficient, local wavelet cross-spectrum", } @Article{li-ulr:well-log, title = "Well-log analysis using localized transforms", author = "Li, X. G. and Ulrych, T. J.", journal = "Journal of Seismic Exploration", volume = "8", number = "3", year = "1999", pages = "243--260", abstract = "The wavelet transform (WT) is used to analyze and characterize well-logs in location and scale. In the WT domain, a well-log can be decomposed into deterministic and statistical components. The deterministic component consists of smooth WT coefficients at the largest scale and large WT coefficients at the rest of the scales. The remaining coefficients represent the statistical component which can be modeled as a fractional Brownian motion (FBM). A well-log is used to illustrate this decomposition. To test the fractal model, we have used both 1-D and 2-D wavelet transforms to simulate FBM processes. These simulated FBM series look like well-logs, which verifies the proposed approach. Both orthogonal and continuous WT's are used for analyzing fractal parameters of FBM processes. The orthogonal WT is used to compute a fractal parameter for a particular time series,and the continuous WT is used to estimate the time variant fractal parameter.", } @Article{li:identification, title = "The wavelet identification of thresholds and time delay of threshold autoregressive models", author = "Li, Y.", journal = SSin, volume = "9", number = "1", year = "1999", pages = "153--166", keywords = "thresholds time delay wavelets", abstract = "In this paper, we consider identification of the thresholds and time delay of threshold autoregressive models with p- dependence and an unknown number of thresholds. By checking p different empirical wavelets of the data to see which have significantly large absolute values, the time delay is identified first. By further checking the empirical wavelets corresponding to the time delay across the fine scale levels, the thresholds and their number are identified. All estimators are shown to be strongly consistent.", } @InProceedings{lia-li-kuo:image, title = "Image compression with embedded multiwavelet coding", author = "Kai-Chieh Liang and Jin Li and C.-C. Jay Kuo", pages = "", crossref = "szu:wavelet3", URL = "http://sipi.usc.edu/~lijin/paper/aero96_mwt.ps", abstract = "", } @InProceedings{lia-par:2d, title = "A two-dimensional translation invariant wavelet representation and its applications", author = "Jie Liang and Thomas W. Parks", booktitle = "Proceedings ICIP-94", volume = "1", pages = "66--70", year = "1994", keywords = "two-dimensional translation invariant wavelet representation. sensitivity. two-dimensional signals. fast algorithm. circular translates. input image. optimal translate. decomposition. quadtree search algorithm. complexity. translation invariant wavelet transform. data compression.", abstract = "Addresses the problem of the sensitivity of wavelet representations to translations for two-dimensional signals. The authors describe a fast algorithm to calculate the two-dimensional wavelet transforms for all the circular translates of an input image. They select the optimal translate for the decomposition using a quadtree search algorithm. The resulted wavelet representation is invariant under translations measured by an additive cost criterion. The complexity of the whole algorithm is O(N/sup 2/ log N) for a N*N input block. They apply this translation invariant wavelet transform to data compression. The results show that by taking into account the effect of translations, additional compression can be achieved beyond that achieved by a standard wavelet transform.", } @Article{lia-par:image, title = "Image coding using translation invariant wavelet transforms with symmetric extensions", author = "Jie Liang and Thomas W. Parks", journal = IEEETIP, volume = "7", number = "5", year = "1998", pages = "762--769", keywords = "representation", abstract = "In this correspondence, we address the problem of translation sensitivity of conventional wavelet transforms for two-dimensional (2-D) signals. We propose wavelet transform algorithms that achieve the following desirable properties simultaneously: i) translation invariance, ii) reduced edge effects, and iii) sice-limitedness. We apply this translation invariant biorthogonal wavelet transform with symmetric extensions to image coding applications with good results.", } @Article{lia-par:translation, title = "A translation-invariant wavelet representation algorithm with applications", author = "Jie Liang and Thomas W. Parks", journal = IEEETSP, volume = "44", number = "2", year = "1996", pages = "225--232", abstract = "We address the time-varying problem of wavelet transforms, and a new translation-invariant wavelet representation algorithm is proposed. Using the algorithm introduced by Beylkin (see SIAM J. Numer. Anal., vol. 29, p.1716-1740, 1992), we compute the wavelet transform for all the circular time shifts of a length- N signal in O(N log N) operations. The wavelet coefficients of the time shift with minimal cost are selected as the best representation of the signal using a binary tree search algorithm with an appropriate cost function. We apply the translation-invariant representation algorithm to a geoacoustic data compression application. The results show that the new algorithm can reduce the distortion (the squared error in our case) substantially, if the input signals are transients that are sensitive to time shifts.", } @Article{lil-par:multiwavelet, title = "Multiwavelet spectral and polarization analyses of seismic records", author = "J. M. Lilly and J. Park", journal = GJI, volume = "122", number = "3", year = "1995", pages = "1001--1021", keywords = "multiwavelet spectral analysis polarization analysis seismic records wavelet transform multiple taper spectral analysis low-variance spectrum estimate nonstationary data process multiwavelet algorithm mutually orthogonal Slepian wavelets time-varying spectral density matrix three-component seismic data three-component motion singular value decomposition seismic body waves elliptical polarization polarization estimators earthquakes", abstract = "Presents an algorithm, based on the wavelet transform and multiple taper spectral analysis, for providing a low-variance spectrum estimate of a non-stationary data process. The `multiwavelet' algorithm uses, within each frequency band, a number of mutually orthogonal Slepian wavelets, optimally concentrated in frequency. The sum of squared wavelet transforms with the Slepian wavelets results in a spectrum estimate that is both low-variance and resistant to broad-band bias. The multiwavelet algorithm is used to estimate the time-varying spectral density matrix S(f,t) for two or more time series, in particular for three-component seismic data. Coherent three-component motion is described by motion along a single trajectory, with appropriate projections onto the three component axes. This trajectory is found by applying a singular value decomposition (SVD) to a matrix M(f,t) of wavelet transform values. The normalized first singular value of the SVD determines whether a correlation among the three components of the seismogram is statistically significant. Where significant, coherent particle motion is reconstructed by a linear combination of the wavelets with coefficients specified by the first left-singular vector. The polarization of this motion with respect to the coordinate axes is given by the first right-singular vector. Where the wavelets are real-valued, the usefulness of this method is limited to cases in which the three components of the seismic record oscillate in phase with each other, as is often the case for seismic body waves. Elliptical polarization is handled by pairing even and odd Slepian wavelets into complex-valued wavelets, capable of detecting phase shifts between components. The authors demonstrate the multiwavelet spectrum and polarization estimators on seismic data from a large shallow earthquake in the Solomon Islands, and from the deep earthquakes beneath Fiji (1994 March 9) and Bolivia (1994 June 9).", } @Article{lin-per-rot:ice, title = "The Discrete Wavelet Transform and the Scale Analysis of the Surface Properties of Sea Ice", author = "Ronald W. Lindsay and Donald B. Percival and D. Andrew Rothrock", journal = IEEETGRS, volume = "34", number = "3", year = "1996", pages = "771--787", keywords = "Sea ice; Radar remote sensing; Discrete wavelet transform; Scale analysis; Daubechies wavelet filter; scale-dependent wavelet variance; Wavelet covariance; Mallat orthogonal-pyramid algorithm; Spring pack ice; Beaufort Sea", abstract = "The formalism of the one-dimensional discrete wavelet transform (DWT) based on Daubechies wavelet filters is outlined in terms of finite vectors and matrices. Both the scale-dependent wavelet variance and wavelet covariance are considered and confidence intervals for each are determined. The variance estimates are more accurately determined with a maximal-overlap version of the wavelet transform. The properties of several Daubechies wavelet filters and the associated basis vectors are discussed. Both the Mallat orthogonal-pyramid algorithm for determining the DWT and a pyramid algorithm for determining the maximal-overlap version of the transform are presented in terms of finite vectors. As an example, the authors investigate the scales of variability of the surface temperature and albedo of spring pack ice in the Beaufort Sea. The data analyzed are from individual lines of a Landsat TM image (25-m sample interval) and include both reflective (channel 3, 30-m resolution) and thermal (channel 6, 120-m resolution) data. The wavelet variance and covariance estimates are presented and more than half of the variance is accounted for by scales of less than 800 m. A wavelet-based technique for enhancing the lower-resolution thermal data using the reflected data is introduced. The simulated effects of poor instrument resolution on the estimated lead number density and the mean lead width are investigated using a wavelet-based smooth of the observations.", } @InCollection{liu:new-perspective, title = "Wavelet Transform and New Perspective on Coastal and Ocean Engineering Data Analysis", booktitle = "Advances in Coastal and Ocean Engineering", author = "P. C. Liu", editor = "P. L-F Liu", volume = "6", pages = "???--???", year = "2000", email = "liu@glerl.noaa.gov", abstract = "", } @InCollection{liu:wavelet-spectrum, title = "Wavelet Spectrum Analysis and Ocean Wind Waves", author = "Paul C. Liu", pages = "151--166", email = "liu@glerl.noaa.gov", crossref = "fou-kum:geophysics", abstract = "", } @InCollection{lum-etal:optimization, title = "Optimization of Bias-Variance Trade-Off in Non Parametric Spectral Analysis by Decomposition into Wavelet Packets", author = "B. Lumeau and J. C. Pesquet and J. F. Bercher and L. Louveau", pages = "285--290", crossref = "mey-roq:progress", } @TechReport{ma-str-vid:first-moment, title = "The First Moment of Wavelet Random Variables", author = "Yanyuan Ma and Gilbert Strang and Brani Vidakovic", number = "97-10", institution = "Institute of Statistics and Decision Sciences, Duke University", year = "1997", URL = "ftp://ftp.isds.duke.edu/pub/WorkingPapers/97-10.ps", } @Article{mah:eddy, title = "Eddy asymmetry in the sheared heated boundary layer", author = "L. Mahrt", journal = JAS, volume = "48", number = "3", year = "1991", pages = "472--482", abstract = "Statistical measures are developed to study the influence of mean shear on the asymmetry of eddy updrafts as observed from low-level aircraft flights in HAPEX, FIFE, and SESAME. This asymmetry involves formation of microfronts between updrafts with slow horizontal motion and downdrafts with faster horizontal motion. The variance of the Haar-wavelet transform (step-function basis) is found to be a superior indicator of the dominant scales of such eddies compared to the structure function. For those analyses where scale dependence is not of interest, the simpler structure function is applied. The coherent structures at the dominant scale are examined by computing eigenvectors of the lagged correlation matrix based on conditionally sampled events.", } @Article{mak:sst, title = "Orthogonal wavelet analysis: {I}nterannual variability in the sea surface temperature", author = "Mankin Mak", journal = BAMetS, volume = "76", year = "1995", pages = "2179--2186", keywords = "ocean, environmental aspects, signal processing technique, ocean temperature analysis", abstract = "The unique capability of orthogonal wavelets, which have attractive time-frequency localization properties as exemplified by the Meyer wavelet, is demonstrated in a diagnosis of the interannual variability using a 44-year dataset of the sea surface temperature (SST). This wavelet analysis is performed in conjunction with an empirical orthogonal function analysis and a Fourier analysis to illustrate their complementary capability. The focus of this article is on the equatorial Pacific SST, which is known to have far-reaching impacts on short-term climate variability. The Meyer spectrum brings to light intriguing episodic characteristics of three separate sequences of El Ni{\~n}o (abnormally warm) and La Ni{\~n}a (abnormally cold) events during the past 42 years. It quantifies the relative contributions to the variability associated with different frequency ranges at different times. Through a wavelet cross-spectral analysis with the SST at an equatorial location and at a midlatitude location in the Pacific Ocean, the planetary character of the SST field associated with such events is also illustrated.", } @Article{mal-hwa:singularity, title = "Singularity detection and processing with wavelets", author = "S. G. Mallat and W. L. Hwang", journal = IEEETIT, volume = "38", number = "2", year = "1992", pages = "617--643", URL = "ftp://cs.nyu.edu/pub/tech-reports/tr549-R245.ps.Z", keywords = "one dimensional signals, signal analysis, signal processing, image edge location, white noise removal, image processing, singularities, Lipschitz exponents, wavelet transform, irregular structures, fast oscillations, modulus maxima, two-dimensional signals", abstract = "The mathematical characterization of singularities with Lipschitz exponents is reviewed. Theorems that estimate local Lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed. It is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their Lipschitz exponents. The wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately. The local frequency of such oscillations is measured from the wavelet transform modulus maxima. It has been shown numerically that one- and two-dimensional signals can be reconstructed, with a good approximation, from the local maxima of their wavelet transform modulus. As an application, an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales. In two dimensions, the wavelet transform maxima indicate the location of edges in images.", } @Article{mal-pap-zha:covariance, title = "Adaptive covariance estimation of locally stationary processes", author = "S. Mallat and G. Papicolaou and Z. Zhang", journal = AofS, volume = "26", number = "1", year = "1998", pages = "1--47", URL = "ftp://math.Stanford.EDU/pub/papers/papanicolaou/lsb.ps.gz", keywords = "", abstract = "", } @Article{mal-zha:matching, title = "Matching pursuits with time-frequency dictionaries", author = "S. Mallat and Z. Zhang", journal = IEEETSP, volume = "41", number = "12", year = "1993", pages = "3397--3415", keywords = "", abstract = "", } @Article{mal-zho:characterization, title = "Characterization of signals from multiscale edges", author = "S. Mallat and S. Zhong", journal = IEEETPAMI, volume = "14", number = "7", year = "1992", pages = "710--732", keywords = "1D signals, 2D signals, picture processing, multiscale Canny edge detection, local maxima, wavelet theory, pattern recognition, multiscale edge representation, image coding", abstract = "A multiscale Canny edge detection is equivalent to finding the local maxima of a wavelet transform. The authors study the properties of multiscale edges through the wavelet theory. For pattern recognition, one often needs to discriminate different types of edges. They show that the evolution of wavelet local maxima across scales characterize the local shape of irregular structures. Numerical descriptors of edge types are derived. The completeness of a multiscale edge representation is also studied. The authors describe an algorithm that reconstructs a close approximation of 1-D and 2-D signals from their multiscale edges. For images, the reconstruction errors are below visual sensitivity. As an application, a compact image coding algorithm that selects important edges and compresses the image data by factors over 30 has been implemented.", } @Article{mal:multiresolution, title = "A theory for multiresolution signal decomposition: {T}he wavelet representation", author = "S. Mallat", journal = IEEETPAMI, volume = "11", number = "7", year = "1989", pages = "674--693", keywords = "picture processing, encoding, pattern recognition, multiresolution signal decomposition, wavelet representation, pyramidal algorithm, convolutions, quadrature mirror filters, data compression, image coding, texture discrimination, fractal analysis", abstract = "Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions. In L/sup 2/(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed.", } @Book{mal:tour, title = "A Wavelet Tour of Signal Processing", author = "St\'{e}phane Mallat", publisher = "Academic Press", address = "San Diego", year = "1998", keywords = "", abstract = "A Wavelet Tour of Signal Processing begins with a presentation of the wonders of the Fourier transform, and then describes its failures for transient signal processing. It presents local time-frequency methods and the related mathematical tools. The book uses an intuitive approach to important mathematical results, and emphasizes practical applications rather than proofs. It describes numerical discrete algorithms as well as some applications to information processing, fractal analysis, noise removal, and compact signal coding. A Wavelet Tour of Signal Processing is intended for signal processing engineers who want to discover the potential applications of recent mathematical advances in time-frequency signal representations. Of interest to researchers in applied mathematics, the book highlights the applications of these new techniques and also provides an overview of signal processing problems", } @Article{mal:vision, title = "{W}avelets for a vision", author = "S. Mallat", journal = PIEEE, volume = "84", number = "4", year = "1996", pages = "604--614", abstract = "Early on, computer vision researchers have realized that multiscale transforms are important to analyze the information content of images. The wavelet theory gives a stable mathematical foundation to understand the properties of such multiscale algorithms. This tutorial describes major applications to multiresolution search, multiscale edge detection, and texture discrimination.", } @Article{mal:zero, title = "Zero-crossings of a wavelet transform", author = "S. G. Mallat", journal = IEEETIT, volume = "37", number = "4", year = "1991", pages = "1019--1033", keywords = "signal reconstruction, wavelet transform, completeness, stability, pattern recognition, multiscale representation, zero-crossings, projection algorithm, fast convergence, iteration, coarse-to-fine stereo-matching algorithm", abstract = "The completeness, stability, and application to pattern recognition of a multiscale representation based on zero-crossings is discussed. An alternative projection algorithm is described that reconstructs a signal from a zero-crossing representation, which is stabilized by keeping the value of the wavelet transform integral between each pair of consecutive zero-crossings. The reconstruction algorithm has a fast convergence and each iteration requires O(N log/sup 2/ (N)) computation for a signal of N samples. The zero-crossings of a wavelet transform define a representation which is particularly well adapted for solving pattern recognition problems. As an example, the implementation and results of a coarse-to-fine stereo-matching algorithm are described.", } @Proceedings{man-hus:asymptotic, title = "Asymptotic Statistics: Proceedings of the fifth Prague Symposium", booktitle = "Asymptotic Statistics: Proceedings of the fifth Prague Symposium", editor = "Petr Mandl and Marie Huskova", publisher = "Physica-Verlag", address = "Heidelberg", series = "Contributions to Statistics", year = "1994", } @Article{man-lee:robust, title = "Robust Estimation of Background Noise and Signal Detection in Climatic Time Series", author = "Michael E. Mann and Jonathan M. Lees", journal = "Climate Change", volume = "33", year = "1996", pages = "409--445", URL = "http://love.geology.yale.edu/mtm/", abstract = "We present a new technique for isolating climate signals in time series with a characteristic 'red' noise background which arises from temporal persistence. This background is estimated by a 'robust' procedure that, unlike conventional techniques, is largely unbiased by the presence of signals immersed in the noise. Making use of multiple-taper spectral analysis methods, the technique further provides for a distinction between purely harmonic (periodic) signals, and broader-band ('quasiperiodic') signals. The effectiveness of our signal detection procedure is demonstrated with synthetic examples that simulate a variety of possible periodic and quasiperiodic signals immersed in red noise. We apply our methodology to historical climate and paleoclimate time series examples. Analysis of a approximate to 3 million year sediment core reveals significant periodic components at known astronomical forcing periodicities and a significant quasiperiodic 100 year peak. Analysis of a roughly 1500 year tree-ring reconstruction of Scandinavian summer temperatures suggests significant quasiperiodic signals on a near-century timescale, an interdecadal 16-18 year timescale, within the interannual El Ninio/Southem Oscillation (ENSO) band, and on a quasibiennial timescale. Analysis of the 144 year record of Great Salt Lake monthly volume change reveals a significant broad band of significant interdecadal variability, ENSO-timescale peaks, an annual cycle and its harmonics. Focusing in detail on the historical estimated global-average surface temperature record, we find a highly significant secular trend relative to the estimated red noise background, and weakly significant quasiperiodic signals within the ENSO band. Decadal and quasibiennial signals are marginally significant in this series.", } @Article{man-wal:stochastic, title = "On stochastic limit and order relationships", author = "H. B. Mann and A. Wald", journal = AofMS, volume = "14", year = "1943", pages = "217--226", } @Article{man:multifractal, title = "A Multifractal Walk Down Wall Street", author = "Benoit B. Mandelbrot", journal = "Scientific American", month = "February", year = "1999", pages = "50--53", } @Unpublished{mar-ada-joh-neu-pat:exact, title = "Exact Risk Analysis of Wavelet Regression", author = "S. J. Marron and S. Adak and Iain Johnstone and Michael H. Neumann and P. Patil", year = "1996", note = "To appear in {\em Journal of Computational and Graphical Statistics}", } @Article{mar-wil:indirect, title = "Indirect estimation of {ARFIMA} and {VARFIMA} models", author = "Martin, V. L. and Wilkins, N. P.", journal = JEcon, volume = "93", number = "1", year = "1999", pages = "149--175", abstract = "Indirect estimation methods are proposed for estimating ARFIMA, as well as more complex VARFIMA models. A general framework for conducting indirect estimation of fractional models is developed that covers simulation methods, choice of auxiliary model and estimation algorithm. Special attention is given to comparing the finite sampling properties of the indirect estimator with Sowell's (1992a) exact time domain maximum-likelihood estimator, the spectral maximum-likelihood estimator of Fox and Taqqu (1986) and the Geweke and Porter-Hudak (1983) spectral regression estimator. The indirect estimator can be computationally faster than the exact time domain maximum-likelihood estimator while generating similar small sample properties. The computational gains of the indirect estimator over maximum likelihood increase as the complexity of the data generating process increases." } @Article{mar:effective, title = "Effective writing in mathematical statistics", author = "J. S. Marron", journal = SN, volume = "53", number = "1", year = "1999", pages = "68--75", } @Article{mas:application, title = "The wavelet transform of stochastic processes with stationary increments and its application to fractional {B}rownian motion", author = "Elias Masry", journal = IEEETIT, volume = "39", number = "1", year = "1993", pages = "260--264", keywords = "wavelet transform, stochastic processes, fractional Brownian motion, random processes, wide-sense stationary increments, correlation function, spectral distribution, orthonormal series expansion, spectral analysis", abstract = "The wavelet transform of random processes with wide-sense stationary increments is shown to be a wide-sense stationary process whose correlation function and spectral distribution are determined. The second-order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained. Applications to the spectral analysis and to the synthesis of fractional Brownian motion are given.", } @Article{mas:convergence, title = "Convergence properties of wavelet series expansions of fractional {B}rownian motion", author = "Elias Masry", journal = ACHA, volume = "3", number = "3", year = "1996", pages = "239--253", keywords = "convergence properties, wavelet series expansions, Brownian motion, approximation error, finite intervals, convergence rates, signal processing", abstract = "We consider the approximation of a fractional Brownian motion by a wavelet series expansion at resolution $2^{-l}$. The approximation error is measured in the integrated mean squared sense over finite intervals and we obtain its expansion as a sum of terms with increasing rates of convergence. The dependence of the coefficients in the expansion of the error on the scale function is explicitly determined.", } @TechReport{mcc-per-wal:phase, title = "On the Phase of Least-Asymmetric Scaling and Wavelet Filters", author = "Emma J. McCoy and Donald B. Percival and Andrew T. Walden", institution = "Dept. of Mathematics, Imperial College of Science, Technology and Medicine", number = "TR-95-15", year = "1995", note = "Submitted to {\em IEEE Transactions on Signal Processing}", URL = "ftp://ftp.statsci.com/pub/WAVELETS/papers/wavephase.ps.gz", abstract = "The advance applied to Daubechies' least-asymmetric wavelet filters at each scale, in order to obtain near zero phase, is derived. The appropriate advance depends on whether half the length of each of the original quadrature mirror filters is even or odd. The departures from zero phase are illustrated.", } @Article{mcc-wal:long-memory, title = "Wavelet Analysis and Synthesis of Stationary Long-Memory Processes", author = "Emma J. McCoy and Andrew T. Walden", journal = JCGS, volume = "5", number = "1", year = "1996", pages = "26--56", abstract = "The discrete wavelet transform (DWT) can be interpreted as a filtering of a time series by a set of octave band filters such that the width of each band as a proportion of its center frequency is constant. A long-memory process having a power spectrum that plots as a straight line on log-frequency/log-power scales over many octaves of frequency is intrinsically related to such a structure. As an example of such processes, we focus on one class of discrete-time, stationary, long-memory processes, the fractionally differenced Gaussian white noise processes (fdGn). We show how the DWT breaks down a fdGn, and show the exact correlation structure of the resulting coefficients for different wavelets (Daubechies' minimum-phase and least-asymmetric and Haar). The DWT is an impressive ``whitening filter.'' A discrete wavelet-based scheme for simulating fdGn's is discussed and is shown to be equivalent to a spectral decomposition of the covariance matrix of the process; however, it can be carried out using only information on the nature of the spectrum of the process --- that is, time-domain information is not required. It produces results comparable with theexact Hosking method. We then show that, using wavelet methods, the spectral slope parameter $d$ can be estimated as well, or better, than when using the best Fourier-based method known to us, namely regression on multitaper spectral ordinates. Since wavelet analysis and synthesis methods can be applied to a much wider variety of empirical or theoretical long-memory processes, wavelet methods could prove a valuable tool in the future in the analysis and synthesis of stochastic processes.", } @PhdThesis{mcc:thesis, title = "Some New Statistical Approaches to the Analysis of Long Memory Processes", author = "Emma J. McCoy", school = "Imperial College, UK, Deptartment of Mathematics", year = "1994", postscript = "http://www.ma.ic.ac.uk/statistics/links/stafflinks/ejm.link/papers/report.ps.gz", } @TechReport{mcn-mau:applications, title = "Applications of wavelet analysis for determining glucose concentration of aqueous solutions using {NIR} spectroscopy", author = "Christopher S. McNulty and Ganapati Mauze", number = "HPL-98-53", institution = "Hewlett-Packard Laboratories", year = "1998", URL = "http://www.hpl.hp.com/techreports/98/HPL-98-53.ps", } @Article{mee-esc:algorithm, title = "An algorithm to compute the {CDF} of the product of two normal random variables", author = "William Q. Meeker and Luis A. Escobar", journal = CSA, volume = "23", number = "1", year = "1994", pages = "271--280", } @Article{men:analysis, title = "Analysis of turbulence in the orthonormal wavelet representation", author = "C. Meneveau", journal = JFM, volume = "232", year = "1991", pages = "469--520", keywords = "turbulence, orthonormal wavelet, turbulent velocity fields, wavelet-transformed Navier-Stokes equations, kinetic energy, one-dimensional velocity signals, boundary layer, turbulent wake, circular cylinder, three-dimensional velocity fields, multifractal scaling", abstract = "A decomposition of turbulent velocity fields into modes that exhibit both localization in wavenumber and physical space is performed. The author reviews some basic properties of such a decomposition, the wavelet transform. The wavelet-transformed Navier-Stokes equations are derived, and he defines new quantities such as e(r,x), t(r,x) and pi (r,x) which are the kinetic energy, the transfer of kinetic energy and the flux of kinetic energy through scale r at position x. The discrete version of e(r,x) is computed from laboratory one-dimensional velocity signals in a boundary layer and in a turbulent wake behind a circular cylinder. The author also computes (r,x), t(r,x) and pi (r,x) from three-dimensional velocity fields obtained from direct numerical simulations. His findings are that the localized kinetic energies become very intermittent in x at small scales and exhibit multifractal scaling. The transfer and flux of kinetic energy are found to fluctuate greatly in physical space for scales between the energy containing scale and the dissipative scale.", } @InProceedings{men:mixed, title = "Wavelet analysis of turbulence: {T}he mixed energy cascade", author = "C. Meneveau", pages = "251--264", crossref = "far-hun-vas:wavelets", keywords = "wavelets, turbulence", abstract = "The wavelet-transformed Navier-Stokes equations are used to define quantities such as the transfer of kinetic energy and the flux of kinetic energy by scale and position. Direct numerical simulations are performed which show large spatial variability at every scale and non-Gaussian statistics. The local energy flux exhibits large spatial intermittency and is often negative, indicating local inverse cascades.", } @Article{mey-sel-taq:wavelets, title = "Wavelets, generalized white noise and fractional integration: {T}he synthesis of fractional {B}rownian motion", author = "Meyer, Y. and Sellan, F. and Taqqu, M. S.", journal = JFAA, volume = "5", number = "5", year = "1999", pages = "465--494", abstract = "We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes Levy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve all independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.", } @Book{mey-roq:progress, title = "Progress in Wavelet Analysis and Applications", booktitle = "Progress in Wavelet Analysis and Applications", editor = "Yves Meyer and Sylvie Roques", publisher = "Editions Fronti{\`e}res", address = "Paris", year = "1993", pages = "785", } @Book{mey:algorithms, title = "Wavelets: Algorithms \& Applications", author = "Yves Meyer", publisher = SIAM, address = "Philadelphia", year = "1993", keywords = "Wavelets, Mathematics", note = "Translated and revised by Robert D. Ryan", loc = "QA403.3 .M4913 1993", } @Book{mey:wavelets, title = "Wavelets and Operators", author = "Yves Meyer", publisher = "Cambridge University Press", series = "Cambridge Studies in Advanced Mathematics 37", year = "1992", note = "Translated to English by D. H. Salinger", ISBN = "0-521-42000-8", keywords = "Fourier Series and Integrals, Filtering and Sampling / Multiresolution Approximation of L2(Rn) / Orthonormal Wavelet Bases / Non-orthogonal Wavelets / Wavelets the Hardy Space H1 and Its Dual BMO / Wavelets and Spaces of Functions and Distributions", URL = "http://www.cup.org/Titles/42/0521420008.html", abstract = "The first book in English to provide a comprehensive account of the mathematical theory of wavelets which has proved to be a powerful tool for harmonic analysts, and an alternative to the standard theory of Fourier analysis", } @Article{mih-etal:low-complexity, title = "Low-complexity image denoising based on statistical modeling of wavelet coefficients", author = "Mihcak, M. K. and Kozintsev, I. and Ramchandran, K. and Moulin, P.", journal = IEEESPL, volume = "6", number = "12", year = "1999", pages = "300--303", abstract = "We introduce a simple spatially adaptive statistical model for wavelet image coefficients and apply it to image denoising, Our model is inspired by a recent wavelet image compression algorithm, the estimation-quantization (EQ) coder. We model wavelet image coefficients as zero-mean Gaussian random variables with high local correlation. We assume a marginal prior distribution on wavelet coefficients variances and estimate them using an approximate maximum a posteriori probability rule. Then we apply an approximate minimum mean squared error estimation procedure to restore the noisy wavelet image coefficients. Despite the simplicity of our method, both in its concept and implementation, our denoising results are among the best reported in the literature." } @Article{mor-per:mbdtw, title = "Minimum-bandwidth discrete-time wavelets", author = "Morris, J. M. and Peravali, R.", journal = SP, volume = "76", number = "2", year = "1999", pages = "181--193", keywords = "minimum bandwidth wavelet transform discrete-time wavelets adaptive simulated annealing", abstract = "In this paper we present a class of minimum-bandwidth, discrete-time orthonormal wavelets (MBDTWs), The wavelets were generated via the filter bank framework and were optimized using the global optimization technique, adaptive simulated annealing (ASA). The objective function is the average normalized bandwidth of the wavelets over all scales as obtained from the filter bank structure. We tabulate the wavelet-defining low-pass filter coefficients {g(n)} for filter lengths of N = 4, 8, 10, 12, 14, 16, 18, 24 and 32 and for L = 2, 3 and 4. We provide comparisons with Daubeohies' discrete wavelets and other classes of optimum wavelets. Finally, we present examples that demonstrate the advantage of our MBDTWs for certain narrowband applications: de-noising of an ECG signal, and compression of an ECG signal and a bird call signal. We compare the performance of our wavelets in these examples with that of Daubechies' least-asymmetric wavelets which are closest to the MBDTWs with respect to our bandwidth measure.", } @InProceedings{mor:from, title = "From Fourier to Wavelet Analysis of Time Series", author = "Pedro A. Morettin", booktitle = "Proceedings in Computational Statistics", editor = "A. Prat", year = "1996", pages = "111--122", } @Article{mou:power, title = "Wavelet Thresholding Techniques for Power Spectrum Estimation", author = "Pierre Moulin", journal = IEEETSP, volume = "42", number = "11", year = "1994", pages = "3126--3136", URL = "ftp://flash.bellcore.com/pub/moulin/wavSE.tar.Z", abstract = "Estimation of the power spectrum S(f) of a stationary random process can be viewed as a nonparametric statistical estimation problem. We introduce a nonparametric approach based on a wavelet representation for the logarithm of the unknown S(f). This approach offers the ability to capture statistically significant components of ln S(f) at different resolution levels and guarantees nonnegativity of the spectrum estimator. The spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive non-Gaussian noise. We propose a wavelet thresholding technique to solve this problem under specified noise/resolution tradeoffs and show that the wavelet coefficients of the additive noise may be treated as independent random variables. The thresholds are computed using a saddle-point approximation to the distribution of the noise coefficients.", } @TechReport{mul-vid:density-estimation, title = "Bayesian Inference with Wavelets: {D}ensity Estimation", author = "Peter M{\"u}ller and Brani Vidakovic", year = "1995", number = "95-34", institution = "Institute of Statisics and Decision Sciences, Duke University", URL = "http://www.isds.duke.edu/~brani/papers/ModMixWav.ps", } @TechReport{mul-vid:MCMC, title = "{MCMC} Methods in Wavelet Shrinkage: {N}on-Equally Spaced Regression, Density and Spectral Density Estimation", author = "Peter M{\"u}ller and Brani Vidakovic", year = "1999", number = "99-01", institution = "Institute of Statisics and Decision Sciences, Duke University", URL = "ftp://ftp.isds.duke.edu/pub/WorkingPapers/99-01.ps", } @Unpublished{mur-aus:using, title = "Using the Wavelet Transform for Multivariate Data Analysis and Time Series Forecasting", author = "Fionn Murtagh and Alex Aussem", year = "1996", note = "Proc. IFCS'96, Kobe, Springer-Verlag, accepted (subject to minor revision)", URL = "ftp://ftp.infm.ulst.ac.uk/pub/Papers/neural-nets/kobe.ps", } @Unpublished{mur:wedding, title = "Wedding the Wavelet Transform and Multivariate Data Analysis", author = "Fionn Murtagh", year = "1996", note = "To appear {\em Journal of Classification}", URL = "ftp://ftp.infm.ulst.ac.uk/pub/Papers/neural-nets/wt-mda-new.ps", } @Book{mye:book, title = "Classical and Modern Regression with Applications", author = "Raymond H. Myers", edition = "2", publisher = "PWS--KENT", address = "Boston", series = "The Duxbury Advanced Series in Statistics and Decision Sciences", year = "1990", } @TechReport{nas-sap-saw:modelling, title = "Statistical modelling of time series using non-decimated wavelet representations", author = "G. P. Nason and T. Sapatinas and A. Sawczenko", institution = "Department of Mathematics, University of Bristol, Bristol", year = "1997", URL = "http://www.stats.bris.ac.uk/pub/reports/Wavelets/smotsunwr.ps.gz", } @InProceedings{nas-sil:stationary, title = "The Stationary Wavelet Transform and some Statistical Applications", author = "Guy P. Nason and Bernard W. Silverman", pages = "281--300", URL = "http://www.stats.bris.ac.uk:81/pub/reports/Silverman/swtsa.ps.gz", crossref = "ant-opp:wavelets", abstract = "", } @Article{nas-sil:transform, title = "The Discrete Wavelet Transform in {S}", author = "Guy P. Nason and Bernard W. Silverman", journal = JCGS, volume = "3", number = "2", pages = "163--191", year = "1994", URL = "http://www.stats.bris.ac.uk:81/pub/reports/Wavelets/tdwtis.ps.gz", keywords = "Curve estimation; Image compression; Nonparametric regression; Smoothing; Software; Statistical algorithms; Thresholding", abstract = "The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called {\tt wavethresh} that works within the statistical language S to perform one- and two-dimensional discrete wavelet transforms. The transforms and their inverses can be computed using any particular wavelet selected from a range of different families of wavelets. Pictures can be drawn of any of the one- or two-dimensional wavelets available in the package. The wavelet coefficients can be presented in a variety of ways to aid in the interpretation of data. The package's wavelet transform ``engine'' is written in C for speed and the object-oriented functionality of S makes {\tt wavethresh} easy to use. We provide a tutorial introduction to wavelets and the {\tt wavethresh} software. We also discuss how the software may be used to carry out nonlinear regression and image compression. In particular, thresholding of wavelet coefficients is a method for attempting to extract signal from noise and {\tt wavethresh} includes functions to perform thresholding according to methods in the literature.", } @InCollection{nas-sil:wavelets, title = "Wavelets for regression and other statistical problems", author = "Guy P. Nason and Bernard W. Silverman", booktitle = "Smoothing and Regression: Approaches, Computation and Application", editor = "M. G. Schimek", publisher = "Wiley", year = "1997", } @Article{nas-von:witsa, title = "Wavelets in time series analysis", author = "Guy P. Nason and Rainer {von Sachs}", journal = PTRSLA, volume = "357", number = "1760", year = "1999", pages = "2511--2526", } @Unpublished{nas-von-kro:evolutionary, title = "Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum", author = "Guy P. Nason and Rainer {von Sachs} and Gerald Kroisandt", year = "2000", URL = "http://playfair.Stanford.EDU/reports/rvs/NvSK.ps.Z", note = "{\em Journal of the Royal Statistical Society Series B}, to appear", } @InProceedings{nas:choice, title = "Choice of the threshold parameter in wavelet function estimation", author = "Guy P. Nason", pages = "261--280", crossref = "ant-opp:wavelets", URL = "", note = "", abstract = "", } @TechReport{nas:regression, title = "Wavelet regression by cross-validation", author = "Guy P. Nason", year = "1994", institution = "Deptartment of Mathematics, University of Bristol", URL = "http://www.stats.bris.ac.uk:81/pub/reports/Wavelets/wvcx.ps.gz", pages = "45", keywords = "wavelets", abstract = "This paper is about using wavelets for regression. The main aim is to introduce and develop a cross-validation method for selecting a wavelet regression threshold that produces good estimates with respect to $L_2$ error. The selected threshold determines which coefficients to keep in an orthogonal wavelet expansion of noisy data and acts in a similar way to a smoothing parameter in non-parametric regression.", } @Article{nas:shrinkage, title = "Wavelet shrinkage by cross-validation", author = "Guy P. Nason", journal = JRSSB, volume = "58", year = "1996", pages = "463--479", URL = "http://www.stats.bris.ac.uk:81/pub/reports/Wavelets/wvcxPAPER.ps.gz", abstract = "Wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics. This paper concentrates on the estimation of functions and images from noisy data by using wavelet shrinkage. A modified form of twofold cross-validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of 2. The cross-validation algorithm is then extended to data sets of any length and to multidimensional data sets.The algorithms are compared with established threshold choosers by using simulation. An application to a real data set arising from anaesthesia is presented.", keywords = "ADAPTIVE ESTIMATION ANAESTHETICS NONPARAMETRIC REGRESSION SMOOTHING PARAMETER SPATIAL ADAPTATION THRESHOLD", } @Article{nas:wavelets, title = "Wavelets", author = "Guy P. Nason", journal = "New Electronics", month = apr, year = "1997", URL = "http://www.stats.bris.ac.uk:81/pub/reports/Wavelets/ne.ps.gz", } @Unpublished{nav-bro:dependence, title = "Dependence Structure of Random Wavelets Coefficients in function of higher Cumulants for Non-Gaussian and Non-Linear processes", author = "Philippe Naveau and Peter Brockwell", year = "1999", note = "Geophysical Statistics Project, National Center for Atmospheric Research", } @Unpublished{nav-etal:exploratory, title = "Exploratory statistical analysis of tropical oceanic convection using discrete wavelet transforms", author = "Philippe Naveau and Mitchell Moncrieff and Jun-Ichi Yano and Xiaoqing Wu", year = "1999", note = "Submitted in the book ``Case Studies in Statistics and the Atmospheric Sciences''", } @Article{nen-rid-zei:infrared, title = "Wavelet analysis of low altitude infrared transmission in the coastal environment", author = "Nener, B. D. and Ridsdill-Smith, T. A. and Zeisse, C.", journal = "Infrared Physics and Technology", volume = "40", number = "5", year = "1999", pages = "399--409", abstract = "Analysis of medium and long wavelength infrared light transmission measurements collected during an intensive experimental investigation over San Diego Bay, CA, USA, has been performed using Morlet wavelets. Wavelet analysis shows locally periodic trends in signals and shows how these trends evolve in time. The analysis has shown the effect of the relative humidity and the windspeed on the transmission in both mid- and long-wave bands of the LR. The effect of air mass parameter does not appear to be strongly correlated. The wavelet representations of the data given in the figures provide a useful overview of the effects of the atmosphere on infrared transmission.", } @Article{neu-von:anisotropic, title = "Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra", author = "Michael H. Neumann and Ranier von Sachs", journal = AofS, volume = "25", number = "1", year = "1997", pages = "38--76", abstract = "", } @InCollection{neu-von:beyond, title = "Wavelet thresholding: {B}eyond the {G}aussian {I.I.D} situation", author = "Michael H. Neumann and Ranier von Sachs", pages = "301--329", crossref = "ant-opp:wavelets", abstract = "", } @TechReport{neu:spectral94, title = "Spectral Density Estimation via Nonlinear Wavelet Methods for Stationary Non-Gaussian Time Series", author = "Michael H. Neumann", year = "1994", institution = "Statistics Research Report SRR 028-94, CMA, Australian National University, Canberra", URL = "http://wilton.anu.edu.au/research.reports/94srr.html", } @Article{neu:spectral96, title = "Spectral Density Estimation via Nonlinear Wavelet Methods for Stationary Non-Gaussian Time Series", author = "Michael H. Neumann", journal = JTSA, volume = "17", number = "6", year = "1996", pages = "601--633", abstract = "In the present paper we consider nonlinear wavelet estimators of the spectral density f of a zero mean, not necessarily Gaussian, stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of empirical coefficients that arise from an orthonormal series expansion according to a wavelet basis. The main goal of this paper is to transfer these methods to spectral density estimation. This is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram.Using these results we can show the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. The resulting estimator of f keeps all interesting properties such as high spatial adaptivity that are already known for wavelet estimators in the case of Gaussian regression. It turns out that appropriately tuned versions of this estimator attain the optimal uniform rate of convergence of their L 2 risk in a wide variety of Besov smoothness classes, including classes where linear estimators (kernel, spline) are not able to attain this rate. Some simulations indicate the usefulness of the new method in cases of high spatial inhomogeneity.", keywords = "Spectral density estimation wavelet estimators nonlinear wavelet shrinkage asymptotic normality large deviations", } @Article{new:analysis1, title = "{W}avelet analysis of vibration, {P}art {I}: theory", journal = "Transactions of the ASME. Journal of Vibration and Acoustics", volume = "116", number = "4", year = "1994", pages = "409--416", author = "D. E. Newland", abstract = "Wavelets provide a new tool for the analysis of vibration records. They allow the changing spectral composition of a nonstationary signal to be measured and presented in the form of a time-frequency map. The purpose of this paper, which is Part I of a pair, is to introduce and review the theory of orthogonal wavelets and their application to signal analysis. It includes the theory of dilation wavelets, which have been developed over a period of about ten years, and of harmonic wavelets which have been proposed recently by the author. Part II is about presenting the results on wavelet maps and gives a selection of examples. The papers will interest those who work in the field of vibration measurement and analysis and who are in positions where it is necessary to understand and interpret vibration data.", keywords = "wavelet analysis. vibration. changing spectral composition. nonstationary signal. time-frequency map. orthogonal wavelets. signal analysis. dilation wavelets. harmonic wavelets. wavelet maps. vibration measurement.", } @Article{new:analysis2, title = "{W}avelet analysis of vibration, {P}art 2: wavelet maps", journal = "Transactions of the ASME. Journal of Vibration and Acoustics", volume = "116", number = "4", year = "1994", pages = "417--25", author = "D. E. Newland", abstract = "For pt. 1, see ibid., vol. 116, p. 409-16, (1994). Wavelet maps provide a graphical picture of the frequency composition of a vibration signal. This paper, which is Part 2 of a pair, describes their construction and properties. In the case of harmonic wavelets, there are close similarities between wavelet maps and sonograms. A range of practical examples illustrate how the wavelet method may be applied to vibration analysis and some of its advantages.", keywords = "wavelet maps. graphical picture. frequency composition. vibration signal. harmonic wavelets. sonograms. vibration analysis.", } @Book{new:book3, title = "An Introduction to Random Vibrations, Spectral \& Wavelet Analysis", author = "David Edward Newland", edition = "3", publisher = "Longman Scientific \& Technical", address = NY, year = "1993", pages = "477", ISBN = "0-470-22153-4", keywords = "Random vibration, Spectral theory, Mathematics, Data processing, Wavelets", } @Article{new:harmonic, title = "{H}armonic wavelet analysis", journal = PRSLA, volume = "443", number = "1917", year = "1993", pages = "203--225", author = "D. E. Newland", keywords = "harmonic wavelet. frequency spectrum. octave band. fast Fourier transform. Fourier coefficients. FFT. inverse transform.", abstract = "A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x)=(exp(i4 pi x)-exp(i2 pi x))/i2 pi x. This function w(x) is concentrated locally around x=0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.", } @Article{new:properties, title = "Some properties of discrete wavelet maps", author = "D. E. Newland", journal = PEM, volume = "9", number = "1", year = "1994", pages = "59--69", } @Article{nin:estimation, title = "Estimation of 1/f noise", author = "B. Ninness", journal = IEEETIT, volume = "44", number = "1", pages = "32--46", year = "1998", keywords = "flicker noise fractional Brownian motion Hurst exponent maximum-likelihood estimation 1/f noise wavelet analysis", abstract = "Several models have emerged for describing 1/f(gamma) noise processes. Based on these, various techniques for estimating the properties of such processes have been developed. This paper provides theoretical analysis of a new wavelet-based approach which has the advantages of having low computational complexity and being able to handle the case where the 1/f(gamma) noise might be embedded in a further white-noise process. However, the analysis conducted here shows that these advantages are balanced by the fact that the wavelet-based scheme is only consistent for spectral exponents gamma in the range gamma is an element of (0, 1). This is in contradiction to the results suggested in previous empirical studies. When gamma is an element of (0, 1) this paper also establishes that wavelet-based maximum-likelihood methods are asymptotically Gaussian and efficient. Finally, the asymptotic rate of mean-square convergence of the parameter estimates Is established and is shown to slow as gamma approaches one. Combined with a survey of non-wavelet-based methods, these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1/f(gamma) noise processes", } @Unpublished{now:shift-invariant, title = "Shift invariant wavelet-based statistical models and $1/f$ processes", author = "R. D. Nowak", note = "IEEE DSP Workshop, Bryce Canyon, UT", year = "1998", } @Unpublished{ode-bur:class, title = "New Class of Wavelets for Signal Approximation", author = "Jan E. Odegard and C. Signey Burrus", note = "Department of Electrical and Computer Engineering, Rice University", year = "1996", email = "odegard@rice.edu, csb@rice.edu", } @InProceedings{ogd-che:abrupt, title = "Testing for Abrupt Jumps with Wavelets", author = "R. Todd Ogden and Cheng Cheng", booktitle = "Proceedings of the 1997 Conference on the Interface of Statistics and Computer Science", pages = "", year = "1997", } @Article{ogd-hil:2D, title = "Data Analytic Wavelet Threshold Selection in 2-{D} Signal Denoising", author = "R. Todd Ogden and M. Hilton", journal = IEEETSP, volume = "45", number = "2", year = "1997", pages = "496--500", URL = "http://www.stat.sc.edu/~ogden/papers/technote.ps", abstract = "A data adaptive scheme for wavelet shrinkage-based noise removal is developed. The method involves a statistical test of hypothesis that takes into account the wavelet coefficients' magnitudes and relative positions. The amount of smoothing performed during noise removal is controlled by the user-supplied confidence level of the tests.", } @InBook{ogd-lyn:bayesian, title = "Bayesian analysis of change-point models", authors = "R. Todd Ogden and James D. Lynch", crossref = "mul-vid:biwbm", pages = "???--???", URL = "http://www.stat.sc.edu/~ogden/papers/springer4.ps", } @Article{ogd-par:change-point, title = "Change-point Approach to Data Analytic Wavelet Thresholding", author = "R. Todd Ogden and Emanuel Parzen", journal = SC, volume = "6", number = "2", year = "1996", pages = "93--99", URL = "http://www.stat.sc.edu/~ogden/papers/datana.ps", } @Article{ogd-par:data-dependent, title = "Data Dependent Wavelet Thresholding in Nonparametric Regression with Change-point Applications", author = "R. Todd Ogden and Emanuel Parzen", journal = CSDA, volume = "22", year = "1996", pages = "53--70", URL = "http://www.stat.sc.edu/~ogden/papers/datdep.ps", } @Unpublished{ogd:bayesian, title = "Wavelets in {B}ayesian Change-Point Analysis", author = "R. Todd Ogden", note = "Department of Statistics, University of South Carolina", year = "1996", URL = "http://www.stat.sc.edu/~ogden/papers/bayes.ps", } @Book{ogd:book, title = "Essential Wavelets for Statistical Applications and Data Analysis", author = "R. Todd Ogden", year = "1996", publisher = "Birkhauser", address = "Boston", URL = "http://www.birkhauser.com/cgi-win/ISBN/0-8176-3864-4", abstract = "Exciting new developments in wavelet theory have attracted much attention and sparked new research in many fields of applied mathematics. New tools are available for efficient data compression, image analysis, and signal processing, and there is a great deal of activity in developing new wavelet methods. The same features that make wavelets useful in these fields also make wavelets a natural and attractive choice in many areas of statistical data analysis. {\em Essential Wavelets for Statistical Applications and Data Analysis} presents an accesible, introductory survey for new wavelet analysis tools and how they can be applied to fundamental data analysis problems. A variety of problems in statistics are discussed in a non-theoretical style, with an emphasis on understanding of wavelet methods. The only technical prerequisite is basic knowledge of undergraduate calculus, linear algebra, and basic statistical theory.", } @Article{ogd:preconditioning, title = "On Preconditioning the Data for the Wavelet Transform When the Sample Size is Not a Power of Two", author = "R. Todd Ogden", journal = CSB, volume = "26", year = "1997", pages = "267--285", URL = "http://www.stat.sc.edu/~ogden/papers/notpow2.ps", } @PhdThesis{ogd:thesis, title = "Wavelet Thresholding in Nonparametric Regression with Change-Point Applications", author = "R. Todd Ogden", school = "Texas A\&M University", year = "1994", postscript = "http://www.stat.sc.edu/~ogden/diss.ps", } @Article{ohs-iji-kur:online, title = "Online detection of pulse sequence in random noise using a wavelet", author = "Ohsumi, A and Ijima, H. and Kuroishi, T", journal = IEEETSP, volume = "47", number = "9", pages = "2526--2531", year = "1999", abstract = "Two types of wavelet-based algorithms are proposed for an online detection of a train of unknown pulse signals corrupted by random noise. The mechanism of detecting singularities hidden in the noisy observation data is analyzed, and the performance of the proposed signal detectors is evaluated. Simulation studies are provided to confirm the effectiveness of the algorithms." } @Unpublished{omb-etal:automatic, title = "Automatic Statistical Analysis of Bivariate Non-Stationary Time Series", author = "Hernando C. Ombao and Jonathan A. Raz and Rainer {von Sachs} and Beth A. Malow", year = "1999", note = "Submitted to {\em JASA}", } @Article{oon:detecting, title = "A Wavelet Method for Detecting S-Waves in Seismic Data", author = "P. J. Oonicnx", journal = "Computational Geosciences", volume = "3", pages = "111--134", year = "1999", } @Article{pan-fan:wavelet-spectrum, title = "Discrete wavelet transform power spectrum estimator", author = "Jes{\'u}s Pando and Li-Zhi Fang", journal = PRE, volume = "57", number = "3", year = "1998", pages = "3593--3601", URL = "http://ojps.aip.org/PLEEE8-bt/html/57_3.html", keywords = "nucleus-nucleus collisions high-energies distributions fluctuations turbulence", abstract = "A method for measuring the spectrum of a density field by the discrete wavelet transform (DWT) is studied. We show how the Fourier power spectrum can be detected by using the wavelet function coefficients (WFC) of the DWT. This method can successfully measure the power spectrum in samples for which traditional methods often fail because the samples are finite sized, have a complex geometry, or are varyingly sampled. We demonstrate that the spectrum features, such as the power law index, the magnitude, and the typical scales can be determined by the DWT reconstructed spectrum. We apply this method to analyze the power spectrum of the spatial distribution of the Ly-alpha clouds. The two popular data sets used for the spectrum detection have quite different geometries and samplings, yet the one-dimensional (1D) power spectra and their 3D reconstruction given by the DWT estimator show the same features. The analysis makes clear that the DWT estimator is a sensitive tool in revealing common and physical properties from diverse data sets.", } @Article{pan-muk:tidal, title = "Wavelet analysis on transient behaviour of tidal amplitude fluctuations observed by meteor radar in the lower thermosphere above Bulgaria", author = "Pancheva, D. and Mukhtarov, P.", journal = AG, volume = "18", number = "3", year = "2000", pages = "316--331", keywords = "Meteorology and atmospheric dynamics (middle atmosphere dynamics; waves and tides) - Radio science (nonlinear phenomena)" abstract = "On the basis of bispectral analysis applied to the hourly data set of neutral wind measured by meteor radar: in the MLT region above Bulgaria it was demonstrated that nonlinear processes are frequently and regularly acting in the mesopause region. They contribute significantly to the short-term tidal variability and are apparently responsible for the observed complicated behavior of the tidal characteristics. A Morlet wavelet transform is proposed as a technique for studying nonstationary signals. By simulated data it. was revealed that the Morlet wavelet transform is especially convenient for analyzing signals with: (I) a wide range of dominant frequencies which are-localized in different time intervals; (2) amplitude and frequency modulated spectral components, and (3) singular, wavelike events, observed in the neutral wind of the MLT region and connected mainly with large-scale disturbances propagated from below. By applying a Morlet wavelet transform to the hourly values of the amplitudes of diurnal and semidiurnal tides the basic oscillations with periods of planetary waves (1.5-20 days), as well as their development in time, are obtained. A cross-wavelet analysis is used to clarify the relation between the tidal and mean neutral wind variability. The results of bispectral analysis indicate: which planetary waves participated in the nonlinear coupling with the atmospheric tides, while the results of cross-wavelet analysis outline their time intervals if these interactions are local.", } @Article{pan-wan:stochastic, title = "A Stochastic Nonlinear Regression Estimator Using Wavelets", author = "Pan, Zuohong and Wang, Xiaodi", journal = CE, volume = "11", number = "1-2", year = "1998", pages = "89--102", keywords = "capital markets empirical studies including regulation", abstract = "A new wavelet-based estimator is introduced which combines the state-space model with the wavelet transform in an effort to explore the stock market inefficiency. The new estimator possesses some superior qualities that are illustrated through its actual performance in forecasting the S&P 500.", } @Article{pap-sik-wei:characterization, title = "The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts", author = "Papadakis, M. and \v{S}iki\'{c}, H. and Weiss, G.", journal = JFAA, volume = "5", number = "5", year = "1999", pages = "495--521", abstract = "The ``classical'' wavelets, those psi is an element of L-2(R) such that {2(j/2)psi(2(j)x - k)}, j, k is an element of Z, is an orthonormal basis for L-2(R), are known to be characterized by two simple equations satisfied by <(psi)over cap>. The ``multiresolution analysis'' wavelets (briefly, the MRA wavelets) have a simple characterization and so do the scaling functions that produce these wavelets. Only certain smooth classes of the low pass filters that are determined by these scaling functions, however, appear to be characterized in the literature (see Chapter 7 of [3] for an account of these matters). In this paper we present a complete characterization of all these filters. This somewhat technical result does provide a method for simple constructions of low pass filters whose only smoothness assumption is a Holder condition at the origin. We also obtain a characterization of all scaling sets and, in particular a description of all bounded scaling sets as well as a detailed description of the class of scaling functions.", } @InProceedings{pap-sol-was:segmentation, title = "Segmentation-independent estimates of turbulence parameters", author = "G. C. Papanicolaou and Knut S{\o}lna and Donald C. Washburn", booktitle = "Airborne Laser Advanced Technology", editor = "T. D. Steiner and P. H. Merritt", series = "Proceedings of the SPIE", volume = "3381", pages = "256--267", year = "1998", URL = "ftp://math.Stanford.EDU/pub/papers/papanicolaou/maint.ps.gz", abstract = "We present a new approach for analyzing local power law processes and apply it to temperature measurements from the upper atmosphere. We segment the data and use the wavelet scale spectrum to estimate the parameters of the power law, the scale factor and the exponent. These parameters vary from segment to segment. Part of this variation is due to the non-stationary of the data. Another part is due to estimation errors that depend on the segmentation. In this paper show how to remove effectively these segmentation dependent variations.", } @Unpublished{pap-sol:local, title = "Wavelet based estimation of local {K}olmogorov turbulence", author = "G. Papanicolaou and K. S{\o}lna", year = "1999", URL = "ftp://math.Stanford.EDU/pub/papers/papanicolaou/main.ps.gz", note = "Submitted to the {\em Journal of the American Statistical Association}", } @Article{par-man:interannual, title = "Interannual Temperature Events and Shifts in Global Temperature: A ``Multiwavelet'' Correlation Approach", author = "Jeffrey Park and Michael E. Mann", journal = EI, volume = "4", year = "2000", pages = "???--???", } @InProceedings{pat-sim:texture, title = "Texture modelling and synthesis using joint statistics of complex wavelet coefficients", author = "J. Portilla and E. P. Simoncelli", booktitle = "IEEE Workshop on Statistical and Computational Theories of Vision", year = "1999", pages = "", URL = "ftp://ftp.cns.nyu.edu/pub/eero/portilla99a.ps.gz", abstract = "We present a statistical characterization of texture images in the context of an overcomplete complex wavelet transform. The characterization is based on empirical observations of statistical regularities in such images, and parameterized by (1) the local auto-correlation of the coefficients in each subband; (2) both the local auto-correlation and cross-correlation of coefficient magnitudes at other orientations and spatial scales; and (3) the first few moments of the image pixel histogram. We develop an efficient algorithm for synthesizing random images subject to these constraints using alternated projections, and demonstrate its effectiveness on a wide range of synthetic and natural textures. In particular, we show that many important structural elements in textures (e.g., edges, repeated patterns or alternated patches of simpler texture), can be captured through joint second order statistics of the coefficient magnitudes. We also show the flexibility of the representation, by applying to a variety of tasks which can be viewed as constrained image synthesis problems, such as spatial and spectral extrapolation.", } @TechReport{pen-vid:non-equally, title = "On non-equally spaced wavelet regression", author = "Marianna Pensky and Brani Vidakovic", number = "98-06", institution = "Institute of Statistics and Decision Sciences, Duke University", year = "1998", URL = "ftp://ftp.isds.duke.edu/pub/WorkingPapers/98-06.ps", abstract = "Wavelet-based regression analysis is widely used mostly for equally-spaced designs. For such designs wavelets are superior to other traditional orthonormal bases because of their versatility and ability to parsimoniously describe irregular functions. If the regression design is random, an automatic solution is not available. Given the observations (X_i, Y_i), i = 1,..., n, we estimate the regression function m(x)=E(Y|X=x) as a series \sum_k \hat c_{jk} \phi_{jk}(x) where \{ \phi_{jk}(x), ~k \in Z \} are scaling functions spanning the multiresolution subspace V_j. We propose a method that utilizes a probabilistic model on X_i's in defining the empirical coefficients \hat c_{jk}. The paper deals with both theoretical and practical aspects of the proposed estimator. We explore MSE convergence rates of the estimator. The performance of the estimator is compared to that of some traditional regression methods.", } @TechReport{per-bru:approximate, title = "Wavelet-Based Approximate Maximum Likelihood Estimation for Trend-Contaminated Fractional Difference Processes", author = "Donald B. Percival and Andrew G. Bruce", number = "67", institution = "MathSoft, Inc., 1700 Westlake Avenue N., Seattle, WA 98109-9891", year = "1998", URL = "ftp://ftp.statsci.com/pub/longmem/wavelet-mle.ps", } @TechReport{per-bru:estimation, title = "Estimation of Long Memory Processes with Missing Data", author = "Donald B. Percival and Andrew G. Bruce", number = "64", institution = "MathSoft, Inc., 1700 Westlake Avenue N., Seattle, WA 98109-9891", year = "1997", URL = "ftp://ftp.statsci.com/pub/longmem/missing.ps", } @Unpublished{per-sar-dav:wavestrapping, title = "Wavestrapping Time Series: {A}daptive Wavelet-Based Bootstrapping", author = "Donald B. Percival and Sylvain Sardy and Anthony Davision", year = "1999", note = "Isaac Newton Institute for Mathematical Sciences", } @InCollection{per-gut:introduction, title = "An Introduction to Spectral Analysis and Wavelets", author = "Donald B. Percival and Peter Guttorp", pages = "175--186", crossref = "cia-cox-mon-pav:advanced", URL = "", abstract = "", } @InCollection{per-gut:long-memory, title = "Long-Memory Processes, the {A}llan Variance and Wavelets", author = "Donald B. Percival and Peter Guttorp", pages = "325--344", crossref = "fou-kum:geophysics", URL = "", abstract = "", } @Article{per-mof:subtidal, title = "Analysis of Subtidal Coastal Sea Level Fluctuations Using Wavelets", author = "Donald B. Percival and Harold O. Mofjeld", journal = JASA, volume = "92", number = "439", year = "1997", pages = "868--880", keywords = "coastal sea level variability, discrete wavelet transform, natural hazards, time series analysis, tsunamis", abstract = "Subtidal coastal sea level fluctuations affect coastal ecosystems and the consequences of destructive events such as tsunamis. We analyze a time series of subtidal fluctuations at Crescent City, California, during 1980-1991 using the maximal overlap discrete wavelet transform (MODWT). Our analysis shows that the variability in these fluctuations depends on the season for scales of 32 days and less. We show how the MODWT characterizes nonstationary behavior succinctly and how this characterization can be used to improve forecasts of inundation during tsunamis and storm surges. Pie provide pseudocode and enough details so that data analysts in other disciplines can readily apply MODWT analysis to other nonstationary time series.", } @Article{per-phi-bas:compared, title = "Wavelet spectra compared to {F}ourier spectra", author = "Val\'{e}rie Perrier and Thierry Philipovitch and Claude Basdevant", journal = "Journal of Mathematical Physics", volume = "36", number = "3", year = "1995", pages = "1506-1519", abstract = "The relation between Fourier spectra and spectra obtained from wavelet analysis is established. Small scale asymptotic analysis shows that the wavelet spectrum is meaningful only when the analyzing wavelet has enough vanishing moments. These results are related to regularity theorems in Besov spaces. For the analysis of infinitely regular signals, a new wavelet, with an infinite number of cancellations is proposed.", } @Book{per-wal:wmtsa, title = "Wavelet Methods for Time Series Analysis", author = "Donald B. Percival and Andrew T. Walden", year = "2000", publisher = "Cambridge University Press", address = "Cambridge", ISBN = "", keywords = "", URL = "http://weber.u.washington.edu/~dbp/", abstract = "", note = "Forthcoming", } @Article{per:characterization, title = "{C}haracterization of frequency stability: frequency-domain estimation of stability measures", journal = PIEEE, volume = "79", number = "7", year = "1991", pages = "961--972", author = "Donald B. Percival", abstract = "The author focuses on the frequency domain approach, which provides a complete characterization of frequency. The standard characterization of frequency stability in the frequency domain is the spectral density function (SDF). The author describes SDFs that model sampled frequency stability data and that are related to the SDFs of the standard characterization. On the basis of standard techniques in spectral analysis, he outlines a systematic way of estimating SDFs typical of frequency stability data. The recommended procedure is to check for broadband bias in the periodogram using a sequence of data tapers and, if bias is in evidence, to design an autoregressive prewhitening filter to prewhiten the data. The author considers the relationship between the Allan variance and the SDF and outlines two nonparametric ways of translating stability measures between the two domains-one based upon pilot analysis and the other upon J. Rutman's bandpass variance (1978).", keywords = "frequency stability. frequency domain. spectral density function. spectral analysis. broadband bias. periodogram. sequence of data tapers. autoregressive prewhitening filter. Allan variance. pilot analysis. bandpass variance.", } @Article{per:variance, title = "On estimation of the wavelet variance", author = "Donald B. Percival", journal = BKA, volume = "82", number = "3", year = "1995", pages = "619--631", URL = "ftp://ftp.statsci.com/pub/WAVELETS/papers/wavevar.ps.gz", abstract = "Thw wavelet variance decomposes the variance of a time series into components associated with differen scales. We consider two estimators of the wavelet variance: the first based upon the discrete wavelet transform, and the second, called the maximal-overlap estimator, based upon a filtering interpretation of wavelets. We determine the large sample distribution for both estimatorsand show that the maximal-overlap estimator ismore efficient for a class of processes of interest in the physical sciences. We discuss methods for determining an approximate confidence interval for the wavelet variance. We demonstrate through Monte Carlo experiments that the large sample distribution for the maximal-overlap estimator is a reasonable approximation even for the moderate sample size of 128 observations. We apply our proposed methodology to a series of observations related to vertical shear in the ocean.", } @Article{pes-kri-car:time-invariant, title = "Time-invariant orthonormal wavelet representations", author = "Jean-Christophe Pesquet and Hamid Krim and Herv{\'e} Carfantan", journal = IEEETSP, volume = "44", number = "8", year = "1996", pages = "1964--1970", abstract = "A simple construction of an orthonormal basis starting with a so-called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptanceand generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. We address the time-invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it,is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation.", } @InProceedings{pes-kri-lep-ham:bayesian, title = "{B}ayesian approach to best basis selection", booktitle = "IEEE International Conference on Acoustics, Speech, and Signal Processing", volume = "5", year = "1996", pages = "2634--2637", author = "J. C. Pesquet and H. Krim and D. Leporini and E. Hamman", note = "7-10 May 1996, Atlanta, GA, USA", abstract = "Wavelet packets and local trigonometric bases provide an efficient framework and fast algorithms to obtain a `best basis' or `best representation' of deterministic signals. Applying these deterministic techniques to stochastic processes may, however, lead to variable results. We revisit this problem and introduce a prior model on the underlying signal in noise and account for the contaminating noise model as well. We thus develop a Bayesian-based approach to the best basis problem, while preserving the classical tree search efficiency.", keywords = "deterministic signals, Bayesian approach, wavelet packets, local trigonometric bases, fast algorithms, best basis selection, best signal representation, deterministic techniques, stochastic processes, stochastic signals, contaminating noise model, classical tree search efficiency, Bernoulli-Gaussian mixtures, Bernoulli-Gaussian priors", } @Article{pes:statistical, title = "Statistical properties of the wavelet decomposition of certain non-{G}aussian self-similar processes", author = "Pesquet-Popescu B.", journal = SP, volume = "75", number = "3", pages = "303--322", year = "1999", email = "bpopescu@csi.com", URL = "http://www.elsevier.nl/cas/tree/store/sigpro/sub/1999/75/3/1386.pdf", keywords = "non-stationary signals self-similarity wavelet analysis higher-order statistics alpha-stable processes lower-order statistics long-range dependence", abstract = "Self-similar processes have recently received increasing attention in the signal processing community, due to their wide applicability in modeling natural phenomena which exhibit ``1/f'' spectra and/or long-range dependence. At the same time, wavelet decomposition has become a very useful tool in describing nonstationary self-similar processes. In this paper, we consider extensions of existing results to non-Gaussian self-similar processes. We first investigate the existence and properties of higher-order statistics of wavelet decomposition for self-similar processes with finite variance. We then consider certain self-similar processes with infinite variance, and study the statistical properties of their wavelet coefficients.", } @InProceedings{pet-ben:uranus, title = "A New Insight in {U}ranus Rings: {A} Wavelet Analysis of the {V}oyager 2 Data", author = "J. M. Petit and Ph. Bendjoya", booktitle = "Completing the Inventory of the Solar System", editor = "Terrence W. Rettig and Joseph M. Hahn", volume = "107", series = "Astronomical Society of the Pacific Conference Proceedings", pages = "137--146", year = "1996", keywords = "wavelet analysis, uranus rings", abstract = "A new signal processing analysis, based on the wavelet transform has been developed. It allows the detection and the reconstruction of fine structures in a very noisy signal. It removes the noise and gives a quantified level of detection of the structures against chance fluctuations. This powerful method has been applied on the PPS Voyager 2 data on the Uranus rings. A preliminary catalog of structures found in the $\sigma$ Sagitarii occultation experiment, is proposed here.", } @TechReport{pet-ste:gamma, title = "{EDF} statistics for testing for the {G}amma distribution", author = "N. A. Pettitt and M. A. Stephens", institution = "Department of Statistics, Stanford University", number = "323", year = "1982", } @TechReport{pet:bayesian, title = "Bayesian Spectral Analysis of Long Memory Time Series", author = "Giovanni Petris", number = "97-08", institution = "Institute of Statistics and Decision Sciences, Duke University", year = "1997", URL = "ftp://ftp.isds.duke.edu/pub/WorkingPapers/97-08.ps", } @Article{pet:non-parametric, title = "A non-parametric approach to the change point problem", author = "N. A. Pettitt", journal = AS, volume = "28", number = "", year = "1979", pages = "126--135", } @Article{pet:some, title = "Some results on estimating a change-point using non-parametric type statistics", author = "N. A. Pettitt", journal = JSCS, volume = "11", number = "", year = "1980", pages = "261--272", } @PhdThesis{pet:thesis, title = "Bayesian Analysis of Long Memory Time Series", author = "Giovanni Petris", year = "1997", school = "Institute of Statistics and Decision Sciences, Duke University", postscript = "ftp://ftp.isds.duke.edu/pub/Theses/giovanni.ps.gz", } @Book{pie-etal:quadpack, title = "\nobreak{QUADPACK}: A Subroutine Package for Automatic Integration", booktitle = "\nobreak{QUADPACK}: A Subroutine Package for Automatic Integration", author = "R. Piessons and E. {de Doncker-Kapenga} and C. W. {\"U}berhuber and D. K. Kahaner", series = "Springer Series in Computational Mathematics", volume = "1", publisher = "Springer-Verlag", address = "Heidelberg", year = "1983", } @Article{pin-vid:estimating, title = "Estimating the square root of a density via compactly supported wavelets", author = "A. Pinheiro and B. Vidakovic", journal = CSDA, volume = "25", number = "4", year = "1997", pages = "399--415", } @InCollection{plo-str:from, title = "From Wavelets to Multiwavelets", author = "Gerlind Plonka and Vasily Strela", booktitle = "Mathamatical Methods for Curves and Surfaces II", editor = "M. Dahlem and T. Lyche and L. Shumaker", publisher = "Vanderbilt University Press", year = "1998", } @Book{pre-teu-vet-fla:numerical, title = "Numerical Recipes in {C}: The Art of Scientific Computing", author = "William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery", edition = "2", year = "1992", publisher = "Cambridge University Press", address = "Cambridge", URL = "http://cfata2.harvard.edu/nr/", } @Article{pri:wavelets, title = "Wavelets and Time-Dependent Spectral Analysis", author = "M. B. Priestley", journal = JTSA, volume = "17", number = "1", year = "1996", pages = "85--104", abstract = "One of the key features of wavelet analysis is its potential use for effecting time-frequency decompositions of non-stationary signals. The relationship between wavelet analysis and timedependent spectral analysis has so far rested mainly on heuristic reasoning: in this paper we examine the relationship in a more precise mathematical form. A crucial feature of this analysis is the need to define carefully the notion of `frequency' when applied to non-stationary signals.", keywords = "Wavelets wavelet transforms discrete wavelet transforms multiresolution analysis Fourier transforms windowed Fourier transforms spectral analysis uncertainty principle evolutionary spectra", } @InProceedings{pro-smi:multichannel, title = "Multichannel time-series modelling and prediction by wavelet networks", author = "Ale\v{s} Proch{\'a}zka and Jonathan Smith", booktitle = "VIII European Signal Processing Conference EUSIPCO-96", year = "1996", pages = "???--???", URL = "http://rex.vscht.cz/prochaz/ps/eusip96.ps", } @Article{pro-vei:trends, title = "Trends, cycles and nonstationarities in isotope signals of {P}hanerozoic seawater", author = "Prokoph, A. and Veizer, J.", journal = CG, volume = "161", number = "1-3", year = "1999", pages = "225--240", abstract = "The new set of Sr-87/Sr-86, delta(18)O and delta(13)C experimental data for Phanerozoic seawater, the ``Bochum/Ottawa Isotope Dataset'', has been tested by wavelet, discontinuity and sliding window correlation dimension analyses for cyclicities and nonstationarities in the isotope signal. The tests indicate discontinuities in the strontium isotope signal at similar to 500, 340, 288, 210, 65 and 28 Ma, while for the oxygen and carbon isotopes they are at similar to 500, 385, 290, 210 and 65 Ma. These discontinuities, often coincident with major stage boundaries, reflect mostly single (likely tectonic) events that do not affect the structure of the underlying system. The two most pronounced nonstationarities in all isotope systematics are at similar to 65 and 210 Ma, respectively, that is at the K/T and Norian/Rhaetian transitions. Wavelet analysis for all three isotope systems yields a long-term quasi-periodicity at similar to 94-125 Ma, best developed during the Paleozoic, with superimposed intermittent 48-57 and 29-35 Ma oscillations, all likely a reflection of plate reorganizations within the Caledonian, Hercynian and Alpine tectonic cycles.", } @Article{qiu-er:wavspect, title = "Wavelet spectrogram of noisy signals", author = "Lunji Qiu and Meng Hwa Er", journal = IJE, volume = "79", number = "5", year = "1995", pages = "665--677", email = "elqiu@ntuvax.ntu.ac.sg", abstract = "The wavelet transform is of interest for analysing non- stationary signals. The squared modulus of the wavelet transform leads to the wavelet spectrogram or scalogram. When signals are embedded in additive noise, it is important to study the estimation accuracy in terms of bias and variance. The mean and variance statistical properties of the wavelet spectrogram of a signal embedded in additive gaussian white noise are derived in this paper. Examples and simulation results are also presented.", } @Article{qiu-u-sha:leakeage, title = "The leakage problem of orthonormal wavelet transforms when applied to atmospheric turbulence", author = "Jie Qiu and Kyaw Tha Paw U and Roger H. Shaw", journal = JGRA, volume = "100", number = "D12", year = "1995", pages = "25,769--25,779", abstract = "Orthonormal wavelet transforms are becoming common in the study of turbulence phenomena. Although they are powerful tools in representing a signal, their use as tools to study the characteristics of turbulent structures can create appreciable errors in interpretation. It is shown here that although the orthonormal wavelet transform is computationally economical by taking advantage of multiresolution analysis, it has insufficient resolution in both scale and location to resolve detailed information of turbulence structures. Lacking in resolution, the energy at a particular frequency (or wavelength) may leak into neighboring frequencies and may pass down to smaller scales to produce an artificial ``cascade'' of energy (with a slope close to -2/3). The choice of wavelet basis function is important to the wavelet spectrum, especially in the study of turbulence flows dominated by coherent structures, since the method most accurately senses energy contained in pulses that have a similar pattern to the wavelet function. To use the method as a filter can be problematic owing to the low resolution of the orthonormal wavelet transform; nonorthonormal wavelet analysis should be employed when high resolution is important. When orthonormal wavelet transforms have to be used for signal analysis, segmented averaging should be employed.", } @Book{rai:book, title = "Special Functions", author = "Earl D. Rainville", year = "1960", publisher = "The Macmillan Company", address = NY, } @Article{rai:minimax, title = "Minimax estimation of sharp change points", author = "Raimondo, M.", journal = AofS, volume = "26", number = "4", year = "1998", pages = "1379--1397", keywords = "change point cusp jump minimax estimation nonparametric regression wavelets", abstract = "We define the sharp change point problem as an extension of earlier problems in change point analysis related to nonparametric regression. As particular cases, these include estimation of jump points in smooth curves. More generally, we give a systematic treatment of the correct rate of convergence for estimating the position of a `cusp' of an arbitrary order. We propose a test function for the local regularity of a signal sample implementation of our method, from observations of the signal at discrete time positions i/n, i = 1,..., n, we use a wavelet transformation to approximate the position of the change point in the no-noise case. We study the noise effect, in the worst case scenario over a wide class of functions having a unique irregularity of `order alpha' and propose a sequence of estimators which converge at the rate n(-1/(1+2 alpha)), as n tends to infinity. Finally we analyze the likelihood ratio of the problem and show that this is actually the minimax rate of convergence. Examples of thresholding empirical wavelet coefficients to estimate the position of sharp change points are also presented.", } @Article{ram-lam:decomposition, title = "Decomposition of economic relationships by timescale using wavelets - {M}oney and income", author = "Ramsey, J. B. and Lampart, C.", journal = "Macroeconomic Dynamics", volume = "2", number = "1", year = "1998", pages = "49--71", keywords = "wavelets timescale velocity money income permanent income hypothesis money-income causality", abstract = "Economists have long known that timescale matters in that the structure of decisions as to the relevant time horizon, degree of time aggregation, strength of relationship, and even the relevant variables differ by timescale. Unfortunately, until recently it was difficult to decompose economic time series into orthogonal timescale components except for the shea or long run in which the former is dominated by noise. Wavelets are used to produce an orthogonal decomposition of some economic variables by timescale over six different timescales. The relationship of interest is that between money and income, i.e., velocity. We confirm that timescale decomposition is very important for analyzing economic relationships. The analysis indicates the importance of recognizing variations in phase between variables when investigating the relationships between them and throws considerable light on the conflicting results that have been obtained in the literature using Granger causality tests.", } @Article{ram-lam:decomposition2, title = "The Decomposition of economic relationships by time scale using wavelets: {E}xpenditure and income", author = "Ramsey, J. B. and Lampart, C.", journal = "Studies in Nonlinear Dynamics and Econometrics", volume = "3", number = "1", year = "1998", pages = "23--42", keywords = "permanent income consumption hypothesis", abstract = "Economists have long known that time scale matters, in that the structure of decisions as to the relevant time horizon, degree of time aggregation, strength of relationship, and even the relevant variables differ by time scale. Unfortunately, until recently it was difficult to decompose economic time series into orthogonal time-scale components except for the short and long run, in which the former is dominated by noise. This paper uses wavelets to produce an orthogonal decomposition of some economic variables by time scale over six different time scales. The relationship of interest is the permanent income hypothesis. We confirm that time-scale decomposition is very important for analyzing economic relationships and that a number of anomalies previously noted in the literature are explained by these means. The analysis indicates the importance of recognizing variations in phase between variables when investigating the economic relationships.", } @Article{ram-usi-zas:us-stock, title = "An analysis of {U.S.} Stock Price Behavior Using Wavelets", author = "Ramsey, James B. and Uskinov, Daniel and Zaslavsky, George M.", journal = "Fractals", volume = "3", number = "2", year = "1995", pages = "377--389", keywords = "", abstract = "", } @Article{ram-zei:fBm, title = "On the wavelet transform of fractional {B}rownian motion", author = "J. Ramanathan and O. Zeitouni", journal = IEEETIT, volume = "37", number = "4", year = "1991", pages = "1156--1158", keywords = "wavelet transform, fractional Brownian motion, covariance structure, Gaussian processes", abstract = "A theorem characterizing fractional Brownian motion by the covariance structure of its wavelet transform is established. The authors examine whether there are alternate Gaussian processes whose wavelet transforms have a natural covariance structure. In addition, the authors examine if there are any Gaussian processes whose wavelet transform is stationary with respect to the affine group (i.e. the statistics of the wavelet transform do not depend on translations and dilations of the process).", } @Article{ram-zha:analysis, title = "The analysis of foreign exchange data using waveform dictionaries", author = "James B. Ramsey and Zhifeng Zhang", journal = "Journal of Empirical Finance", volume = "4", year = "1997", pages = "341--372", } @InCollection{ram-zha:application, title = "The application of wave form dictionaries to stock market index data", author = "Ramsey, J. B. and Zhang, Z. F.", booktitle = "Predictability of Dynamical Systems", editor = "Kravstov, Y. A. and Kadtke, J. B.", publisher = "Springer Verlag", address = NY, volume = "69", year = "1996", pages = "189--205", } @Article{ram:contribution, title = "The contribution of wavelets to the anlaysis of economic and financial data", author = "James B. Ramsey", journal = PTRSLA, volume = "357", number = "1760", year = "1999", pages = "2593--2606", } @Unpublished{ram:regression, title = "Regression over Time Scale Decompositions: {A} Sampling Analysis of Distributional Properties", author = "James B. Ramsey", year = "1998", note = "New York University", } @Article{rei-etal:multifractal, title = "A multifractal wavelet model with application to network traffic", author = "Riedi, R. H. and Crouse, M. S. and Ribeiro, V. J. and Baraniuk, R. G.", journal = IEEETIT, volume = "45", number = "3", year = "1999", pages = "992--1018", URL = "", keywords = "long-range dependence multifractals network traffic positive 1/f noise wavelets", abstract = "In this paper, we develop a new multiscale modeling framework for characterizing positive-valued data with long-range-dependent correlations (1/f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing N-point data sets. We study both the second-order and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variance-time plots and moment scaling) and queuing behavior, Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.", } @InProceedings{rib-etal:simulating, title = "Simulation of non{G}aussian Long-Range Dependent Traffic using Wavelets", author = "V. J. Ribeiro and R. H. Riedi and M. S. Crouse and R. G. Baraniuk", booktitle = "ACM SIGMETRICS Conference on the Measurement and Modeling of Computer Systems", year = "1999", note = "1-4 May 1999, Atlanta, Georgia", abstract = "In this paper, we develop a simple and powerful multiscale model for the synthesis of nonGaussian, long-range dependent (LRD) network traffic. Although wavelets effectively decorrelate LRD data, wavelet-based models have generally been restricted by a Gaussianity assumption that can be unrealistic for traffic. Using a multiplicative superstructure on top of the Haar wavelet transform, we exploit the decorrelating properties of wavelets while simultaneously capturing the positivity and ``spikiness'' of nonGaussian traffic. This leads to a swift O(N) algorithm for fitting and synthesizing N-point data sets. The resulting model belongs to the class of multifractal cascades, a set of processes with rich statistical properties. We elucidate our model's ability to capture the covariance structure of real data and then fit it to real traffic traces. Queueing experiments demonstrate the accuracy of the model for matching real data. Our results indicate that the nonGaussian nature of traffic has a significant effect on queuing.", } @Unpublished{rid-den:aeromagnetic, title = "The wavelet transform in aeromagnetic processing", author = "T. A. Ridsdill-Smith and M. C. Dentith", journal = "GEOPHYSICS", volume = "64", number = "4", year = "1999", pages = "1067--1078", URL = "http://geolpc42.geol.uwa.edu.au/papers/wavemag.zip", abstract = "The phase-shift method of wavefield extrapolation applies a phase shift in the Fourier domain to deduce a scalar wavefield at one depth level given its value at another. The phase-shift operator varies with frequency and wavenumber, and assumes constant velocity across the extrapolation step. We use nonstationary filter theory to generalize this method to nonstationary phase shift (NSPS), which allows the phase shift to vary laterally depending upon the local propagation velocity. For comparison, we derive an analytic form for the popular phase shift plus interpolation (PSPI) method in the limit of an exhaustive set of reference velocities. NSPS and this limiting form of PSPI can be written as generalized Fourier integrals which reduce to ordinary phase shift in the constant velocity limit. In the (x, omega) domain, these processes are the transpose of each other; however, only NSPS has the physical interpretation of forming the scaled, linear superposition of laterally-variable impulse responses (i.e., Huygen's wavelets).The difference between NSPS and PSPI is clear when they are compared in the case of a piecewise constant velocity variation. Define a set of windows such that the jth window is unity when the propagation velocity is the jth distinct velocity and is zero otherwise. NSPS can be computed by applying the window set to the input data to create a set of windowed wavefields, which are individually phase-shift extrapolated with the corresponding constant velocity, and the extrapolated set is superimposed. PSPI proceeds by phase-shift extrapolating the input data for each distinct velocity, applying the jth window to the jth extrapolation, and superimposing. Though neither process is fully correct, PSPI has the unphysical limit that discontinuities in the lateral velocity variation cause discontinuities in the wavefield, whereas NSPS shows the expected wavefront ``healing.'' We then formulate a finite aperture compensation for NSPS which has the practical result of absorbing lateral boundaries for all incidence angles. Wavefield extrapolation can be regarded as the crosscorrelation of the wavefield with the expected response of a point diffractor at the new depth level. Aperture compensation simply applies a laterally varying window to the infinite, theoretical diffraction response. The crosscorrelation becomes spatially variant, even for constant velocity, and hence is a nonstationary filter. The nonstationary effects of aperture compensation can be simultaneously applied with the NSPS extrapolation through a laterally variable velocity field.", } @Article{rid:separating, title = "Separation filtering of aeromagnetic data using filter-banks", author = "T. A. Ridsdill-Smith", journal = "Exploration Geophysics", volume = "29", number = "3-4", year = "1998", pages = "577--583", URL = "http://geolpc42.geol.uwa.edu.au/papers/aseg98trs.zip", abstract = "", } @Article{rie-sid:adaptive, title = "Adaptive smoothing of the log-spectrum with multiple tapering", author = "Kurt S. Riedel and Alexander Sidorenko", journal = IEEETSP, volume = "44", number = "7", year = "1996", pages = "1794--1800", abstract = "A hybrid estimator of the log-spectral density of a stationary time series is proposed, First, a multiple taper estimate is performed, followed by kernel smoothing the log-multiple taper estimate, This procedure reduces the expected mean square error by (pi(2)/4)(4/5) over simply smoothing the log tapered periodogram, A data-adaptive implementation of a variable-bandwidth kernel smoother is given.", } @Article{rie-etal:multifractal-wavelet, title = "A Multifractal Wavelet Model with Application to Network Traffic", author = "R. H. Riedi and M. S. Crouse and V. J. Ribeiro and R. G. Baraniuk", journal = IEEETIT, volume = "45", number = "4", year = "1999", pages = "992--1018", URL = "http://www-dsp.rice.edu/~riedi/cv_frame.html", keywords = "Multifractals, long-range dependence, positive 1/f noise, wavelets, network traffic", abstract = "In this paper, we develop a new multiscale modeling framework for characterizing positive-valued data with long-range-dependent correlations (1/f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing N-point data sets. We study both the second-order and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variance-time plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.", } @InProceedings{rie:geophysics, title = "Multifractals and Wavelets: {A} potential tool in Geophysics", author = "R. H. Riedi", booktitle = "Society of Exploration Geophysicists Annual Meeting", address = "New Orleans", year = "1998", pages = "???--???", URL = "http://www-dsp.rice.edu/~riedi/cv_frame.html", } @TechReport{rie:introduction, title = "Introduction to Multifractals", author = "R. H. Riedi", number = "99-06", institution = "ECE Department, Rice University", year = "1999", } @Article{rio-fla:time-scale, title = "Time-Scale Energy Distributions: {A} General Class Extending Wavelet Transforms", author = "Olivier Rioul and Patrick Flandrin", journal = IEEETSP, volume = "40", number = "7", year = "1992", pages = "1746--1757", abstract = "A proposed theoretical framework for time-scale energy representation is based on local frequency which is covariant under modulations and time scaling which is covariant under dilations or contractions. The frameworks seeks to illustrate the relationship between scalograms and spectrograms. Results show that, from the Wigner-Ville distribution, it is possible to shift continuously to either a scalogram or a spectrogram. The approach simultaneously maintains a balance between time-frequency resolution and cross-terms reduction in both time-scale and time-frequency representations.", } @Article{rio-vet:wavelets, title = "Wavelets and Signal Processing", author = "Olivier Rioul and Martin Vetterli", journal = IEEESPM, volume = "8", number = "4", year = "1991", pages = "14--38", abstract = "A simple, nonrigorous, synthetic view of wavelet theory is presented for both review and tutorial purposes. The discussion includes nonstationary signal analysis, scale versus frequency, wavelet analysis and synthesis, scalograms, wavelet frames and orthonormal bases, the discrete-time case, and applications of wavelets in signal processing. The main definitions and properties of wavelet transforms are covered, and connections among the various fields where results have been developed are shown.", } @TechReport{rio-vid:random, title = "Wavelet-Based Random Densities", author = "David Insua Rios and Brani Vidakovi\'{c}", institution = "Institute of Statisics and Decision Sciences, Duke University", year = "1997", note = "Discussion Paper 97-05", URL = "ftp://ftp.isds.duke.edu/pub/WorkingPapers/97-05.ps", } @Article{rob-par-alv:extraction, title = "Extraction of impulse response data via wavelet transform for structural system identification", author = "A. N. Robertson and K. C. Park and K. F. Alvin", journal = JVA, volume = "120", number = "1", year = "1998", pages = "252--260", keywords = "vibration", abstract = "This paper presents a wavelet transform-based method of extracting the impulse response characteristics from the measured disturbances and response histories of linear structural dynamic systems. The proposed method is found to be effective in determining the impulse response functions for systems subjected to harmonic (narrow frequency-band) input signals and signals with sharp discontinuities, thus alleviating the Gibbs phenomenon encountered in FFT methods. When the system is subjected to random burst input signals for which the FFT methods are known to perform well, the proposed wavelet method performs equally well with a fewer number of ensembles than FFT-based methods. For completely random input signals, both the wavelet and FFT methods experience difficulties, although the wavelet method appears to perform somewhat better in tracing the fundamental response modes.", } @Article{rob-par-alv:identification, title = "Identification of structural dynamics models using wavelet-generated impulse response data", author = "A. N. Robertson and K. C. Park and K. F. Alvin", journal = JVA, volume = "120", number = "1", year = "1998", pages = "261--266", keywords = "vibration", abstract = "", } @InProceedings{rom-cho-bar:bayesian, title = "{B}ayesian Tree-Structured Image Modeling using Wavelet-domain Hidden {M}arkov Models", author = "J. K. Romberg and H. Choi and R. G. Baraniuk", booktitle = "SPIE Technical Conference on Mathematical Modeling, Bayesian Estimation, and Inverse Problems", volume = "3816", year = "1999", pages = "31--44", email = "jrom@rice.edu, choi@ece.rice.edu, richb@rice.edu", URL = "http://www-dsp.rice.edu/publications/pub/jrom99spie.ps.Z", abstract = "Wavelet-domain hidden Markov models have proven to be useful tools for statistical signal and image processing. The hidden Markov tree model captures the key features of the joint density of the wavelet coefficients of real-world data. One potential drawback to the HMT framework is the need for computationally expensive iterative training. In this paper, we prose two reduced-parameter HMT models that capture the class of real-world images. In the image HMT (iHMT) model we use the fact that for a large class of images the structure of the HMT is self-similar across scale. This allows us to reduce the complexity of the iHMT to just nine easily trained parameters. In the universal HMT (uHMT) we take a Bayesian approach and fix these nine parameters. The uHMT requires no training of any kind. While simple, we show using a series of image estimation/denoising experiments that these two new models retain nearly all of the key structure modeled by the full HMT. Finally, we propose a fast shift-invariant HMT estimation algorithm that outperforms all other wavelet- based estimators in the current literature, both in mean- square error and visual metrics.", } @InProceedings{rom-cho-bar:bayesian2, title = "{B}ayesian Wavelet Domain Image Modeling using Hidden {M}arkov Trees", author = "J. K. Romberg and H. Choi and R. G. Baraniuk", booktitle = "Proceedings of the IEEE International Conference on Image Processing", volume = "", year = "1999", pages = "???--???", email = "jrom@rice.edu, choi@ece.rice.edu, richb@rice.edu", URL = "http://www-dsp.rice.edu/publications/pub/jrom-icip99.ps.Z", } @InProceedings{rom-cho-bar:shift-invariant, title = "Shift-Invariant Denoising using Wavelet-Domain Hidden {M}arkov Trees", author = "J. K. Romberg and H. Choi and R. G. Baraniuk", booktitle = "Conference Record of The Thirty-Third Asilomar Conference on Signals, Systems and Computers", volume = "", year = "1999", pages = "???--???", email = "jrom@rice.edu, choi@ece.rice.edu, richb@rice.edu", URL = "http://www-dsp.rice.edu/publications/pub/jrom-asilomar99.ps.Z", } @Unpublished{rou-vei:measuring, title = "Measuring Long-Range Dependence Under Changing Traffic Conditions", author = "Matthew Roughan and Darryl Veitch", year = "1999", note = "preprint", } @InProceedings{rou-vei-abr:on-line, title = "On-line estimation of the parameters of long-range dependence", author = "Matthew Roughan and Darryl Veitch and Patrice Abry", booktitle = "Proceedings Globecom '98", address = "Sydney", volume = "6", year = "1998", pages = "3716--3721", } @Article{rug-vid:bayesian, title = "A Bayesian decision theoretic approach to the choice of thresholding parameter", author = "Ruggeri, F. and Vidakovic, B.", journal = SSin, volume = "9", number = "1", year = "1999", pages = "183--197", keywords = "Bayes rule hard-thresholding wavelets", abstract = "Thresholding rules recently became of considerable interest when Donoho and Johnstone applied them in the wavelet shrinkage context. Analytically simple, such rules are very efficient in data denoising and data compression problems. In this paper we find hard thresholding decision rules that minimize Bayes risk for broad classes of underlying models. Standard Donoho-Johnstone test signals are used to evaluate performance of such rules. We show that an optimal Bayesian decision theoretic (BDT) hard thresholding rule can achieve smaller mean squared error than some standard wavelet thresholding methods, if the prior information on the noise level is precise.", } @Book{rus-etal:wavelets, title = "Wavelets and Their Applications", booktitle = "Wavelets and Their Applications", editor = "Mary Beth Ruskai and Gregory Beylkin and Ronald Coifman and Ingrid Daubechies and Stephane Mallat and Yves Meyer and Louise Raphael", publisher = "Jones and Bartlett Publishers", year = "1992", pages = "474", URL = "http://www.jbpub.com/catalog/Detail.CFM?titles__ISBN=0867202254", loc = "QA 403.3 .W385 1992", ISBN = "0-86720-225-4", } @Article{rus-lab-les:commodity, title = "Wavelet Analysis of Commodity Price Behavior", author = "Davidson, Russell and Labys, Walter C. and Lesourd, Jean-Baptiste", journal = CE, volume = "11", number = "1-2", year = "1998", pages = "103--128", abstract = "We propose a form of semi-nonparametric regression based on wavelet analysis. Traditional time series methods usually involve either the time or the frequency domain, but wavelets can combine the information from both of these. While wavelet transforms are typically restricted to equally spaced observations an integer power of 2 in number, we show how to go beyond these constraints. We use our methods to construct ``patios'' for twenty-one important international commodity price series. These graph the magnitude of the variations in the series at different time scales for various subperiods of the full sample.", } @Article{sac-pra:coherent, title = "Coherent modes in multiscale variability of streamflow over the {U}nited {S}tates", author = "Saco, P. and Kumar, P.", journal = WRR, volume = "36", number = "4", year = "2000", pages = "1049--1067", URL = "", abstract = "Motivated by the need to understand large-scale hydrologic response, significant research has been directed toward the identification of coherent regions using characteristics of streamflow variability. Typically, these regions are delineated using principal component analysis on streamflow. This method does not account for differences in temporal scales of fluctuations embedded in the time series. To capture this, we use wavelet spectral analysis. Wavelet spectra from the specific streamflow series are obtained for outflow binned at 3 degrees-length segments along the border of the conterminous United States. Rotated principal component analysis is performed on the wavelet spectra to obtain clusters of segments that exhibit similar distribution of variability across scales. Three physically distinct modes explain over 89% of the variability. Two of the modes identified are associated with high variability at seasonal scales, and the third is associated with high variability at small timescales. The runoff generation mechanisms underlying the observed modes of multiscale variability of various regions are also discussed. Each of these coherent modes of multiscale variability indicate the existence of regions with similar scales of fluctuations that are located geographically apart, as well as regions located geographically close with dissimilar scales of fluctuations.", } @Article{sai-bey:autocorrelation, title = "Multiresolution representations using the autocorrelation functions of compactly supported wavelets", author = "Naoki Saito and Gregory Beylkin", journal = IEEETSP, volume = "41", number = "12", year = "1993", pages = "3584--3590", URL = "http://math.ucdavis.edu/~saito/publications/miniframe.html", abstract = "Proposes a shift-invariant multiresolution representation of signals or images using dilations and translations of the autocorrelation functions of compactly supported wavelets. Although these functions do not form an orthonormal basis, their properties make them useful for signal and image analysis. Unlike wavelet-based orthonormal representations, the present representation has (1) symmetric analyzing functions, (2) shift-invariance, (3) associated iterative interpolation schemes, and (4) a simple algorithm for finding the locations of the multiscale edges as zero-crossings. The authors also develop a noniterative method for reconstructing signals from their zero-crossings (and slopes at these zero-crossings) in their representation. This method reduces the reconstruction problem to that of solving a system of linear algebraic equations.", } @InProceedings{sai:asilomar98, title = "The least statistically-dependent basis and its applications", author = "Naoki Saito", booktitle = "Conference Record of The Thirty-Second Asilomar Conference on Signals, Systems and Computers", year = "1998", pages = "732--736", URL = "http://math.ucdavis.edu/~saito/publications/asilomar98.html", } @Unpublished{sai:image, title = "Image approximation and modeling via least statistically-dependent bases", author = "Naoki Saito", note = "Submitted to {\em Pattern Recognition}", year = "1999", abstract = "Statistical independence is one of the most desirable properties of a coordinate system for representing and modeling images. In reality, however, truly independent coordinates may not exist for a given set of images, or it may be too difficult to compute them in practice. Therefore, it makes sense to obtain the least statistically-dependent coordinate system efficiently. To achieve this goal, we use the best-basis algorithm with new criterion that can rapidly select the least statistically-dependent basis (LSDB) from a basis dictionary (e.g., the local cosine or wavelet packet dictionaries) containing a huge number of orthonormal (or biorthogonal) bases. Our new basis selection criterion is minimization of the mutual information of the distributions of the basis coefficients as a measure of statistical dependence, which in turn is equivalent to minimization of the sum of the differential entropy of each coordinate in the basis dictionary. We show that this criterion, combined with the best-basis algorithm, can find the coordinates closest to the statistical independence from all possible bases searchable in a basis dictionary with O(n [log n]^p), where n is the dimensionality of the image (the number of pixels in each image), and p=1 for the wavelet packet dictionaries, and p=2 for the local cosine/sine dictionaries. In this sense, we can view this LSDB algorithm as the best-basis version of the Independent Component Analysis (ICA), which is increasingly gaining popularity. This criterion is different from that of the Joint Best Basis (JBB) proposed by Wickerhauser, which can be viewed as the best-basis version of the Karhunen-Loeve basis (KLB). We demonstrate the application of the LSDB to image approximation and modeling and compare its performance with that of KLB and JBB using a collection of real geophysical acoustic waveforms and an image database of human faces. For these datasets, the LSDB provides the best approximation in terms of the average relative l^2 errors among various bases including the KLB, JBB, DCT, and wavelet basis. For image modeling, we propose two simple stochastic models for a given class of signals or images based on the LSDB coordinates. The first model is to assume the statistical independence among the LSDB coordinates, which allows us to sample typical coefficients of each coordinate separately using the empirical distribution estimated from the available training coefficients of that coordinate, which in turn easily allows us to simulate new images at our disposal. For the geophysical acoustic waveforms, this first model turned out to be good enough. The second model is based on the ``second rotation'' by the KLB computed from the top m LSDB coordinates. This model gives us the decorrelated coordinates built on top of the LSDB coordinates. The simulation results on the human face database using the second model suggest that this second rotation can further reduce the statistical dependency among the coordinates, and allows better modeling for a class of complicated images.", } @InProceedings{sai:least, title = "Least statistically-dependent basis and its applications to image modeling", author = "Naoki Saito", pages = "24--37", crossref = "lai-uns-ald:wavelet6", URL = "http://math.ucdavis.edu/~saito/publications/lsdb_spie2.html", } @PhdThesis{sai:thesis, title = "Local Feature Extraction and Its Applications Using a Library of Bases", author = "Naoki Saito", school = "Yale University", year = "1994", URL = "http://math.ucdavis.edu/~saito/publications/saito_phd.html", } @Article{sak:pseudo, title = "Pseudodiagonalization of the autocorrelation of a stochastic process by an over-complete wavelet system", author = "F. Sakaguchi", journal = ECJ3, volume = "78", number = "4", year = "1995", pages = "16--27", note = "Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. 77-A, No. 8, August 1994, pp. 1065--1074", abstract = "If a stochastic process can be regarded as a superposition of the wavelets which arise randomly and independently of one another, the random-wavelet picture of a stochastic process is intuitive and convenient. This paper investigates theoretically in what case the picture can be used; i.e., in what case the autocorrelation of the stochastic process can be diagonalized by using the over-complete wavelet system. A general method for calculating the pseudodiagonal form from an arbitrarily given autocorrelation function using the operator algebra is proposed. Next, some properties of stationary wavelet-diagonal processes are investigated where it is shown that the power spectra of these processes are related to the spectral estimates under the circumstances in which the number of the time points are constrained to a constant finite number.", } @Article{sar-brz:shift-invariant, title = "A Shift-Invariant Discrete Wavelet Transform", author = "Hamed Sari-Sarraf and Dragana Brzakovic", journal = IEEETSP, volume = "45", number = "10", year = "1997", pages = "2621--2626", abstract = "This correspondence presents a unifying approach to the derivation and implementation of a shift-invariant wavelet transform of one-and two-dimensional (1-D and 2-D) discrete signals, Starting with Mallat's multiresolution wavelet representation (MRWAR), the correspondence presents an analytical process through which a shift-invariant, orthogonal, discrete wavelet transform called the multiscale wavelet representation (MSWAR) is obtained, The coefficients in MSWAR are shown to be inclusive of those in MRWAR with the implication that the derived representation is invertible. The computational complexity of MSWAR is quantified in terms of the required convolutions, and its implementation is shown to be equivalent to the filter upsampling technique.", } @Article{sar-etal:unequal, title = "Wavelet DeNoising for Unequally Spaced Data", author = "Sylvain Sardy and Donald B. Percival and Andrew G. Bruce and Hong-Ye Gao and Werner Stuetzle", journal = SC, volume = "9", number = "1", year = "1999", pages = "65--75", URL = "ftp://ftp.statsci.com/pub/WAVELETS/papers/unequal.ps.gz", keywords = "Nonparametric regression; Wavelet transform; WaveShrink", abstract = "Wavelet shrinkage (WaveShrink) is a relatively new technique for nonparametric function estimation that has been shown to have asymptotic near-optimality properties over a wide class of functions. As originally formulated by Donoho and Johnstone, WaveShrink assumes equally spaced data. Because so many statistical applications (e.g., scatterplot smoothing) naturally involve unequally spaced data, we investigate in this paper how WaveShrink can be adapted to handle such data. Focusing on the Haar wavelet, we propose four approaches that extend the Haar wavelet transform to the unequally spaced case. Each approach is formulated in terms of continuous wavelet basis functions applied to a piecewise constant interpolation of the observed data, and each approach leads to wavelet coefficients that can be computed via a matrix transform of the original data. For each approach, we propose a practical way of adapting WaveShrink. We compare the four approaches in a Monte Carlo study and find them to be quite comparable in performance. The computationally simplest approach (isometric wavelets) has an appealing justification in terms of a weighted mean square error criterion and readily generalizes to wavelets of higher order than the Haar.", } @Article{sar:minimax, title = "Minimax threshold for denoising complex signals with waveshrink", author = "Sylvain Sardy", journal = IEEETSP, volume = "48", number = "4", year = "2000", pages = "1023--1028", URL = "http://dmawww.epfl.ch/~sardy/PS/article-minimax.ps", abstract = "For the problem of signal extraction from noisy data, Waveshrink has proven to be a powerful tool, both from an empirical and an asymptotic point of view. Waveshrink is especially efficient at estimating spatially inhomogeneous signals, A key step of the procedure is the selection of the threshold parameter. Donoho and Johnstone propose a selection of the threshold based on a minimax principle, Their derivation is specifically for real signals and real wavelet transforms. In this paper, we propose to extend the use of Waveshrink to denoising complex signals with complex wavelet transforms. We illustrate the problem of denoising complex signals with an electronic surveillance application.", } @Unpublished{sar-tse-bru:robust, title = "Robust Wavelet Denoising", author = "S. Sardy and P. Tseng and A. G. Bruce", year = "1998", note = "Submitted to {\em Journal of the American Statistical Association}", URL = "http://dmawww.epfl.ch/~sardy/PS/article-robust.ps", } @Article{sca-etal:quasi, title = "{T}he quasi-periodic oscillations and very low frequency noise of {S}corpius {X}-1 as transient chaos: a dripping handrail?", journal = "Astrophysical Journal Letters", volume = "411", number = "2", year = "1993", pages = "91--94", author = "J. D. Scargle and T. {Steiman, Cameron} and K. Young and D. L. Donoho and J. P. Crutchfield and J. Imamura", abstract = "The authors present evidence that the quasi-periodic oscillations (QPO) and very low frequency noise (VLFN) characteristic of many accretion sources are different aspects of the same physical process. They analyzed a long, high time resolution EXOSAT observation of the low-mass X-ray binary (LMXB) Sco X-1. The X-ray luminosity varies stochastically on time scales from milliseconds to hours. The nature of this variability-as quantified with both power spectrum analysis and a new wavelet technique, the scalegram-agrees well with the dripping handrail accretion model, a simple dynamical system which exhibits transient chaos. In this method both the QPO and VLFN are produced by radiation from blobs with a wide size distribution, resulting from accretion and subsequent diffusion of hot gas, the density of which is limited by an unspecified instability to lie below a threshold.", keywords = "V818 Sco. 4U 1617-15. variable star. quasi-periodic oscillations. very low frequency noise. transient chaos. accretion sources. low-mass X-ray binary. Sco X-1. X-ray luminosity. power spectrum analysis. wavelet technique. scalegram. dripping handrail accretion model. blobs. size distribution. diffusion. hot gas. density. instability.", } @InCollection{sca:astronomical, title = "Wavelet Methods in Astronomical Time Series Analysis", author = "Jeffrey D. Scargle", email = "jeffrey@sunshine.arc.nasa.gov", pages = "226--248", crossref = "rao-pri-les:applications", abstract = "", } @InProceedings{sca:astronomical2, title = "Astronomical Time Series Analysis: {N}ew Methods for Studying Periodic and Aperiodic Systems", author = "Jeffrey D. Scargle", booktitle = "The Weise Observatory 25th Anniversary Symposium: Astronomical Time Series", email = "jeffrey@sunshine.arc.nasa.gov", year = "1996", abstract = "", } @InCollection{sca:detection, title = "Detection and modeling of chaotic dynamics using wavelet techniques", author = "Jeffrey D. Scargle", email = "jeffrey@sunshine.arc.nasa.gov", crossref = "szu:wavelet1", note = "", URL = "", abstract = "Powerful new data analysis techniques based on wavelets are proving extremely useful in the reduction and interpretation of time series data. The goals of these methods include denoising, characterizing, modeling, and compressing of time series data. The multi-scale nature of wavelet analysis makes it especially useful for detection and characterization of self-similar or 'scaling' behavior, such as is common for chaotic processes. This paper describes how wavelet techniques led to a transient-chaos model for a rapidly fluctuating celestial X-ray source. The methods described here are freely available in a new software package called TeachWave, developed by David Donoho and Iain Johnstone of Stanford University (anonymous ftp to playfair.stanford.edu; the software is in directory /pub/software/wavelets, and a number of related technical papers are in /pub/reports).", } @InProceedings{sch-swe:sphere, author = "Peter Schr{\"o}der and Wim Sweldens", title = "Spherical Wavelets: {E}fficiently Representing Functions on the Sphere", booktitle = "Computer Graphics Proceedings (SIGGRAPH 95)", year = "1995", publisher = "ACM Siggraph", pages = "161--172", URL = "http://cm.bell-labs.com/who/wim/papers/sphere.ps.gz", abstract = "Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allofw fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bi-directional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.", } @Book{sch-web:recent, title = "Recent Advances in Wavelet Analysis", booktitle = "Recent Advances in Wavelet Analysis", editor = "Larry L. Schumaker and Glenn Webb", volume = "3", series = "Wavelet Analysis and its Applications", year = "1993", publisher = "Academic Press, Inc.", ISBN = "0-12-632370-4", abstract = "Recent Advances in Wavelet Analysis is the third volume in the WAVELET ANALYSIS AND ITS APPLICATIONS series. This edited volume features ten timely and important articles authored by various experts in their respective fields, including such notable contributors as David L. Donoho, Ingrid Daubechies (MacArthur grant awardees in ‘91 and ‘92, respectively), Phillippe Tchamitchian, Patrick Flandrin (both featured speakers at the ‘92 International Wavelets Conference in Toulouse), Charles Chui, and Bjorn Jawerth (one of the editors of the Wavelet Digest). This book covers recent advances in wavelet analysis and applications in areas including wavelets on bounded intervals, wavelet decomposition of special interest to statisticians, wavelets approach to differential and integral equations, analysis of subdivision operators, and wavelets related to problems in engineering and physics. Anyone interested in the ever-evolving field of wavelets will find this book an excellent addition to the series and to the literature overall.", } @Article{sch:investigation, title = "On the investigation of hidden periodicities with application to a supposed 26-day period of meterological phenomena", author = "A. Schuster", journal = "Terrestrial Magnetism", volume = "3", year = "1898", pages = "13--41", } @Article{sch:optimal, title = "Fast and statistically optimal period search in uneven sampled observations", author = "A. Schwarzenberg-Czerny", journal = ApJ, volume = "460", number = "2", year = "1996", pages = "107--110", URL = "http://www.journals.uchicago.edu/ApJ/journal/issues/ApJL/v460n2/5758/5758.html", abstract = "The classical methods for searching for a periodicity in uneven sampled observations suffer from a poor match of the model and true signals and/or use of a statistic with poor properties. We present a new method employing periodic orthogonal polynomials to fit the observations and the analysis of variance (ANOVA) statistic to evaluate the quality of the fit. The orthogonal polynomials constitute a flexible and numerically efficient model of the observations. Among all popular statistics, ANOVA has optimum detection properties as the uniformly most powerful test. Our recurrence algorithm for expansion of the observations into the orthogonal polynomials is fast and numerically stable. The expansion is equivalent to an expansion into Fourier series. Aside from its use of an inefficient statistic, the Lomb-Scargle power spectrum can be considered a special case of our method. Tests of our new method on simulated and real light curves of nonsinusoidal pulsators demonstrate its excellent performance. In particular, dramatic improvements are gained in detection sensitivity and in the damping of alias periods.", } @Article{sch:translation-invariant, author = "Scholl, D. J.", title = "Translation-invariant data visualization with orthogonal discrete wavelets", journal = IEEETSP, volume = "46", number = "7", year = "1998", pages = "2031--2034", abstract = "Orthogonal discrete wavelet transforms, can be made translation-invariant by adding redundant wavelet coefficients through repeated shifting operations. Othogonality is lost, but isometry and compact time support can be preserved. The practical application to data visualization of scalograms based on such transforms is discussed and illustrated with measured transient signals.", } @Article{sel:balanced, title = "Balanced multiwavelet bases based on symmetric {FIR} filters", author = "Selesnick, Ivan W.", journal = IEEETSP, volume = "48", number = "1", year = "2000", pages = "184--191", abstract = "This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming we wish to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r(2) scaling functions phi(i,j). This paper also introduces new K-balanced orthogonal multiwavelet bases based on symmetric FIR filters. The nonlinear design equations arising in this work are solved using the Grobner basis. The minimal-length K-balanced multiwavelet bases based on even-length symmetric FIR filters are better behaved than those based on odd-length symmetric FIR filters, as illustrated by special relations they satisfy and by examples constructed.", } @Article{sel:interpolating, title = "Interpolating multiwavelet bases and the sampling theorem", author = "Selesnick, Ivan W.", journal = IEEETSP, volume = "47", number = "6", year = "1999", pages = "1615--1621", email = "seles@taco.poly.edu", keywords = "filter banks multiwavelet bases sampling wavelet transforms", abstract = "This paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal (interpolating). They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which is not continuous, This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal. The scaling functions thereby support a Shannon-like sampling theorem, Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator.", } @Article{sel:slantlet, title = "The Slantlet transform", author = "Selesnick, Ivan W.", journal = IEEETSP, volume = "47", number = "5", year = "1999", pages = "1304--1313", email = "seles@taco.poly.edu", keywords = "wavelet transform bases L(2)", abstract = "The discrete wavelet transform (DWT) is usually carried out by filterbank iteration; however, for a fixed number of zero moments, this does not yield a discrete-time basis that is optimal with respect to time localization. This paper discusses the implementation and properties of an orthogonal DWT, with two zero moments and with improved time localization. The basis is not based on filterbank iteration; instead, different filters are used for each scale. For coarse scales, the support of the discrete-time basis functions approaches two thirds that of the corresponding functions obtained by filterbank iteration, This basis, which is a special case of a class of bases described by Alpert, retains the octave-hand characteristic and is piecewise linear (but discontinuous). Closed-form expressions for the filters are given, an efficient implementation of the transform is described, and improvement in a denoising example is shown. This basis, being piecewise linear, is reminiscent of the slant transform, to which it is compared.", } @Article{ser-wal-per:properties, title = "Statistical Properties of the Wavelet Variance Estimator for Non-{G}aussian/Non-Linear time series", author = "Abdeslam Serroukh and Andrew T. Walden and Donald B. Percival", journal = JASA, volume = "95", number = "449", year = "2000", pages = "184--196", URL = "http://www.ma.ic.ac.uk/~atw/Wavar.ps.gz", } @Article{ser-wal:bivariate1, title = "Wavelet scale analysis of bivariate time series {I}: {M}otivation and estimation", author = "Abdeslam Serroukh and Andrew T. Walden", journal = JNS, volume = "", number = "", year = "2000", pages = "", note = "to appear", } @Article{ser-wal:bivariate2, title = "Wavelet scale analysis of bivariate time series {II}: {S}tatistical properties for linear processes", author = "Abdeslam Serroukh and Andrew T. Walden", journal = JNS, volume = "", number = "", year = "2000", pages = "", note = "to appear", } @Article{ses-cso-ste:exponential, title = "Tests for the Exponential Distribution using {K}olmogorov-type Statistics", author = "V. Seshadri and M. Cs{\"o}rg{\H o} and M. A. Stephens", journal = JRSSB, volume = "31", number = "3", year = "1969", pages = "499--509", } @Article{sha-hou:on-line, title = "An on-line wavelet transform for de-noising of high performance liquid chromatograms", author = "Xueguang Shao and Shunquan Hou", journal = "Analytical Letters", volume = "32", number = "12", year = "1999", pages = "2507--2520", } @Article{sha-xia:wavelet, title = "The Wavelet Analysis Method of Stationary Random Processes", author = "Luo Shaoming and Zhang Xiangwei", journal = "Applied Mathematics and Mechanics", volume = "19", number = "10", year = "1998", pages = "929--935", note = "English Edition", } @Article{sha-yen:exact, title = "On the exact values of orthonormal scaling coefficients of lengths 8 and 10", author = "Shann, W. C. and Yen, C. C.", journal = ACHA, volume = "6", number = "1", year = "1999", pages = "109--112", abstract = "We show the exact values of the scaling coefficients of length 8 and 10 for Daubechies' orthonormal scaling functions.", } @Article{she-rob-szu:optical, title = "Optical Wavelet Transform", author = "Yunlong Sheng and Donny Roberge and Harold H. Szu", journal = OE, volume = "31", number = "9", year = "1992", pages = "1840--1845", abstract = "The wavelet transform is implemented using an optical multichannel correlator with a bank of wavelet transform filters. This approach provides a shift-invariant wavelet transform with continuous translation and discrete dilation parameters. The wavelet transform filters can be in many cases simply optical transmittance masks. Experimental results show detection of the frequency transition of the input signal by the optical wavelet transform.", } @Article{she-str:asymptotics, title = "Asymptotics of {D}aubechies filters, scaling functions, and wavelets", author = "Shen, J. H. and Strang, G.", journal = ACHA, volume = "5", number = "3", year = "1998", pages = "312--331", abstract = "We study the asymptotic form as p --> infinity of the Daubechies orthogonal minimum phase filter h(p)[n], scaling function phi(p)(t), and wavelet w(p)(t). Kateb and Lemarie calculated the leading term in the phase of the frequency response H-p(omega). The infinite product <(phi)over cap>(p)(omega) = Pi H-p(omega/2(k)) leads us to a problem in stationary phase, for an oscillatory integral with parameter t. The leading terms change form with tau = t/p and we find three regions for phi(p)(tau): (1) An Airy function up to near tau(0): root 42 pi/p Ai(-root 42 pi p(2)(tau - tau(0))) + o(p(-1/3)) (2) An oscillating region root 2/pi pG'(omega(tau))cos [p(G((-1))(omega(tau)) - G(omega(tau))omega(tau)) + pi/4] + o(p(-1/2)) (3) A rapid decay after tau(1): (1/p pi)(1/(tau - tau(1)))sin[p(G((-1))(pi) - tau pi)] + o(p(-1)) The numbers tau(0) similar or equal to 0.1817 and tau(1) similar or equal to 0.3515 are known constants. The function G and its integral G((-1)) are independent of p. Regions 1 and 2 are matched over the interval p(-2/3) much less than tau - tau(0) much less than 1. The wavelets have a simpler asymptotic expression because the Airy wavefront is removed by the highpass filter. We also find the asymptotics of the impulse response h(p)[n] -a different function g(omega) controls the three regions. The difficulty throughout is to estimate the phase.", } @Article{she:dwt, title = "The discrete wavelet transform: {W}edding the \`{a} trous and {M}allat algorithms", author = "Mark J. Shensa", journal = IEEETSP, volume = "40", number = "10", year = "1992", pages = "2464--2482", abstract = "Two separately motivated implementations of the wavelet transform are brought together. It is observed that these algorithms are both special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by the choice of filters. In fact, the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact. Moreover, it is shown that the commonly used Lagrange a trous filters are in one-to-one correspondence with the convolutional squares of the Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, and conditions are derived under which it computes the continuous wavelet transform exactly. Suitable filter constraints for finite energy and boundedness of the discrete transform are also derived. Relevant signal processing parameters are examined, and it is observed that orthonormality is balanced by restrictions on resolution.", } @Article{she:inverse, title = "Discrete inverses for nonorthogonal wavelet transforms", author = "M. J. Shensa", journal = IEEETSP, volume = "44", number = "4", year = "1996", pages = "798--807", URL = "ftp://ftp.nosc.mil/pub/Shensa/WTinverse_TR1621.ps.Z", keywords = "discrete inverses, nonorthogonal wavelet transforms, signal processing, resolution, standard inversion procedure, finite expansion, algorithms, Morlet wavelets", abstract = "Discrete nonorthogonal wavelet transforms play an important role in signal processing by offering finer resolution in time and scale than their orthogonal counterparts. The standard inversion procedure for such transforms is a finite expansion in terms of the analyzing wavelet. While this approximation works quite well for many signals, it fails to achieve good accuracy or requires an excessive number of scales for others. This paper proposes several algorithms that provide more adequate inversion and compares them in the case of Morlet wavelets. In the process, both practical and theoretical issues for the inversion of nonorthogonal wavelet transforms are discussed.", } @Article{shu-fed:shannon, title = "Analysis and synthesis of $1/f$ processes via Shannon wavelets", author = "Shusterman, E. and Feder, M.", journal = IEEETSP, volume = "46", number = "6", year = "1998", pages = "1698--1702", keywords = "nonstationary processes 1/f noise spectral analysis wavelets", abstract = "1/f processes can he very useful in modeling processes with long-term correlation. We propose analysis and synthesis procedures to express these processes in terms of the Shannon wavelet. Unlike previous techniques, our analysis procedure generates uncorrelated decomposition coefficients for the 1/f process. This is done hy taking onto account, and then removing, the residual correlation between the wavelet components. The analysis procedure is the major contribution of this work. The proposed synthesis algorithm, which is a byproduct of the proposed analysis algorithm, is competitive with other techniques.", } @Article{sim-fre-ade-hee:shiftable, title = "Shiftable multiscale transforms", author = "Eero P. Simoncelli and William T. Freeman and Edward H. Adelson and David J. Heeger", journal = IEEETIT, volume = "38", number = "2", year = "1992", pages = "587--607", URL = "ftp://ftp.cis.upenn.edu/pub/eero/simoncelli91.ps.Z", keywords = "scale domain. orientation domain. signal analysis. multiscale transforms. orthogonal wavelet transforms. translation invariance. wavelet subbands. shiftability. spatial domain. sampling theorem. shiftable transforms. 1-D transform. jointly shiftable. scale-space analysis. stereo disparity measurement. image enhancement", abstract = "One of the major drawbacks of orthogonal wavelet transforms is their lack of translation invariance: the content of wavelet subbands is unstable under translations of the input signal. Wavelet transforms are also unstable with respect to dilations of the input signal and, in two dimensions, rotations of the input signal. The authors formalize these problems by defining a type of translation invariance called shiftability. In the spatial domain, shiftability corresponds to a lack of aliasing; thus, the conditions under which the property holds are specified by the sampling theorem. Shiftability may also be applied in the context of other domains, particularly orientation and scale. Jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored. Two examples of jointly shiftable transforms are designed and implemented: a 1-D transform that is jointly shiftable in position and scale, and a 2-D transform that is jointly shiftable in position and orientation. The usefulness of these image representations for scale-space analysis, stereo disparity measurement, and image enhancement is demonstrated.", } @InProceedings{sim-fre:steerable, title = "The Steerable Pyramid: {A} Flexible Architecture for Multi-Scale Derivative Computation", author = "Eero P. Simoncelli and William T. Freeman", booktitle = "International Conference on Image Processing", volume = "3", address = "23-26 Oct. 1995, Washington, DC, USA", month = oct, year = "1995", pages = "444--447", URL = "ftp://ftp.cis.upenn.edu/pub/eero/simoncelli95b.ps.Z", keywords = "flexible architecture multiscale derivative computation steerable pyramid linear image decomposition orientation subbands scale subbands basis functions directional derivative operators transform filters Fourier domain", abstract = "We describe an architecture for efficient and accurate linear decomposition of an image into scale and orientation subbands. The basis functions of this decomposition are directional derivative operators of any desired order. We describe the construction and implementation of the transform.", } @Article{sim-han-nes:determination, title = "Determination of the Hurst exponent by use of wavelet transforms", author = "Simonsen, I. and Hansen, A. and Nes, O. M.", journal = PRE, volume = "58", number = "3", year = "1998", pages = "2779--2787", keywords = "inverse fractal problem surfaces roughness growth", abstract = "We propose a method for (global) Hurst exponent determination based on wavelets. Using this method, we analyze synthetic data with predefined Hurst exponents, fracture surfaces, and data from economy. The results are compared to those obtained with Fourier spectral analysis. When many samples are available, the wavelet and Fourier methods are comparable in accuracy. However, when one or only a few samples are available, the wavelet method outperforms the Fourier method by a large margin.", } @TechReport{sli-etal:intraseasonal, title = "Intraseasonal oscillations in 15 atmospheric general circulation models (results from an {AMIP} diagnostic subproject)", author = "J. M. Slingo and K. R. Sperber and J. S. Boyle and J.-P. Ceron and M. Dix and B. Dugas and W. Ebisuzaki and J. Fyfe and D. Gregory and J.-F. Gueremy and J. Hack and A. Harzallah and P. Inness and A. Kitoh and W. K.-M. Lau and B. McAvaney and R. Madden and A. Matthews and T. N. Palmer and C.-K. Park and D. Randell and N. Renno", number = "661", institution = "World Meteorological Organization", year = "1995", } @Article{son-wu:multiscale, title = "Multiscale interrelations between air temperature in southeast {C}hina and {E}l {N}ino: {W}avelet analysis", author = "Sonechkin, D. M. and Wu, H. B.", journal = "Izvestiya Akademii Nauk Fizika Atmosfery I Okeana", volume = "35", number = "2", year = "1999", pages = "250--258", keywords = "seasonal cycle oscillation transform chaos", abstract = "Multiscale manifestations of the El Nino-Southern Oscillation in the dynamics of air temperature in various regions of the earth were examined, using southeast China as an example. The wavelet transforms of surface air temperature time series observed in Guangzhou (southeast China) and sea surface temperature time series observed in the Nine 1 + 2 and Nino 3 regions were analyzed. An intense interdecadal oscillation was revealed in these time series. The subsequent correlation of the transformed series separately for each time scale, which is a new technique designed to analyze interrelations of nonstationary stochastic processes, showed that the interrelations between these series are complex in character. This manifests itself in positive and negative correlations between the series on interannual and interdecadal scales, respectively. Since the interdecadal variability is determined. by a sequence of extremely intense El Nino events, it follows that the response of the Guangzhou temperature to intense El Nino is other than that to routine El Nino.", note = "In Russian", } @Article{spo:adaptive, title = "Adaptive hypothesis testing using wavelets", author = "V. G. Spokoiny", journal = AofS, volume = "24", number = "6", year = "1996", pages = "??--??", abstract = "", } @Article{sta-zee:filter-banks, title = "Two-dimensional orthogonal filter banks and wavelets with linear phase", author = "Stanhill, D. and Zeevi, Y. Y.", journal = IEEETSP, volume = "46", number = "1", year = "1998", pages = "183--190", keywords = "filter banks linear phase symmetry 2-D wavelets", URL = "ftp://ftp.technion.ac.il/pub/supported/ee/Signal_processing/2dlp.ps.gz", abstract = "Two-dimensional (2-D) compactly supported, orthogonal wavelets and filter banks having linear phase are presented, Two cases are discussed: wavelets with two-fold symmetry (centrosymmetric) and wavelets with four-fold symmetry that are symmetric (or anti-symmetric) about the vertical and horizontal axes. We show that imposing the requirement of linear phase in the case of order-factorable wavelets imposes a simple constraint on each of its polynomial order-1 factors, We thus obtain a simple and complete method of constructing orthogonal order-factorable wavelets with linear phase, This method is exemplified by design in the case of four-band separable sampling. An interesting result that is similar to the one well-known in the one-dimensional (1-D) case is obtained: Orthogonal order-factorable wavelets cannot be both continuous and have four-fold symmetry.", } @Article{sta-zee:wavelets, title = "Two-dimensional orthogonal wavelets with vanishing moments", author = "Stanhill, D. and Zeevi, Y. Y.", journal = IEEETSP, volume = "44", number = "10", year = "1996", pages = "2579--2590", keywords = "reconstruction filter banks regularity matrices design", URL = "ftp://ftp.technion.ac.il/pub/supported/ee/Signal_processing/2dvm.ps.gz", abstract = "We investigate a very general subset of 2-D, orthogonal, compactly supported wavelets, This subset includes all the wavelets with a corresponding wavelet (polyphase) matrix that can be factored as a product of factors of degree-1 in one variable. In this paper, we consider, in particular, wavelets with vanishing moments, The number of vanishing moments that can be achieved increases with the increase in the McMillan degrees of the wavelet matrix. We design wavelets with the maximal number of vanishing moments for given McMillan degrees by solving a set of nonlinear constraints on the free parameters defining the wavelet matrix and discuss their relation to regular, smooth wavelets, Design examples are given for two fundamental sampling schemes: the quincunx and the four-band separable sampling, The relation of the wavelets to the well-known 1-D Daubechies wavelets with vanishing moments is discussed.", } @InProceedings{sta-mur-bij:multi, title = "Multiresolution and Astronomical Image Processing", author = "J. L. Stark and F. Murtagh and A. Bijaoui", booktitle = "Astronomical Data Analysis Software and Systems IV", editor = "R. A. Shaw and H. E. Payne and J. J. E. Hayes", series = "ASP Conference Series", volume = "77", year = "1995", pages = "279--288", URL = "http://http.hq.eso.org/~fmurtagh/adass-jls.ps", abstract = "We present several wavelet transform algorithms and their applications in astronomical image processing (restoration, object detection, compression, etc.).", } @Article{sta-sie-gre:spectral, title = "Spectral analysis using the wavelet transform", author = "Starck J. L. and Siebenmorgen R. and Gredel R.", journal = ApJ, volume = "482", number = "2", pages = "1011--1020", year = "1997", URL = "http://www.journals.uchicago.edu/ApJ/cgi-bin/resolve?1997ApJ...482.1011SMPS", keywords = "infrared, stars. methods, observational. stars, individual (HH 100). stars, pre-main-sequence. techniques, spectroscopic.", abstract = "We introduce a new signal processing technique to analyze noisy spectra. The method is based on the wavelet transform and employs the a trous algorithm. Noise determination and detection criteria are discussed in detail, together with pitfalls related to the use of wavelets in the analysis of spectra. Simulations are presented to demonstrate the power and the shortcomings of our method. We apply our technique to the case of continuum sources that show superposed interstellar or circumstellar absorption or emission bands that are shallow and broad. In particular, we analyze an L-band spectrum of the Herbig-Haro energy source HH 100 IRS. The analysis indicates the presence of a shallow emission band near 3.51 mu m that is tentatively assigned to arise from aliphatic (CH2) vibrations.", } @Article{ste-pav-boj:nao, title = "Is the {N}orth {A}tlantic {O}scillation a random walk?", author = "Stephenson, D. B. and Pavan, V. and Bojariu, R.", journal = IJC, volume = "20", number = "1", year = "2000", pages = "1--18", abstract = "The North Atlantic Oscillation (NAO) is a major mode of large-scale climate variability which contains a broad spectrum of variations. There are substantial contributions from short-term 2-5 year variations, which have clearly marked teleconnections. Decadal trends are also apparent in the historical record of the NAO and may be due to either stochastic or deterministic processes. Evidence is presented that suggests the NAO exhibits 'long-range' dependence having winter values residually correlated over many years. Several simple stochastic models have been used to fit the NAO SLP (sea-level pressure) wintertime index over the period 1864-1998, and their performance at predicting the following year has been assessed. Long-range fractionally integrated noise provides a better fit than does either stationary red noise or a non-stationary random walk.", } @Article{ste:edf, title = "{EDF} Statistics for Goodness of Fit and Some Comparisons", author = "M. A. Stephens", journal = JASA, volume = "69", number = "347", pages = "730--737", year = "1974", keywords = "Normal distribution; Exponential distribution; Power; Anderson-Darling statistic; Cram{\'e}r-von Mises test; Kolmogorov-Smirnov test; EDF = empirical distribution function", } @Article{ste:estimation, title = "Estimation of the mean of a multivariate normal distribution", author = "Charles M. Stein", journal = AofS, volume = "9", number = "6", year = "1981", pages = "1135--1151", } @InCollection{ste:tests-edf, title = "Tests Based on {EDF} Statistics", author = "Michael A. Stephens", pages = "97--193", crossref = "dag-ste:goodness", note = "", URL = "", abstract = "", } @InCollection{ste:tests-exp, title = "Tests for the Exponential Distribution", author = "Michael A. Stephens", pages = "421--459", crossref = "dag-ste:goodness", note = "", URL = "", abstract = "", } @Article{ste:use, title = "Use of the {K}olmogorov--{S}mirnov, {C}ram{\'e}r--von {M}ises and related statistics without extensive tables", author = "Michael A. Stephens", journal = JRSSB, volume = "32", number = "1", year = "1970", pages = "115--122", } @Article{sto-lan-ngy:practical, title = "Practical synthesis of accurate fractal images", author = "M. A. Stosik and R. G. Lane and D. T. Nguyen", journal = "Graphical Models and Image Processing", volume = "57", number = "3", year = "1995", pages = "206--219", abstract = "This paper compares the synthesis of fractal images using both wavelets and a modified form of the random midpoint displacement algorithm. The accuracy of the generated fractal is investigated by an analysis of its second-order temporal statistics. It is shown that, although wavelets can be used to produce a good approximation to fractional Brownian motion, a technique based on the random midpoint displacement algorithm is in practice much simpler to implement, faster to generate, and results in a comparable accuracy. Furthermore the proposed method is shown to be considerably more efficient computationally.", } @Unpublished{sto:remarks, title = "Remarks on the Unsubsampled Wavelet Transform and the Lifting Scheme", author = "Alexander Stoffel", year = "1998", note = "Submitted to {\em Signal Processing}", } @Unpublished{str-etal:multiwavelet, title = "The Application of Multiwavelet Filter Banks to Image Processing", author = "V. Strela and P. N. Heller and G. Strang and P. Topiwala and C. Heil", year = "1995", note = "Submitted to {\em IEEE Transactions on Image Processing}", URL = "http://www-math.mit.edu/~gs/papers/mw_journal.ps.gz", } @Article{str-hah:mammograms, title = "Wavelet Transforms for Detecting Microcalcifications in Mammograms", author = "Robin N. Strickland and Hee Il Hahn", journal = IEEETMI, volume = "15", number = "2", year = "1996", pages = "218--229", email = "strickland@ece.arizona.edu", abstract = "Clusters of fine, granular microcalcifications in mammograms may be an early sign of disease. Individual grains are difficult to detect and segment due to size and shape variability and because the background mammogram texture is typically inhomogeneous. We develop a two-stage method based on wavelet transforms for detecting and segmenting calcifications. The first stage is based on an undecimated wavelet transform, which is simply the conventional filter bank implementation without downsampling, so that the low-low (LL), low-high (LH), high-low (HL), and high-high (HH) sub-bands remain at full size. Detection takes place in HH and the combination LH+HL. Four octaves are computed with two inter- octave voices for finer scale resolution. By appropriate selection of the wavelet basis the detection of microcalcifications in the relevant size range can be nearly optimized. In fact, the filters which transform the input image into HH and LH+HL are closely related to prewhitening matched filters for detecting Gaussian objects (idealized microcalcifications) in two common forms of Markov (background) noise. The second stage is designed to overcome the limitations of the simplistic Gaussian assumption and provides an accurate segmentation of calcification boundaries. Detected pixel sites in HH and LH+HL are dilated then weighted before computing the inverse wavelet transform. Individual microcalcifications are greatly enhanced in the output image, to the point where straightforward thresholding can be applied to segment them. FROC curves are computed from tests using a freely distributed database of digitized mammograms.", } @Book{str-ngu:book, author = "Gilbert Strang and Truong Nguyen", title = "Wavelets and Filter Banks", publisher = "Wellesley-Cambridge Press", year = "1996", address = "Wellesley, MA", URL = "http://saigon.ece.wisc.edu/~waveweb/Tutorials/overview.html", abstract = "This new textbook by Gilbert Strang and Truong Nguyen offers a clear and easy-to-understand introduction to two central ideas -- filter banks for discrete signals, and wavelets. The connections are fully explained -- the wavelet is determined by a choice of filter coefficients. All important wavelet properties (orthogonality or biorthogonality, symmetry, accuracy of approximation, and smoothness) come from specific properties of the filters. The text shows how to construct those filters and wavelets. The applications are very widespread -- and they continue to develop rapidly. The book gives a direct approach to signal processing and image processing through filter banks that iterate on the lowpass filter (this is the wavelet idea). Blocking and ringing artifacts are analyzed, along with many MATLAB applications. Wavelets and Filter Banks is written for the very broad audience that uses these ideas: Digital Signal Processing and Speech Processing, Image Processing including Medical Imaging, Scientific and Engineering Applications, Students and Professionals (wanting to understand wavelets!)", } @InCollection{str-sie:haar, title = "The {H}aar Wavelet Transform in the Time Series Similarity Paradigm", author = "Z. R. Struzik and A. P. J. M. Siebes", booktitle = "Principles of Data Mining and Knowledge Discovery", editor = "J. M. Zytkow and J. Rauch", publisher = "Springer-Verlag", address = "Berlin", pages = "12--22", year = "1999", email = "Zbigniew.Struzik@cwi.nl", abstract = "Similarity measures play an important role in many data mining algorithms. To allow the use of such algorithms on non-standard databases, such as databases of financial time series, their similarity measure has to be defined. We present a simple and powerful technique which allows for the rapid evaluation of similarity between time series in large data bases. It is based on the orthonormal decomposition of the time series into the Haar basis. We demonstrate that this approach is capable of providing estimates of the local slope of the time series in the sequence of multi-resolution steps. The Haar representation and a number of related represenations derived from it are suitable for direct comparison, e.g. evaluation of the correlation product. We demonstrate that the distance between such representations closely corresponds to the subjective feeling of similarity between the time series. In order to test the validity of subjective criteria, we test the records of currency exchanges, finding convincing levels of correlation.", } @InProceedings{str-sie:iwoss99, title = "Measuring Time Series' Similarity through Large Singular Features Revealed with Wavelet Transformation", author = "Z. R. Struzik and A. P. J. M. Siebes", booktitle = "Proceedings of the International Workshop on Database and Expert Systems Application", month = "September", publisher = "IEEE Computer Society Press", address = "Florence, Italy", year = "1999", pages = "162--166", abstract = "For the majority of data mining applications, there are no models of data which would facilitate the task of comparing records of time series. We propose a generic approach to comparing noise time series using the largest deviations from consistent statistical behaviour. For this purpose we use a powerful framework based on wavelet decomposition, which allows filtering polynomial bias, while capturing the essential singular behaviour. In addition, we are able to reveal scale-wise ranking of singular events including their scale free characteristic: the Hoelder exponent.", } @InProceedings{str-sie:pakdd98, title = "Wavelet Transform in Similarity Paradigm", author = "Z. R. Struzik and A. P. J. M. Siebes", booktitle = "Proceedings of the Pacific-Asia Conference on Knowledge Discovery and Data Mining", series = "Lecture Notes in Artificial Intelligence", volume = "1394", month = "April", pages = "295--309", year = "1998", address = "Melbourne, Australia", abstract = "Searching for similarity in time series finds still broader applications in data mining. However, due to the very broad spectrum of data involved, there is no possibility of defining one single notion of similarity suitable to serve all applications. We present a powerful framework based on wavelet decomposition, which allows designing and implementing a variety of criteria for the evaluation of similarity between time series. As an example, two main classes of similarity measures are considered. One is the global, statistical similarity, which uses the wavelet transform derived Hurst exponent to classify time series according to their global scaling properties. The second measure estimates similarity locally using the scale-position bifurcation representation.", } @TechReport{str-sie:paradigm1, title = "Wavelet transform in similarity paradigm {I}", author = "Zbigniew R. Struzik and Arno P. J. M. Siebes", number = "INS-R9802", institution = "CWI, Amsterdam, The Netherlands", year = "1998", abstract = "Searching for similarity in time series finds still broader applications in data mining. However, due to the very broad spectrum of data involved, there is no possibility of defining one single notion of similarity suitable to serve all applications.", } @TechReport{str-sie:paradigm2, title = "Wavelet transform in similarity paradigm {II}", author = "Zbigniew R. Struzik and Arno P. J. M. Siebes", number = "INS-R9815", institution = "CWI, Amsterdam, The Netherlands", year = "1998", abstract = "For the majority of data mining applications, there are no models of data which would facilitate the tasks of comparing records of time series, thus leaving one with `noise' as the only description. We propose a generic approach to comparing noise time series using the largest deviations from consistent statistical behaviour. For this purpose we use a powerful framework based on wavelet decomposition, which allows filtering polynomial bias, while capturing the essential singular behaviour. In particular we are able to reveal scale-wise ranking of singular events including their scale-free characteristic: the H{\"o}lder exponent. We use such characteristics to design a compact representation of the time series suitable for direct comparison, e.g. evaluation of the correlation product. We demonstrate that the distance between such representations closely corresponds to the subjective feeling of similarity between the time series. In order to test the validity of subjective criteria, we test the records of currency exchanges, finding convincing levels of (local) correlation.", } @Article{str-str:multiwavelets, title = "Orthogonal multiwavelets with vanishing moments", author = "Gilbert Strang and Vasily Strela", journal = OE, volume = "33", number = "7", year = "1994", pages = "2104--2107", email = "gs@math.mit.edu", URL = "", keywords = "vanishing moments dilation equation scaling function low-pass filter high-pass filter matrices vectors translation finite energy matrix coefficients piecewise linear orthogonal wavelets orthogonal multiwavelets", abstract = "A scaling function is the solution to a dilation equation Phi(t)= Sigma c/sub k/ Phi (2t-k), in which the coefficients come from a low-pass filter. The coefficients in the wavelet W(t)= Sigma d/sub k/ Phi (2t-k) come from a high-pass filter. When these coefficients are matrices, Phi and W are vectors: there are two or more scaling functions and an equal number of wavelets. By dilation and translation of the wavelets, we have an orthogonal basis W/sub ijk/=W/sub i/(2/sup j/t-k) for all functions of finite energy. These ``multiwavelets'' open new possibilities. They can be shorter, with more vanishing moments, than single wavelets. They can be symmetric, which is impossible for scalar wavelets (except for Haar's). We determine the conditions to impose on the matrix coefficients c/sub k/ in the design of multiwavelets, and we construct a new pair of piecewise linear orthogonal wavelets with two vanishing moments.", } @TechReport{str-wal:denoising, title = "Signal and Image denoising via wavelet thresholding: {O}rthogonal and biorthogonal, scalar and multiple wavelet transforms", author = "Vasily Strela and Andrew T. Walden", number = "TR-98-01", institution = "Statistics Section, Department of Mathematics, Imperial College of Science, Technology \& Medicine", year = "1998", } @InProceedings{str-wal:orthogonal, title = "Orthogonal and biorthogonal multiwavelets for signal denoising and image compression", author = "Vasily Strela and Andrew T. Walden", crossref = "szu:wavelet5", } @Article{str:brief, title = "Wavelets and Dilation Equations: {A} Brief Introduction", author = "Gilbert Strang", journal = SIREV, volume = "31", number = "4", year = "1989", pages = "614--627", URL = "http://www-math.mit.edu/~gs/papers/siamrev.ps.gz", abstract = "This is an introduction to the construction of wavelets from the solution to a dilation equation. It discusses the approximation and orthogonal properties of wavelets and describes the recursive algorithms that decompose and reconstruct a function. The object of wavelets is to localize as far as possible in both time and frequency, with efficient algorithms.", } @Unpublished{str:creating, title = "Creating and Comparing Wavelets", author = "Gilbert Strang", year = "1996", note = "Department of Mathematics, Massachusetts Institute of Technology", URL = "http://www-math.mit.edu/~gs/papers/dundee.ps.gz", email = "gs@math.mit.edu", } @InCollection{str:ftae99, title = "Local Effective H{\"o}der Exponent Estimation on the Wavelet Transform Maxima Tree", author = "Z. R. Struzik", booktitle = "Fractals: Theory and Applications in Engineering", editor = "Michel Dekking and Jacques L\'{e}vy V\'{e}hel and Evelyne Lutton and Claude Tricot", address = NY, publisher = "Springer-Verlag", pages = "93--112", year = "1999", abstract = "We present a robust method of estimating an effective Holder exponent locally at an arbitrary resolution. The method is motivated by the multiplicative cascade paradigm, and implemented on the hierarchy of singularities revealed with the wavelet transform modulus maxima tree. In addition, we illustrate the possibility of the direct estimation of the scaling spectrum of the effective Holder exponent, and we link it to the established partition functions based multifractal formalism. We motivate both the local and the global multifractal analysis by showing examples of computer generated and real life time series.", } @Article{str:make, title = "How to Make Wavelets", author = "Robert S. Strichartz", journal = AMM, volume = "100", number = "6", year = "1993", pages = "539--557", keywords = "Fourier analysis, Matrices, Norms, Matrix mechanics", abstract = "Wavelet bases where Haar functions are constructed from a single function by dyadic dilations and integer translations are considered as approximate definitions of a wavelet expansion. First, a scaling function and associated multiresolution analysis are chosen. The orthonormality conditions should be satisfied by generation of a multiresolution analysis of the function. The wavelets are then constructed by solving two algebraic identities and establishing the properties of the wavelet functions.", } @PhdThesis{str:thesis, title = "Multiwavelets: Theory and Applications", author = "Vasily Strela", school = "Massachusetts Institute of Technology", year = "1996", URL = "http://pascal.dartmouth.edu/~strela/vvthmain.ps", } @Article{str:versus, title = "Wavelet transforms versus {F}ourier transforms", author = "Gilbert Strang", journal = BAMathS, volume = "28", number = "2", year = "1993", pages = "288--305", abstract = "An orthogonal basis for piecewise constant functions is constructed by dilation and translation. The wavelength transform maps each function to its coefficients with respect to this basis. The approximation is found to be poor and is improved by dilation equations. Higher-order wavelets are constructed and indirect and recursive methods are used to compute them. The practicality of the wavelet transform and Fourier transform in signal processing are discussed.", } @Article{str:wavelets, title = "Wavelets", author = "G. Strang", journal = "American Scientist", volume = "82", year = "1994", pages = "250--255", URL = "http://www-math.mit.edu/~gs/papers/amsci.ps.gz", abstract = "The transformation of signals into a sum of small, overlapping waves offers a new method for analyzing, storing and transmitting information. The author discusses: Fourier and wavelet transforms; choosing the best basis; higher dimensions; fast wavelet transform; Daubechies wavelets; high-definition television; the future of fingerprints.", } @Unpublished{swe-sch:building, title = "Building Your Own Wavelets at Home", author = "Wim Sweldens and Peter Schr{\"o}der", year = "1996", note = "In ``Wavelets in Computer Graphics'', ACM SIGGRAPH Course Notes", URL = "http://cm.bell-labs.com/who/wim/papers/athome.ps.gz", abstract = "We give an practical overview of three simple techniques to construct wavelets under general circumstances: interpolating subdivision, average interpolation, and lifting. We include examples concerning the construction of wavelets on an interval, weighted wavelets, and wavelets adapted to irregular samples.", } @Article{swe:future, author = "W. Sweldens", title = "Wavelets: {W}hat Next?", journal = "Proc. IEEE", volume = "84", number = "4", pages = "680--685", year = "1996", URL = "http://cm.bell-labs.com/who/wim/papers/future.ps", abstract = "In this concluding article, we want to look ahead and see what the future can bring to wavelet research. We try to find a common denominator for ``wavelets'' and identify promising research directions and challenging problems.", } @Article{swe:lift1, author = "W. Sweldens", title = "The lifting scheme: {A} custom-design construction of biorthogonal wavelets", journal = "Appl. Comput. Harmon. Anal.", volume = "3", number = "2", pages = "186--200", year = "1996", URL = "http://cm.bell-labs.com/who/wim/papers/lift1.ps", abstract = "We present the lifting scheme, a new idea of constructing compactly supported wavelets with compactly supported duals. The lifting scheme provides a simple relationship between all multiresolution analyses with the same scaling function. It isolates the degrees of freedom remaining after fixing the biorthogonality relations. Then one has full control over these degrees of freedom to custom-design the wavelet for a particular application. It also leads to a faster implementation of the fast wavelet transform. We illustrate the use of the lifting scheme in the construction of wavelets with interpolating scaling functions.", } @TechReport{swe:lift2, author = "W. Sweldens", title = "The lifting scheme: {A} construction of second generation wavelets", institution = "Department of Mathematics, University of South Carolina", number = "1995:6", year = "1995", URL = "http://cm.bell-labs.com/who/wim/papers/lift2.ps", abstract = "We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included.", } @InProceedings{swe:spie95, author = "W. Sweldens", title = "The Lifting Scheme: {A} New Philosophy in Biorthogonal Wavelet Constructions", pages = "68--79", crossref = "lai-uns:wavelet3", URL = "http://cm.bell-labs.com/who/wim/papers/spie95.ps", abstract = "In this paper we present the basic idea behind the lifting scheme, a new construction of biorthogonal wavelets which does not use the Fourier transform. In contrast with earlier papers we introduce lifting purely from a wavelet transform point of view and only consider the wavelet basis functions in a later stage. We show how lifting leads to a faster, fully in-place implementation of the wavelet transform. Moreover, it can be used in the construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one function. A typical example of the latter are wavelets on the sphere.", } @Article{sza-gal-kis:application2, title = "{A}pplication of wavelet analysis in variable star research. {II}. {T}he semiregular star {V} {B}ootis", journal = AandA, volume = "308", number = "3", year = "1996", pages = "791--8", author = "K. Szatmary and J. Gal and L. L. Kiss", abstract = "For pt.I see Astron. Astrophys. Suppl. Ser., vol.108, no.2, p.377-94 (1994). Light curve analysis of the SRa-type variable V Boo is presented and discussed. The periods are determined and the stability of these periods as well as their amplitudes are investigated with wavelet analysis. The amplitude decrease is studied with the so-called ridge procedure, which shows that the amplitude of the longer period strongly decreased while the amplitude of the shorter period seems to remain stable. The possible interpretations of this effect are discussed. Using theoretical models and observational relations physical parameters and pulsational modes of V Boo are also estimated.", keywords = "wavelet analysis, variable star research, light curve analysis, sra type variable, periods, period stability, period amplitudes, amplitude decrease, ridge procedure, interpretations, theoretical models, observational relations, physical parameters, pulsational modes.", } @Article{sza-vin-gal:application1, title = "{A}pplication of wavelet analysis in variable star research. {I}. {P}roperties of the wavelet map of simulated variable star light curves", journal = "AASS", volume = "108", number = "2", year = "1994", pages = "377--394", author = "K. Szatmary and J. Vinko and J. Gal", email = "K.Szatmary@physx.u-szeged.hu, vinko@physx.u-szeged.hu", abstract = "A type of the relatively new time-frequency method, the wavelet analysis is studied. Some results of testing this method are presented. The test data series were defined so that they show similarities with the light variations of variable stars. The effects of observational noise and irregularities in data sampling are pointed out. The wavelet analysis seems to be a suitable method for detecting the local behaviour of the light curves, e.g. phase jump or mode switching. The investigation of time-dependent phenomena, e.g. amplitude or frequency modulation, is more available than in the case of standard Fourier analysis. In order to interpret the real wavelet maps of variable stars it is necessary to take into account the properties of the method presented by similar tests.", keywords = "stellar photometry, measurement technique, variable star light curve, wavelet analysis, variable star, wavelet map, light curve, time-frequency method, observational noise, irregularities, method.", } @Article{szi-kat-par-alb-cah:local, title = "The local effect of intermittency on the inertial subrange energy spectrum of the atmospheric surface layer", author = "Jozsef Szilagyi and Gabriel G. Katul and Marc B. Parlange and John D. Albertson and Anthony T. Cahill", journal = BLM, volume = "79", number = "1--2", year = "1996", pages = "35--50", abstract = "Orthonormal wavelet expansions are applied to atmospheric surface layer velocity measurements. The effect of intermittent events on the energy spectrum of the inertial subrange is investigated through analysis of wavelet coefficients. The local nature of the orthonormal wavelet transform in physical space makes it possible to identify a relationship between the inertial subrange slope of the local wavelet spectrum and a simple indicator (i.e. the local variance of the signal) of local intermittency buildup. The slope of the local wavelet energy spectrum in the inertial subrange is shown to be sensitive to the presence of intermittent events. During well-developed intermittent events (coherent structures), the slope of the energy spectrum is somewhat steeper than -5/3, while in less active regions the slope is found to be flatter than -5/3. When the slopes of local wavelet spectra are ensemble averaged, a slope of -5/3 is recovered for the inertial subrange.", } @Article{tac:differentiation, title = "The differentiation by a wavelet and its application to the estimation of a transfer function", author = "Tachibana, Y.", journal = IEICETFECCS, volume = "E81A", number = "6", year = "1998", pages = "1194--1200", keywords = "wavelet digital filter differential filters digital signal processing identification parameters estimation step response", abstract = "This paper deals with a set of differential operators for calculating the differentials of an observed signal by the Daubechies wavelet and its application for the estimation of the transfer function of a linear system by using non-stationary step-like signals. The differential operators are constructed by iterative projections of the differential of the scaling function for a multiresolution analysis into a dilation subspace. By the proposed differential operators we can extract the arbitrary order differentials of a signal. We propose a set of identifiable filters constructed by the sum of multiple filters with the first order lag characteristics. Using the above differentials and the identifiable filters we propose an identification method for the transfer function of a linear system. In order to ensure the appropriateness and effectiveness of the proposed method some numerical simulations are presented.", } @InProceedings{tas:near-best, title = "Near-best basis selection algorithms with non-additive information cost functions", author = "Taswell, C.", booktitle = "Proceedings of the IEEE-SP International Symposium on Time- Frequency and Time-Scale Analysis", editor = "M. G. Amin", publisher = "IEEE Press", address = "Philadelphia, PA", year = "1994", pages = "13--16", URL = "http://www.wavebox.com/NBBSA94.ps", keywords = "near-best basis selection algorithms non-additive information cost functions search algorithms signal decomposition decision criteria wavelet packet transforms additive information costs matching pursuit decomposition time- frequency analysis data compression", abstract = "Search algorithms for finding signal decompositions called near-best bases using decision criteria called non-additive information costs are proposed for selecting bases in wavelet packet transforms. These new methods are compared with the best bases and additive information costs of Coifman and Wickerhauser (see IEEE Trans. Information Theory, vol.38, p.713-18, 1992). All near-best and best bases were also compared with the matching pursuit decomposition of Mallat and Zhang (see IEEE Trans. Signal Processing, vol.41, p.3397-3415, 1993). Preliminary experiments suggest that for the application of time-frequency analysis, a wide variety of results can be obtained with the different methods, and that for the application of data compression, the near-best basis selected with non-additive costs may outperform the best basis selected with additive costs.", } @Article{tas:satisficing, title = "Satisficing search algorithms for selecting near-best bases in adaptive tree-structured wavelet transforms", author = "Taswell, C.", journal = IEEETSP, volume = "44", number = "10", year = "1996", pages = "2423--2438", URL = "http://www.wavebox.com/SSANBB96.ps", keywords = "image compression speech", abstract = "Satisficing search algorithms are proposed for adaptively selecting near-best basis and near-best frame decompositions in redundant tree-structured wavelet transforms, Any of a variety of additive or nonadditive information cost functions can be used as the decision criterion for comparing and selecting nodes when searching through the tree,The algorithms are applicable to tree-structured transforms generated by any kind of wavelet whether orthogonal, biorthogonal, or nonorthogonal, These satisficing search algorithms implement suboptimizing rather than optimizing principles, and acquire the important advantage of reduced computational complexity with significant savings in memory, flops, and time, Despite the suboptimal approach, top-down tree-search algorithms with additive or nonadditive costs that yield near-best bases can be considered, in certain important and practical situations, better than bottom-up tree-search algorithms with additive costs that yield best bases, Here, ''better than'' means that, effectively, the same level of performance can be attained for a relative fraction of the computational work, Experimental results comparing the various information cost functions and basis selection methods are demonstrated for both data compression of real speech and time- frequency analysis of artificial transients.", } @InCollection{tas:top-down, title = "Top-Down and Bottom-Up Tree Search Algorithms for Selecting Bases in Wavelet Packet Transforms", author = "Carl Taswell", crossref = "ant-opp:wavelets", pages = "???--???", URL = "http://www.wavebox.com/TDBUTSA94.ps", keywords = "", abstract = "Search algorithms for finding signal decompositions called near-best bases using decision criteria called non-additive information costs have recently been proposed by Taswell for selecting bases in wavelet packet transforms represented as binary trees. These methods are extended here to distinguish between top-down and bottom-up tree searches. Other new non-additive information cost functions are also proposed. In particular, the near-best basis with the non-additive cost of the Shannon entropy on probabilities is compared against the best basis with the additive cost of the Coifman-Wickerhauser entropy on energies. All wavelet packet basis decompositions are also compared with the nonorthogonal matching pursuit decomposition of Mallat and Zhang and the orthogonal matching pursuit decomposition of Pati et al. Monte Carlo experiments using a constant-bit-rate variable-distortion paradigm for lossy compression suggest that the statistical performance of top-down near-best bases with non-additive costs is superior to that of bottom-up best bases with additive costs. Top-down near-best bases provide a significant increase in computational efficiency with reductions in memory, flops, and time while nevertheless maintaining similar coding efficiency with comparable reconstruction errors measured by l^p-norms. Finally, a new compression scheme called parameterized model coding is introduced and demonstrated with results showing better compression than standard scalar quantization coding at comparable levels of distortion.", } @Book{teo:computational, title = "Computational Signal Processing with Wavelets", author = "A. Teolis", publisher = "Springer-Verlag", year = "1997", pages = "332", keywords = "Introduction * Mathematical Preliminaries * Signal Representation and Frames * Continuous Wavelet Transform * Discrete Wavelet Transform * Non-orthogonal Wavelet Transform * Wavelet Signal Processing * World Wide Web Access", abstract = "Computational Signal Processing with Wavelets examines both theoretical and practical aspects of computational signal processing using wavelets. Theoretically, an emphasis is placed on balancing the accessibility of the material with the level of mathematical rigor which sacrifices as little as possible of both. Computationally, wavelet signal processing algorithms are presented and applied to signal compression, noise suppression, and signal identification. Numerical illustrations of these computational techniques are further provided with interactive software (MATLAB) via an internet accessible WEB site. Starting from basic principles of signal representation with atomic functions, a mathematically well founded theory of the discretization of analog signals is developed. General families are specialized to wavelet families and the discrete representation are specialized to generally non-orthogonal wavelet transforms. The theory leads naturally to the computer implementation of the non-orthogonal wavelet transform. Specific topics covered include general signal representation, continuous and discrete Fourier transforms, orthonormal and biorthogonal bases, frames, wavelet frames and frame reconstruction, discrete wavelet transform, multi-resolution analysis, orthonormal wavelets, continuous wavelet transform, non-orthogonal wavelet transform, and wavelet based signal processing algorithms for compression, noise suppression, and identification. The discussion is at the level of a senior or beginning graduate student level and is accessible to signal processing professionals and practicioners. Dissemination of the material is provided by a hybrid combination of traditional (text) and non-traditional (internet and electronic) media.", } @Article{tet-kri:ocean, title = "{SAR} Ocean Image Representation Using Wavelets", author = "Joseph G. Teti and H. N. Kritikos", journal = IEEETGRS, volume = "30", number = "5", year = "1992", pages = "1089--1094", abstract = "The utility of wavelet analysis as a tool for geophysical research is examined using both continuous and discrete versions of the wavelet transform. In both cases, waveform decomposition and reconstruction is possible using somewhat different computational procedures. The theoretical background of each procedure is briefly described and applied using a specific 'wavelet'. The wavelet used is based on a Gaussian function, and provides simultaneous time-frequency (or space-wavenumber) localization that meets the lower limit of the uncertainty principle. A representation of this type is ideally suited for the analysis of waveforms that arise from nonstationary processes. The properties of wavelet analysis are examined by expanding an FM-chirp waveform and azimuth cuts taken from two different SAR ocean images. The performance and ease of implementation are compared for the continuous and discrete formulations, and the effects of filtering in wavelet phase space using the discrete case are also examined.", } @Article{tew-kim:correlation, title = "Correlation Structure of the Discrete Wavelet Coefficients of Fractional {B}rownian Motion", author = "A. H. Tewfik and M. Kim", journal = IEEETIT, volume = "38", number = "2", year = "1992", pages = "904--909", abstract = "It is shown that the discrete wavelet coefficients of fractional Brownian motion at different scales are correlated and that their auto- and cross-correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional Brownian motion itself. The rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet.", } @Book{tit:book, title = "The Theory of Functions", author = "E. C. Titchmarsh", publisher = "Oxford University Press", address = "Oxford", edition = "2", year = "1939", pages = "454", } @Article{tor-com:practical, title = "A Practical Guide to Wavelet Analysis", author = "Christopher Torrence and Gilbert P. Compo", journal = BAMetS, volume = "79", number = "1", year = "1998", pages = "61--78", URL = "http://paos.colorado.edu/research/wavelets/", keywords = "", abstract = "A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino-Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmöller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of changes in ENSO variance on interdecadal timescales. Using new datasets that extend back to 1871, the Nino3 sea surface temperature and the Southern Oscillation index show significantly higher power during 1880-1920 and 1960-90, and lower power during 1920-60, as well as a possible 15-yr modulation of variance. The power Hovmöller of sea level pressure shows significant variations in 2--8-yr wavelet power in both longitude and time.", } @Unpublished{tor-web:interdecadal, title = "Interdecadal Changes in the {ENSO}-Monsoon System", author = "Christopher Torrence and Peter J. Webster", journal = JC, volume = "12", number = "8", year = "1999", pages = "2679--2690", URL = "http://www.cgd.ucar.edu/~torrence/interdec/", keywords = "", abstract = "The El Nino-Southern Oscillation (ENSO) and Indian monsoon are shown to have undergone significant interdecadal changes in variance and coherency over the last 125 years. Wavelet analysis is applied to indexes of equatorial Pacific sea surface temperature (Nino3 SST), the Southern Oscillation index, and all-India rainfall. Time series of 2-7-yr variance indicate intervals of high ENSO-monsoon variance (1875-1920 and 1960-90) and an interval of low variance (1920-60). The ENSO-monsoon variance also contains a modulation of ENSO-monsoon amplitudes on a 12-20-yr timescale.The annual-cycle (1 yr) variance time series of Nino3 SST and Indian rainfall is negatively correlated with the interannual ENSO signal. The 1-yr variance is larger during 1935-60, suggesting a negative correlation between annual-cycle variance and ENSO variance on interdecadal timescales.The method of wavelet coherency is applied to the ENSO and monsoon indexes. The Nino3 SST and Indian rainfall are found to be highly coherent, especially during intervals of high variance. The Nino3 SST and Indian rainfall are approximately 180 degrees out of phase and show a gradual increase in phase difference versus Fourier period. All of the results are shown to be robust with respect to different datasets and analysis methods.", } @Article{tor-web:persistence, title = "The Annual Cycle of Persistence in the {E}l {N}i\~no-{S}outhern {O}scillation", author = "Christopher Torrence and Peter J. Webster", journal = QJRMS, volume = "125", number = "", year = "1998", pages = "1985--2004", URL = "http://www.cgd.ucar.edu/~torrence/barrier/", keywords = "El Ni\~no-Southern Oscillation, Predictability, Interdecadal Variability", abstract = "A spring `predictability barrier' exists in both data and models of the El Ni\~no-Southern Oscillation (ENSO) phenomenon. In statistical analyses this barrier manifests itself as a drop-off in monthly persistence (lagged correlation) while in coupled ocean-atmosphere models it appears as a decrease in forecast skill. The persistence barrier for ENSO indices is investigated using historical sea surface temperature and sea level pressure data. Simple statistical models are used to show that the persistence barrier occurs because the boreal spring is the transition time from one climate state to another, when the `signal-to-noise' of the system is lowest and the system is most susceptible to perturbations. The strength of the persistence barrier is shown to depend on the degree of phase locking of the ENSO to the annual cycle. The phase locking of the ENSO to the annual cycle, as well as the ENSO variance, is shown to vary on interdecadal time scales. During 1871--1920 and 1960--1990 the ENSO variance was high, while during 1920--1950 the ENSO variance was low. Using wavelet analysis, this interdecadal variability in ENSO is shown to be correlated with changes in Indian summer monsoon strength. Finally, the change in persistence barrier strength between 1960--1979 and 1980--1995 is related to changes in the phase locking of ENSO to the annual cycle. These changes in persistence and phase locking appear to be related to the increased forecast skill seen in recent coupled ocean-atmosphere models.", } @PhdThesis{tor:thesis, title = "The El Ni\~no-Southern Oscillation: Interannual Predictability and Interdecadal Variability", author = "Christopher Torrence", school = "University of Colorado at Boulder", year = "1997", } @Article{tre-and:turbulence, title = "On wavelet analysis of nonstationary turbulence", author = "Beorge Trevi{\~n}o and Edgar L. Andreas", journal = BLM, volume = "81", number = "3-4", year = "1996", pages = "271--288", keywords = "coherent structures, forest canopy, transforms, cascade, motions", abstract = "Wavelets are new tools for turbulence analysis that are yielding important insights into boundary-layer processes. Wavelet analysis, however, has some as yet undiscussed limitations: failure to recognize these can lead to misinterpretation of wavelet analysis results. Here we discuss some limitations of wavelet analysis when applied to nonstationary turbulence. Our main point is that the analysis wavelet must be carefully matched to the phenomenon of interest, because wavelet coefficients obscure significant information in the signal being analyzed. For example, a wavelet that is a second-difference operator can provide no information on the linear trend in a turbulence signal. Wavelet analysis also yields no meaningful information about nonlinear behavior in a signal - contrary to claims in the literature - because, at any instant, a wavelet is a single-scale operator, while nonlinearity involves instantaneous interactions among many scales.", } @InCollection{tri:adaptive, title = "Adaptive Density Estimation", author = "K. Tribouley", pages = "385--395", crossref = "ant-opp:wavelets", URL = "", abstract = "", } @Article{tri:practical, title = "Practical estimation of multivariate densities using wavelet methods", author = "K. Tribouley", journal = SN, volume = "49", number = "1", year = "1995", pages = "41--62", abstract = "This paper describes a practical method for estimating multivariate densities using wavelets. As in kernel methods, wavelet methods depend on two types of parameters. On the one hand we have a functional parameter: the wavelet [phi] (comparable to the kernel K) and on the other hand we have a smoothing parameter: the resolution index (comparable to the bandwidth h). Classically, we determine the resolution index with a cross-validation method. The advantage of wavelet methods compared to kernel methods is that we have a technique for choosing the wavelet [phi] among a fixed family. Moreover, the wavelets method simplifies significantly both the theoretical and the practical computations.", keywords = "Density estimation cross validation wavelet orthonormal bases Besov spaces", } @Article{tso-kum-els-tso:dna, journal = PRE, volume = "53", number = "2", year = "1997", pages = "1828--1834", title = "{W}avelet analysis of {DNA} sequences", author = "A. A. Tsonis and P. Kumar and J. B. Elsner and P. A. Tsonis", abstract = "In this paper we use wavelet analysis in order to probe the localized structure of DNA sequences. We demonstrate that, unlike other conventional approaches, wavelets are able to decompose seemingly homogeneous regions in noncoding sequences into smaller distinct regions that obey their own repetition and construction rules. The significance of this result to gene evolution is discussed.", keywords = "coding sequences, sampling theory, evolution, propagation, gene", } @Article{tur-hal:interpolation, title = "Interpolation Methods for Nonlinear Wavelet Regression with Irregularly Spaced Design", author = "Berwin A. Turlach and Peter Hall", journal = "AS", volume = "25", number = "5", year = "1997", email = "berwin@alphasun.anu.edu.au", abstract = "We suggest and discuss interpolation methods that enable nonlinear wavelet estimators to be employed with stochastic design, or non-dyadic regular design, in problems of nonparametric regression. This approach allows relatively rapid computation, involving dyadic approximations to wavelet-after-interpolation techniques. New types of interpolation are described, enabling first-order variance reduction at the expense of second-order increases in bias. The effect of interpolation on threshold choice is addressed, and appropriate thresholds are suggested for error distributions with as few as four finite moments. A concise account of mean squared error properties is given for interpolation-based wavelet estimators applied to piecewise-smooth functions.", } @Article{uns-ald:biomedical, journal = PIEEE, volume = "84", number = "4", year = "1996", pages = "626--638", title = "{A} review of wavelets in biomedical applications", author = "M. Unser and A. Aldroubi", abstract = "In this paper we present an overview of the various uses of the wavelet transform (WT) in medicine and biology. We start by describing the wavelet properties that are the most important for biomedical applications. In particular, we provide an interpretation of the continuous wavelet transform (CWT) as a prewhitening multiscale matched filter. Me also briefly indicate the analogy between the WT and some of the biological processing that occurs in the early components of the auditory and visual system. We then review the rises of the WT for the analysis of 1-D physiological signals obtained by phonocardiography, electrocardiography (ECC), and electroencephalography (EEG), including evoked response Next, we provide a survey of recent wavelet developments in medical imaging. These include biomedical image processing algorithms (e.g., noise reduction, image enhancement, and detection of microcalcifications in mammograms), image reconstruction and acquisition schemes (tomography, and magnetic resonance imaging (MRI)), and multiresolution methods for the registration and statistical analysis of functional images of the brain (positron emission tomography (PET) and functional MRI (fMRI)). In each case, we provide the reader with some general background information and a brief explanation of how the methods work.", } @Article{uns-the-ald:shift, title = "Shift-orthogonal wavelet bases", author = "Unser, M. and Thevenaz, P. and Aldroubi, A.", journal = IEEETSP, volume = "46", number = "7", year = "1998", pages = "1827--1836", keywords = "multiresolution approximations ondelettes spaces", abstract = "Shift-orthogonal wavelets are a new type of multiresolution wavelet bases that are orthogonal with respect to translation (or shifts) within one level but not with respect to dilations across scales. In this paper, we characterize these wavelets and investigate their main properties by considering two general construction methods. In the first approach, we start by specifying the analysis and synthesis function spaces and obtain the corresponding shift-orthogonal basis functions by suitable orthogonalization. In the second approach, we take the complementary view and start from the digital filterbank. We present several illustrative examples, including a hybrid version of the Battle-Lemarie spline wavelets. We also provide filterbank formulas for the fast wavelet algorithm. A shift-orthogonal wavelet transform is closely related to an orthogonal transform that uses the same primary scaling function; both transforms have essentially the same approximation properties. One experimentally confirmed benefit of relaxing the interscale orthogonality requirement is that we can design wavelets that decay faster than their orthogonal counterpart.", } @TechReport{uyt-bul:red-black, title = "The Red-Black Wavelet Transform", author = "Geert Uytterhoeven and Adhemar Bultheel", number = "271", institution = "Department of Computer Science, Katholieke Universiteit Leuven", year = "1997", URL = "http://www.cs.kuleuven.ac.be/publicaties/rapporten/tw/TW271.ps.gz", } @TechReport{van-bro-fea:long-memory, title = "Wavelet Analysis of Long-memory Processes", author = "Marina Vannucci and Philip J. Brown and Tom Fearn", institution = "Institute of Mathematics and Statistics, University of Kent at Canterbury", note = "UKC/IMS/98/22", year = "1998", email = "M.Vannucci@ukc.ac.uk", URL = "", } @Article{van-cor:covariance, title = "Covariance Structure of Wavelet Coefficients: {T}heory and Models in a Bayesian Perspective", author = "Marina Vannucci and Fabio Corradi", journal = JRSSB, volume = "?", number = "?", year = "1999", pages = "???--???", email = "mvannucci@stat.tamu.edu, corradi@stat.ds.unifi.it", URL = "http://stat.tamu.edu/~mvannucci/webpages/papers/wjrssb.ps", keywords = "", abstract = "", } @InCollection{van-cor:dependence, title = "Modeling Dependence in the Wavelet Domain", author = "Marina Vannucci and Fabio Corradi", pages = "???--???", crossref = "mul-vid:biwbm", URL = "http://stat.tamu.edu/~mvannucci/webpages/papers/deprior.ps", keywords = "", abstract = "", } @Article{van-vid:preventing, title = "Preventing the {D}irac disaster: {W}avelet based density estimation", author = "Marina Vannucci and Brani Vidakovic", journal = JISS, volume = "6", number = "2", year = "1999", pages = "???--???", URL = "http://stat.tamu.edu/~mvannucci/webpages/paper/wjiss.ps", } @PhdThesis{van:thesis, title = "On the Application of Wavelets in Statistics", author = "Marina Vannucci", school = "Dipartimento di Statistica ``G. Parenti'', University of Florence, Italy", year = "1996", postscript = "http://www.isds.duke.edu/~brani/wp/marina.ps", note = "In Italian", } @Unpublished{vei-abr:constancy, title = "A Statistical Test for the Time Constancy of Scaling Exponents", author = "Darryl Veitch and Patrice Abry", year = "1999", note = "Submitted for publication", URL = "http://www.serc.rmit.edu.au/~darryl/A4.ps", } @Article{vei-abr:joint, title = "A wavelet based joint estimator of the parameters of long-range dependence", author = "Darryl Veitch and Patrice Abry", journal = IEEETIT, volume = "45", number = "3", year = "1999", pages = "878--897", URL = "http://www.serc.rmit.edu.au/~darryl/A3.ps", keywords = "Hurst parameter long-range dependence packet traffic parameter estimation telecommunications networks time-scale analysis wavelet decomposition", abstract = "A joint estimator is presented for the two parameters that define the long-range dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a recently proposed wavelet-based estimator of the scaling parameter [4], as well as extending it to include the associated power parameter. An important feature is its conceptual and practical simplicity, consisting essentially in measuring the slope and the intercept of a linear fit after a discrete wavelet transform is performed, a very fast (O(n)) operation. Under well-justified technical idealizations the estimator is shown to be unbiased and of minimum or close to minimum variance for the scale parameter, and asymptotically unbiased and efficient for the second parameter. Through theoretical arguments and numerical simulations it is shown that in practice, even for small data sets, the bias is very small and the variance close to optimal for both parameters. Closed-form expressions are given for the covariance matrix of the estimator as a function of data length, and are shown by simulation to be very accurate even when the technical idealizations are not satisfied, Comparisons are made against two maximum-likelihood estimators. In terms of robustness and computational cost the wavelet estimator is found to be clearly superior and statistically its performance is comparable, We apply the tool to the analysis of Ethernet teletraffic data, completing an earlier study on the scaling parameter alone.", } @Article{vel-ulr:annealing, title = "Simulated annealing wavelet estimation via fourth-order cumulant matching", author = "D. R. Velis and T. J. Ulrych", journal = "Geophysics", volume = "61", number = "6", year = "1996", pages = "1939--1948", keywords = "deconvolution, coefficients, gaussianity, phase", abstract = "The fourth-order cumulant matching method has been developed recently for estimating a mixed-phase wavelet from a convolutional process. Matching between the trace cumulant and the wavelet moment is done in a minimum mean-squared error sense under the assumption of a non-Gaussian, stationary, and statistically independent reflectivity series. This leads to a highly nonlinear optimization problem, usually solved by techniques that require a certain degree of linearization, and that invariably converge to the minimum closest to the initial model. Alternatively, we propose a hybrid strategy that makes use of a simulated annealing algorithm to provide reliability of the numerical solutions by reducing the risk of being trapped in local minima. Beyond the numerical aspect, the reliability of the derived wavelets depends strongly on the amount of data available. However, by using a multidimensional taper to smooth the trace cumulant, we show that the method can be used even in a trace-by-trace implementation, which is very important from the point of view of stationarity and consistency. We demonstrate the viability of the method under several reflectivity models. Finally, we illustrate the hybrid strategy using marine and held real data examples. The consistency of the results is very encouraging because the improved cumulant matching strategy we describe can be effectively used with a limited amount of data.", } @Book{vet-kov:wavelets, title = "Wavelets and Subband Coding", author = "Martin Vetterli and Jelena Kova{\v{c}}evi{\'c}", year = "1995", publisher = "Prentice Hall PTR", address = "New Jersey", URL = "http://www.prenhall.com/allbooks/ptr_0130970808.html", keywords = "discrete-time case, or filter banks; development of wavelets; continuous wavelet and local Fourier transforms; efficient algorithms for filter banks and wavelet computations; and signal compression", } @TechReport{vid-kat-alb:multiscale, title = "Multiscale Denoising of Self Similar Processes", author = "Brani Vidakovic and Gabriel Katul and John Albertson", number = "00-02", institution = "Institute of Statisics and Decision Sciences, Duke University", year = "2000", URL = "http://ftp.isds.duke.edu/WorkingPapers/00-02.ps", abstract = "A practical limitation to investigating self-similarity in geophysical phenomena from their measured state variables is that measured signals are typically convolved with instrumentation noise at multiple scales. This study develops and tests a multiscale Bayesian model (BEFE) for separating a $1/f$-like signal from inherent instrumentation noise and contrasts its performance to the Wiener-type (WAS) and Fourier amplitude (FAS) shrinkage methods. The novel feature in BEFE is that the separation is performed in the wavelet domain and involves the use of a Bayesian inference approach guided by existing theoretical power-laws in the filtered signal energy spectrum. We contrast the performance of all three methods for synthetic fractional Brownian motion ({\it fBm}) signals and turbulent velocity time series collected in the atmospheric boundary layer. A discussion on the advantages and disadvantages of each method is also presented, particularly when the process is not exactly an {\it fBm}.", } @Article{vid-loz:time-dependent, title = "On time-dependent wavelet denoising", author = "Vidakovic, B. and Lozoya, C. B.", journal = IEEETSP, volume = "46", number = "9", year = "1998", pages = "2549--2554", keywords = "denoising image processing wavelet shrinkage", abstract = "In this correspondence, we address the shrinkage of wavelet coefficients and induced denoising in the time domain by taking into consideration the `time' behavior of a noisy signal. We illustrate our time adaptation paradigm in a thresholding procedure utilizing Bayesian hypothesis tests. Both one-dimensional (1-D) and two-dimensional (2-D) signals are considered in examples to motivate and implement our method.", } @Unpublished{vid-mul:kids, title = "Wavelets for Kids: Tutorial Introduction", author = "Brani Vidakovi{\'c} and Peter M{\"u}ller", note = "Institute of Statisics and Decision Sciences, Duke University", year = "1994", email = "brani@isds.duke.edu, pm@isds.duke.edu", URL = "http://www.isds.duke.edu/~brani/papers/wav4kidsA.ps", } @Article{vid:bayes, title = "Nonlinear wavelet shrinkage with {B}ayes rules and {B}ayes factors", author = "Brani Vidakovic", journal = JASA, volume = "93", number = "441", year = "1998", pages = "173--179", URL = "http://www.isds.duke.edu/~brani/papers/WavShrinkBF.ps", keywords = "Bayes model denoising thresholding wavelet regression", abstract = "Wavelet shrinkage, the method proposed by the seminal work of Donoho and Johnstone is a disarmingly simple and efficient way of denoising data. Shrinking wavelet coefficients was proposed from several optimality criteria. In this article a wavelet shrinkage by coherent Bayesian inference in the wavelet domain is proposed. The methods are tested on standard Donoho- Johnstone test functions.", } @Book{vid:book, title = "Statistical Modeling by Wavelets", author = "Brani Vidakovic", publisher = "John Wiley \& Sons", address = NY, year = "1999", pages = "381", ISBN = "0471293652", URL = "http://www.isds.duke.edu/~brani/Wiley.html", keywords = "", abstract = "", } @Article{von-nas-kro:adaptive, title = "Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum", author = "Rainier {von Sachs} and Guy P. Nason and Gerald Kroisandt", journal = JRSSB, volume = "62", number = "2", year = "2000", pages = "271--292", URL = "http://www.stats.bris.ac.uk:81/pub/reports/Wavelets/StanTechRep516.ps.gz", abstract = "", } @InCollection{von-nas-kro:spectral, title = "Spectral representation and estimation for locally stationary wavelet processes", author = "Rainier {von Sachs} and Guy P. Nason and Gerald Kroisandt", booktitle = "Spline Functions and the Theory of Wavelets", editor = "S. Dubuc and G. Deslauriers", series = "CRM Proceedings \& Lecture Notes", publisher = "American Mathematical Society", address = "Montreal, Canada", volume = "18", year = "1999", pages = "381--397", URL = "http://playfair.Stanford.EDU/reports/rvs/vSNK.ps.gz", } @Article{von-neu:stationarity, title = "A Wavelet-based Test for Stationarity", author = "Rainier {von Sachs} and Michael H. Neumann", journal = JTSA, volume = "", number = "", year = "2000", pages = "--", URL = "http://playfair.Stanford.EDU/reports/rvs/Stattest.ps.gz", } @Article{von-sch:evolutionary, title = "Wavelet smoothing of evolutionary spectra by non-linear thresholding", author = "Rainier {von Sachs} and Kai Shneider", journal = ACHA, volume = "3", year = "1996", pages = "268--282", URL = "http://www.mathematik.uni-kl.de/~wwwtecm/preprints/reports/report_106.ps.gz", abstract = "We consider wavelet estimation of the time-dependent (evolutionary) power spectrum of a locally stationary time series in a model which was recently introduced by Dahlhaus [2]. Allowing for departures from stationary proves useful for modelling, e.g., transient phenomena, quasi-oscillating behaviour or spectrum modulation. In contrast to classical parametric and nonparametric (linear) approaches we use nonlinear thresholding of the empirical wavelet coefficients of the evolutionary spectrum. We show how these techniques allow for both adaptively reconstructing the local structure in the time-frequency plane and for denoising the resulting estimates. To this end a threshold choice is derived which is motivated by minimax properties w.r.t. the integrated mean squared error. Our approach is based on a 2-d orthogonal wavelet transform modified by using a cardinal Lagrange interpolation function on the finest scale. As an example, we apply our procedure to a time-varying spectrum motivated from mobile radio propagation.", } @Article{von:modelling, title = "Modelling and Estimation of the Time-varying Structure of Nonstationary Time Series", author = "Rainier {von Sachs}", journal = "Brazilian Journal of Probability and Statistics", volume = "10", number = "2", year = "1996", pages = "181--204", URL = "http://playfair.Stanford.EDU/reports/rvs/brazilpaper.ps.gz", } @TechReport{von:nonparametric, title = "Nonparametric Wavelet Methods for Nonstationary Time Series", author = "Rainier {von Sachs}", number = "98/19", year = "1998", institution = "Institut de Statistique, Universit{\'e} Catholique de Louvain", URL = "http://www.stat.ucl.ac.be/dp/dp98/9819.ps", } @Article{wal-con:matching, title = "Matching pursuit by undecimated discrete wavelet transform for non-stationary time series of arbitrary length", author = "Andrew T. Walden and A. Contreras Cristan", journal = SC, volume = "8", number = "3", year = "1998", pages = "205--219", URL = "http://www.ma.ic.ac.uk/statistics/links/atw.link/matching-pursuit-TR-96-02.ps", keywords = "", abstract = "", } @Article{wal-con:shock, title = "The phase-corrected undecimated discrete wavelet packet transform and the recurrence of high latitude interplanetary shock waves", author = "Andrew T. Walden and Alberto Contreras Cristan", journal = PRSLA, volume = "454", number = "1976", year = "1998", pages = "2243--2266", keywords = "wavelet packet transform wavelet phase non-stationary time series Ulysses spacecraft solar magnetic field heliospheric magnetic-field fourier-analysis representation algorithms ulysses.", URL = "http://www.ma.ic.ac.uk/statistics/links/atw.link/phase-corrected-TR-97-03.ps", abstract = "This paper is concerned with the development and application of the phase-corrected maximal overlap discrete wavelet packet transform (MODWPT). The discrete cyclic filtering steps of the MODWPT are fully explained. Energy preservation is proven. With filter coefficients chosen from Daubechies's least asymmetric class, the optimum time shifts to apply to ensure approximate zero phase filtering at every level of the MODWPT are studied, and applied to the wavelet packet coefficients to give phase corrections which ensure alignment with the original time series. Also, the time series values at each time are decomposed into details associated with each frequency band, and these line up perfectly with features in the original time series since the details are shown to arise through exact zero phase filtering. The phase-corrected MODWPT is applied to a non-stationary time series of hourly averaged Southern Hemisphere solar magnetic field magnitude data acquired by the Ulysses spacecraft. The occurrence times of the shock waves previously determined via manual pattern matching on the raw data match those times in the time-frequency plot where a broadband spectrum is obtained; in other words, the phase-corrected MODWPT provides an approach to picking the location of complicated events. We demonstrate the superiority of the MODWPT in interpreting timing information over two competing methods, namely the cosine packet transform (or 'local cosine transform'), and the short-time Fourier transform." } @Article{wal-per-mcc:multitaper, title = "Spectrum Estimation by Wavelet Thresholding of Multitaper Estimators", author = "Andrew T. Walden and Donald B. Percival and Emma J. McCoy", journal = IEEETSP, volume = "46", number = "12", pages = "3153--3165", year = "1998", abstract = "Current methods for power spectrum estimation by wavelet thresholding use the empirical wavelet coefficients derived from the log periodogram. Unfortunately, the periodogram is a very poor estimate when the true spectrum has a high dynamic range and/or is rapidly varying. In addition, because the distribution of the log periodogram is markedly non-Gaussian, special wavelet-dependent thresholding schemes are needed. These difficulties can be bypassed by starting with a multitaper spectrum estimator. The logarithm of this estimator is close to Gaussian distributed if a moderate number (greater than or equal to 5) of tapers are used. In contrast to the log periodogram, log multitaper estimates are not approximately pairwise uncorrelated at the Fourier frequencies, but the form of the correlation can be accurately and simply approximated. For scale-independent thresholding, the correlation acts in accordance with the wavelet shrinkage paradigm to suppress small-scale `noise spikes' while leaving informative coarse scale coefficients relatively unattenuated. This simple approach to spectrum estimation is demonstrated to work very well in practice. Additionally, the progression of the variance of wavelet coefficients with scale can be accurately calculated, allowing the use of scale-dependent thresholds. This more involved approach also works well in practice but is not uniformly preferable to the scale-independent approach.", } @Book{wal:orthogonal, title = "Wavelets and Other Orthogonal Systems with Applications", author = "Gilbert G. Walter", publisher = "CRC Press Inc.", address = "Boca Raton", pages = "272", year = "1994", URL = "http://www.crcpress.com/prods/7878.htm", abstract = "This book makes accessible to both mathematicians and engineers important elements of the theory, construction, and application of orthogonal wavelets. It is integrated with more traditional orthogonal series, such as Fourier series and orthogonal polynomials. It treats the interaction of both with generalized functions (delta functions), which have played an important part in engineering theory but whose rules are often vaguely presented. Unlike most other books that are excessively technical, this text/reference presents the basic concepts and examples in a readable form. Much of the material on wavelets has not appeared previously in book form. Applications to statistics, sampling theorems, and stochastic processes are given. In particular, the close affinity between wavelets and sampling theorems is explained and developed.", } @Unpublished{wan-cav-son:self-similarity, title = "Self-similarity index estimation via wavelets for locally self-similar processes", author = "Yazhen Wang and Joseph E. Cavanaugh and Changyong Song", year = "1997", note = "Department of Statistics, University of Missouri", } @Article{wan-zho:aseismic, title = "Aseismic designs based on artificial simulations - {H}ow to achieve chirplet-based signal approximation with a strong earthquake ground-model", author = "Jun-Jie Wang and Jing Zhou", journal = IEEESPM, volume = "16", number = "2", month = "March", year = "1999", pages = "94--99", keywords = "motion", } @Unpublished{wan:change-curve, title = "Change curve estimation via wavelets", author = "Yazhen Wang", institution = "Department of Statistics, University of Missouri", year = "1997", note = "{\em Journal of the American Statistical Association}, to be published in 1998", } @Article{wan:indirect, title = "Change-points via wavelets for indirect data", author = "Yazhen Wang", journal = SSin, volume = "9", number = "1", year = "1999", pages = "103--117", abstract = "This article studies change-points of a function for noisy data observed from a transformation of the function. The proposed method uses a wavelet-vaguelette decomposition to extract information about the wavelet transformation of the function from the data and then detect and estimate change-points by the wavelet transformation. Asymptotic theory for the detection and estimation is established. A simulated example is carried out to illustrate the method.", } @Article{wan:jump, title = "Jump and sharp cusp detection by wavelets", author = "Yazhen Wang", journal = BKA, volume = "82", number = "2", year = "1995", pages = "385--397", abstract = "A method proposed to detect jumps and sharp cusps in a function which is observed with noise, by checking if the wavelet transformation of the data has significantly large absolute values across fine scales. Asymptotic theory is established and practical implementation is discussed. The method is tested on simulated examples, and applied to stock market return data.", } @Article{wan:long-range, title = "Function estimation via wavelet shrinkage for long-memory data", author = "Yazhen Wang", journal = AofS, volume = "24", number = "2", year = "1996", pages = "466--484", } @Article{wei-bov:enhancement, title = "Enhancement of compressed images by optimal shift-invariant wavelet packet basis", author = "Wei, D. and Bovik, A. C.", journal = JVCIR, volume = "9", number = "1", year = "1998", pages = "15--24", keywords = "compactly supported wavelets coded images transform reduction algorithms bases", abstract = "A novel postprocessing method based on the optimal shift-invariant wavelet packet (SIWP) representation and wavelet shrinkage is proposed to enhance compressed images. At the encoder, the optimal (in the mean square error sense) SIWP basis is searched using a fast optimization algorithm and the location of the best basis in the entire SIWP library is transmitted as overhead information to the decoder. The selected basis is jointly optimal in terms of both the time-frequency tiling and the relative time-domain offset (or shift) between a signal and its wavelet packet representation. After the decoder reconstructs the compressed image, the postprocessor performs wavelet shrinkage using the optimal basis. Due to its powerful adaptability, the method is shown to achieve a better trade-off between enhancement performance and decoder complexity than both the orthonormal wavelet transform and the undecimated wavelet transform-based methods.", } @Article{wei-dix:underwater, journal = JAcSA, volume = "101", number = "1", year = "1997", pages = "377--383", title = "{W}avelet-based denoising of underwater acoustic signals", author = "L. G. Weiss and T. L. Dixon", abstract = "Underwater environmental measurements of the ocean require signals that are free from unwanted backscatter and clutter. Removing these unwanted signal components usually amounts to applying some form of filtering technique such as a high pass filter, a bandpass filter, a Wiener filter, etc. These approaches however are limited in their abilities to remove acoustic returns that vary spectrally. This paper presents a multiresolution approach to removing unwanted backscatter from high-frequency underwater acoustic signals and compares it to high pass filtering of the same signals. The filtering approach presented applies wavelet transforms for signal recovery and denoising of high-frequency acoustic signals. It is shown that by computing a wavelet transform of the returned signals, applying a denoising technique, and then reconstructing the signal, additional unwanted backscatter can be removed.", keywords = "scattering", } @Unpublished{wei:invariance, title = "Translation Invariance and the Wavelet Transform", author = "John Weiss", year = "1993", note = "Applied Mathematics Group", URL = "http://www.tiac.net/users/nurit/trep3/trep3.html", keywords = "wavelet transform, translation invariance, best basis, transient detection", abstract = "A translation invariant wavelet transform algorithm is defined. The algorithm is an extension of the best basis approach and can be used to define translation invariant best bases and wavelet transforms. The computational cost is a factor of $m$ greater than the usual algorithms, where $m$ is the multiplier of the wavelet system. Some applications to transient detection are presented. A general form of an invariant wavelet transform is presented. This transform is shown to be invariant under a large group of symmetries described, most naturally, by the g-circulant transformations. The symmetries include translation and time-reversal of a periodic data vector. In our construction the expansion coefficients of g-circulant transformations of a data vector areshown to be simply related by periodic shifts of their expansion coefficients. Therefore, under g-circulant transformations the numerical values and ordering are invariant.", } @InCollection{wey-war:de-noising, title = "De-noising using wavelets and cross validation", author = "N. Weyrich and G. T. Warhola", booktitle = "Recent Developments in Approximation Theory, Wavelets and Applications", editor = "S. P. Singh", publisher = "Kluwar", address = "Boston, MA", year = "1995", pages = "523--532", } @Article{wey-war:shrinkage, title = "Wavelet shrinkage and generalized cross validation for image denoising", author = "N. Weyrich and G. T. Warhola", journal = IEEETIP, volume = "7", number = "1", pages = "82--90", year = "1998", abstract = "We present a denoising method based on wavelets and generalized cross validation and apply these methods to image denoising, We describe the method of modified wavelet reconstruction and show that the related shrinkage parameter vector can be chosen without prior knowledge of the noise variance by using the method of generalized cross validation, By doing so, we obtain an estimate of the shrinkage parameter vector and, hence, the image, which is very close to the best achievable mean-squared error result-that given by complete knowledge of the underlying clean image.", } @Unpublished{whi-etal:lmwnr, title = "Testing for Homogeneity of Variance in Time Series: {L}ong Memory, Wavelets and the {N}ile {R}iver", author = "Brandon Whitcher and Simon D. Byers and Peter Guttorp and Donald B. Percival", year = "1998", note = "Submitted for publication", URL = "http://www.eurandom.tue.nl/whitcher/papers/nile.ps", keywords = "Cumulative sum of squares; Discrete wavelet transform, Fractional difference process; Variance change", abstract = "We consider the problem of testing for homogeneity of variance in a time series that has long memory structure. We demonstrate that a test whose null hypothesis is designed to be white noise can in fact be applied, on a scale by scale basis, to the discrete wavelet transform of long memory processes. In particular, we show that evaluating a normalized cumulative sum of squares test statistic using critical levels appropriate for the null hypothesis of white noise yields approximately the same null hypothesis rejection rates when applied to the discrete wavelet transform of samples from a fractional difference process. The point at which the test statistic, using the maximal overlap discrete wavelet transform, achieves its maximum value can be used to estimate the time of the unknown variance change. We apply our proposed test statistic on a time series of Nile River yearly minimum water levels covering the years 622 to 1284 AD. The test confirms an inhomogeneity of variance at short scales and identifies the change point around 720 AD, which coincides closely with the construction of a new device in 715 AD for measuring Nile River water levels.", } @TechReport{whi-gut-per:background, title = "Mathematical Background for Wavelet Estimators for Cross-Covariance and Cross-Correlation", author = "Brandon Whitcher and Peter Guttorp and Donald B. Percival", institution = "National Research Center for Statistics and the Environment", number = "38", year = "1999", URL = "http://www.eurandom.tue.nl/whitcher/papers/background.ps", } @Article{whi-gut-per:covariance, title = "Wavelet Analysis of Covariance with Application to Atmospheric Time Series", author = "Brandon Whitcher and Peter Guttorp and Donald B. Percival", journal = JGRA, volume = "", number = "", year = "2000", pages = "", note = "to appear", URL = "http://www.eurandom.tue.nl/whitcher/papers/wavecov.ps", keywords = "Confidence intervals; Cross-correlation; Cross-covariance; Madden-Julian oscillation; Maximal overlap discrete wavelet transform; Southern Oscillation Index", abstract = "Multi-scale analysis of univariate time series has appeared in the literature at an ever increasing rate. Here we introduce the multi-scale analysis of covariance between two time series using the discrete wavelet transform. The wavelet covariance and wavelet correlation are defined and applied to this problem as an alternative to traditional cross-spectrum analysis. The wavelet covariance is shown to decompose the covariance between two stationary processes on a scale by scale basis. Asymptotic normality is established for estimators of the wavelet covariance and correlation. Both quantities are generalized into the wavelet cross-covariance and cross-correlation in order to investigate possible lead/lag relationships. A thorough analysis of El-Ni\~no--Southern Oscillation events and the Madden--Julian oscillation is performed using a 35+ year record. We show how potentially complicated patterns of cross-correlation are easily decomposed using the wavelet cross-correlation on a scale by scale basis, where each wavelet cross-correlation series is associated with a specific physical time scale.", } @Article{whi-jen:aseg2000, title = "Wavelet Estimation of a Local Long Memory Parameter", author = "Brandon Whitcher and Mark J. Jensen", journal = "Exploration Geophysics", volume = "31", number = "1 \& 2", year = "2000", pages = "89--98", URL = "http://www.eurandom.tue.nl/whitcher/papers/ASEG2000.pdf", keywords = "discrete wavelet transform, least-squares regression, long-range dependence, time series, wavelet variance.", abstract = "Physical processes often exhibit long-range dependence that vary and evolve over time. Various techniques already exist in order to estimate global long-range dependence. That is, these methods assume the level of dependence does not vary over time and hence, fail to adapt to any potential change in the level of long-range dependence. The discrete wavelet transform is well-localized in time-scale space and thus, enables us to consistently estimate a local measure of long-range dependence and the time interval over which this estimate is constant. We utilize the so-called `cone of influence' (level-dependent support of the wavelet filter) induced by the wavelet transform to define an estimator of local variability - the time-dependent wavelet variance. Using a fractionally integrated autoregressive, moving average model, whose differencing parameter changes over time, we estimate the local differencing parameter at time $t$ using a log-log linear relationship between the time-dependent wavelet variance whose support contains $t$ and the scale parameter of the wavelet transform. Simulation studies demonstrate the estimation procedure for a variety of time-varying fractionally integrated autoregressive, moving average processes and we also apply it to a geophysical process.", } @Unpublished{whi-gut-per:multiple, title = "Multiscale Detection and Location of Multiple Variance Changes in the Presence of Long Memory", author = "Brandon Whitcher and Peter Guttorp and Donald B. Percival", year = "1999", note = "{\em Journal of Statistical Computation and Simulation}, to appear", URL = "http://www.eurandom.tue.nl/whitcher/papers/multiple.ps", keywords = "", abstract = "", } @Unpublished{whi:simulating, title = "Simulating {G}aussian Stationary Processes with Unbounded Spectra", author = "Brandon Whitcher", year = "1999", note = "Tentatively accepted for publication in the {\em Journal of Computational and Graphical Statistics}", URL = "http://www.eurandom.tue.nl/whitcher/papers/sim.ps", } @PhdThesis{whi:thesis, title = "Assessing Nonstationary Time Series Using Wavelets", author = "Brandon Whitcher", school = "University of Washington", year = "1998", postscript = "http://www.eurandom.tue.nl/whitcher/papers/thesis.ps.gz", abstract = "The discrete wavelet transform has be used extensively in the field of statistics, mostly in the area of ``denoising signals'' or nonparametric regression. This thesis provides a new application for the discrete wavelet transform, assessing nonstationary events in time series -- especially long memory processes. Long memory processes are those which exhibit substantial correlations between events separated by a long period of time. Departures from stationarity in these heavily autocorrelated time series, such as an abrupt change in the variance at an unknown location or ``bursts'' of increased variability, can be detected and accurately located using discrete wavelet transforms -- both orthogonal and overcomplete. A cumulative sum of squares method, utilizing a Kolomogorov--Smirnov-type test statistic, and an information criterion method are investigated. By analyzing a time series on a scale by scale basis, each scale corresponding to a range of frequencies, the ability to detect and locate a sudden change in the variance in the time series is introduced. Using this same procedure to detect a change in the long memory parameter is also investigated. Applications involve the Nile River minimum water levels and vertical ocean shear measurements. In the atmospheric sciences, broadband features in the spectrum of recorded time series have been hypothesized to be nonstationary events; e.g., the Madden--Julian oscillation. The Madden--Julian oscillation is a result of large-scale circulation cells oriented in the equatorial plane from the Indean Ocean to the central Pacific. The oscillation has been noted to have higher frequencies during warm events in El Nino--Southern Oscillation (ENSO) years. The concepts of wavelet covariance and wavelet correlation are introduced and applied to this problem as an alternative to cross-spectrum analysis. The wavelet covariance is shown to decompose the covariance between two stationary processes on a scale by scale basis. Asymptotic normality of estimators of the wavelet covariance and correlation is shown in order to construct approximate confidence intervals. Both quantities are generalized into the wavelet cross-covariance and cross-correlation in order to investigate possible lead/lag relations in bivariate time series. Atmospheric measurements (such as station pressure and zonal wind speeds) from a single station at Canton Island (2.8S, 171.7W) are analyzed and nicely replicate the results found in Madden and Julian (1971). To highlight that the wavelet methods can provide insight over and above traditional spectral methods (including multitaper techniques) a daily ``Southern Oscillation Index'' and station pressure series from Truk Island (7.4N, 151.8W) are analyzed. The wavelet cross-covariance nicely decomposes the usual cross-covariance into scales which are more easily associated with physical phenomena. The time-varying wavelet covariance is used to show the increase in positive correlation between the SOI and Truk Island station pressure in the first half of each year versus latter half.", } @Book{wic:adapted, title = "Adapted Wavelet Analysis from Theory to Software", author = "Mladen Victor Wickerhauser", publisher = "A K Peters", ISBN = "1-56881-041-5", year = "1994", address = "Wellesley, Massachusetts", abstract = "This detail-oriented text is intended for engineers and applied mathematicians who must write computer programs to perform wavelet and related analyses on real data. It should also be useful to the pure mathematician with questions about wavelet theory applications and to the instructor or student as a textbook in the mathematics and latest techniques in transient signal analysis and processing. Beginning with an overview of mathematical prerequisites, successive chapters rigorously examine the properties of waveforms used in adapted wavelet analysis: discrete ``fast'' Fourier transforms, orthogonal and biorthogonal wavelets, wavelet packets, and localized trigonometric or lapped orthogonal functions. Other chapters discuss the ``best-basis'' method,time-frequency analysis, and combinations of these algorithms useful for signal analysis, de-noising, and data compression. Each chapter discusses the technical aspects of implementation giving examples in pseudocode backed up with a Standard C source code (on optional diskette) and closes with a list of worked exercises.", } @Unpublished{wic:theory-applications, title = "Wavelet Theory and Applications", author = "Mladen Victor Wickerhauser", year = "1997", note = "Department of Mathematics, Washington University of St. Louis", } @Unpublished{won-ip-lua-xie:detection, title = "Wavelet Detection of Jump Points and an Application to Exchange Rates", author = "Heung Wong and Wai-Cheung Ip and Yihui Luan and Zhongjie Xie", year = "1996", note = "The Hong Kong Polytechnic University, Hong Kong", } @Article{won:harmonizable, title = "Wavelet decomposition of harmonizable random processes", author = "P. W. Wong", journal = IEEETIT, volume = "39", number = "1", year = "1993", pages = "7--18", keywords = "pointwise convergence, signal analysis, discrete wavelet decomposition, second-order harmonizable random processes, deterministic wavelet decomposition, complex exponential, bounded convergence, stochastic wavelet decomposition, linear operations, addition, differentiation, linear filtering", abstract = "The discrete wavelet decomposition of second-order harmonizable random processes is considered. The deterministic wavelet decomposition of a complex exponential function is examined, where its pointwise and bounded convergence to the function is proved. This result is then used for establishing the stochastic wavelet decomposition of harmonizable processes. The similarities and differences between the wavelet decompositions of general harmonizable processes and a subclass of processes having no spectral mass at zero frequency, e.g., those that are wide-sense stationary and have continuous power spectral densities, are also investigated. The relationships between the harmonization of a process and that of its wavelet decomposition are examined. Finally, certain linear operations such as addition, differentiation, and linear filtering on stochastic wavelet decompositions are considered. It is shown that certain linear operations can be performed term by term with the decomposition.", } @Article{wor-opp:self-similar, title = "Wavelet-based representations for a class of self-similar signals with application to fractal modulation", author = "Gregory W. Wornell and Alan V. Oppenheim", journal = IEEETIT, volume = "38", number = "2", year = "1992", pages = "785--800", keywords = "spectral characteristics, wavelet transforms, self-similar signals, fractal modulation, deterministic scale-invariance characterization, orthonormal wavelet bases, dy-homogeneous signals, communications-based context, multiple time-scales, multirate modulation strategy, noisy channels", abstract = "A potentially important family of self-similar signals based upon a deterministic scale-invariance characterization is introduced. These signals, which are referred to as 'dy-homogeneous' signals because they generalize the well-known homogeneous functions, have highly convenient representations in terms of orthonormal wavelet bases. In particular, wavelet representations can be exploited to construct orthonormal self-similar bases for these signals. The spectral and fractal characteristics of dy-homogeneous signals make them appealing candidates for use in a number of applications. As one potential example, their use in a communications-based context is considered. Specifically, a strategy for embedding information into a dy-homogeneous waveform on multiple time-scales is developed. This multirate modulation strategy, called fractal modulation, is potentially well-suited for use with noisy channels of simultaneously unknown duration and bandwidth.", } @Book{wor:book, title = "Signal Processing with Fractals: {A} Wavelet Based Approach", author = "Gregory W. Wornell", publisher = "Prentice Hall", address = NJ, year = "1996", URL = "http://www.prenhall.com/books/ptr_013120999x.html", keywords = "Wavelet Transformations, Statistically Self-Similar, Detection and Estimation with Fractal Processes, Deterministically Self-Similar Signals, Fractal Modulation, Linear Self-Similar Systems", abstract = "Fractal signals, derived from wavelet theory, are ideally suited for use in many engineering applications, ranging from communications to remote sensing. This book provides an introduction to wavelet theory from a signal processing perspective, and details fractal signals and a collection of practical wavelet-based techniques for representing and manipulating fractal signals in various applications.", } @Article{wor:karhunen-loeve, title = "A {K}arhunen-{L}o\'{e}ve-like expansion for $1/f$ processes via wavelets", author = "Gregory W. Wornell", journal = IEEETIT, volume = "36", number = "4", year = "1990", pages = "859--861", keywords = "orthonormal wavelet bases, Karhunen-Loeve-like expansion, 1/f processes, scaling processes, uncorrelated random variables", abstract = "While so-called 1/f or scaling processes emerge regularly in modeling a wide range of natural phenomena, as yet no entirely satisfactory framework has been described for the analysis of such processes. Orthonormal wavelet bases are used to provide a new construction for nearly 1/f processes from a set of uncorrelated random variables.", } @Article{wor:representations, title = "Wavelet-based representations for the $1/f$ family of fractal processes", author = "G. W. Wornell", journal = PIEEE, volume = "81", number = "10", year = "1993", pages = "1428--1450", keywords = "signal detection, 1/f family, fractal processes, orthonormal wavelet bases, frequency-domain characterization, Karhunen-Loeve-type expansion, estimation problems", abstract = "It is demonstrated that 1/f fractal processes are, in a broad sense, optimally represented in terms of orthonormal wavelet bases. Specifically, via a useful frequency-domain characterization for 1/f processes, the wavelet expansion's role as a Karhunen-Loeve-type expansion for 1/f processes is developed. As an illustration of potential, it is shown that wavelet-based representations naturally lead to highly efficient solutions to some fundamental detection and estimation problems involving 1/f processes.", } @InProceedings{wu-su:relationship, title = "On the relationship between the self-similarities of fractal signals and wavelet transforms", author = "Bing-Fei Wu and Yu-Lin Su", booktitle = "International Symposium on Signal Processing and its Applications", editor = "B. Boashash and N. Harle and A. A. Zoubir", year = "1996", pages = "736--739", email = "bwu@haeshiuh.cn.nctu.edu.tw", keywords = "fractal signals. wavelet transforms. stochastic processes. correlation functions. power spectra. fractal dimensions. probability measure function. variance function. time series function. time-averaging autocorrelation function. ensemble-averaging autocorrelation function. time-averaging power spectrum. average power spectrum. distribution functions. stationary processes. nonstationary processes. one-dimensional self-similarity. continuous wavelet transform. discrete wavelet transform. quadrature mirror filter structure. fractional Brownian motion process", abstract = "Since many natural phenomena are occasionally defined as stochastic processes and the corresponding fractal characteristics are hidden from their correlation functions or power spectra, the topic would be of interest in signal processing. In this paper, we summarize the fractal dimensions and the relationship of the fractal in probability measure, variance, time series, time-averaging autocorrelation, ensemble-averaging autocorrelation, time-averaging power spectrum, average power spectrum and distribution functions for stationary and nonstationary processes. We also propose that the preservation of the one-dimensional self-similarity of a fractal signal is obtained by using the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT) with the perfect reconstruction quadrature mirror filter structure. Moreover, we extend the results to the two-dimensional case and point out the relationship of the self-similarities between the CWT and DWT of the fractal signals. A fractional Brownian motion process is provided as an example to show the results of this paper.", } @Article{xia-ger-har-sut:design, title = "Design of prefilters for discrete multiwavelet transforms", author = "Xiang-Gen Xia and Jeffrey S. Geronimo and Douglas P. Hardin and Bruce W. Suter", journal = IEEETSP, volume = "44", number = "1", year = "1996", pages = "25--35", keywords = "discrete multiwavelet transforms pyramid algorithm single wavelet transform coefficients tree-structured multirate filter banks premultirate filter banks vector filter banks discrete vector-valued wavelet transform discrete-time vector-valued signals discrete multiwavelet transform energy compaction.", abstract = "The pyramid algorithm for computing single wavelet transform coefficients is well known. The pyramid algorithm can be implemented by using tree-structured multirate filter banks. The authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets. The proposed algorithm can be thought of as a discrete vector-valued wavelet transform for certain discrete-time vector-valued signals. The proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete-time signals. The authors then present some numerical experiments to illustrate the performance of the algorithm, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms.", } @Article{xia-kuo-zha:optimal, title = "{W}avelet coefficient computation with optimal prefiltering", journal = IEEETSP, volume = "42", number = "8", year = "1994", pages = "2191--7", author = "Gen Xia Xiang and C. C. J. Kuo and Zhang Zhen", abstract = "Discrete wavelet transform (DWT) is often used to approximate wavelet series transform (WST) and continuous wavelet transform (CWT), since it can be computed numerically. In this research, we first study the accuracy of the computed DWT coefficients obtained from the Shensa (see ibid., vol.40, no.10, p.2464-2482, 1992) algorithm as an approximate of the WST coefficients. Based on the accuracy analysis, we then propose a procedure to design optimal FIR prefilters used in the Shensa algorithm to reduce the approximation error. Finally, numerical examples are presented to demonstrate the performance of the optimal FIR prefilters.", keywords = "wavelet coefficient. optimal prefiltering. discrete wavelet transform. DWT. wavelet series transform. continuous wavelet transform. WST. CWT. accuracy analysis. optimal FIR prefilters. approximation error reduction. Shensa algorithm.", } @Article{yan-sag-tsu:system, title = "System impulse response identification using a multiresolution neural network", author = "Zi-Jiang Yang and Setsuo Sagara and Teruo Tsuji", journal = "Automatica", volume = "33", number = "7", year = "1997", pages = "1345--1350", URL = "", keywords = "system impulse response identification multiresolution neural network discrete-time impulse response model linear system sampled input-output data I/O data high-frequency components continuous-time impulse response scaling functions wavelet functions genetic algorithm AIC redundant subsystems", abstract = "This paper proposes a new identification method for the discrete-time impulse response model of a linear system from sampled input-output data. Our attention is especially focused on identification of the impulse response, which includes high-frequency components locally. The continuous-time impulse response of the system is approximated by a multiresolution neural network composed of the scaling and wavelet functions. Hence the system under study can be viewed as the weighted sum of a group of subsystems in which the scaling functions and wavelet functions are interpreted as their impulse responses respectively. Then the genetic algorithm and the AIC are introduced to select significant subsystems at each resolution level such that some redundant subsystems that are sensitive to the noise effects are discarded. It is shown through a simulation that the proposed method yields accurate estimate of the impulse response, even in the ill-conditioned cases.", } @Article{yu-mykland:looking, title = "Looking at {M}arkov samplers through cusum path plots: {A} simple diagnostic idea", author = "Bin Yu and Per Mykland", journal = SC, volume = "8", number = "3", year = "1998", pages = "275--286", } @Unpublished{zha:temporal-spatial, title = "Multiresolution analysis for temporal-spatial processes: {A} preliminary report", author = "Jun Zhai", year = "1997", note = "Department of Statistics, North Carolina State University", } @Article{zie-stu:relation, title = "The relation of the {A}llan- and $\Delta$-variance to the continuous wavelet transform", author = "M. Zielinsky and J. Stutzki", journal = AandA, volume = "347", number = "2", year = "1999", pages = "630--633", URL = "http://xxx.lanl.gov/abs/astro-ph/9904190", abstract = "This paper is understood as a supplement to the paper by Stutzki et al. 1998, where we have shown the usefulness of the Allan-variance and its higher dimensional generalization, the Delta-variance, for the characterization of molecular cloud structures. In this study we present the connection between the Allan- and Delta-variance and a more popular structure analysis tool: the wavelet transform. We show that the Allan- and Delta-variances are the variances of wavelet transform coefficients.", } @Book{fou-kum:geophysics, title = "Wavelets in Geophysics", booktitle = "Wavelets in Geophysics", editor = "Efi Foufoula-Georgiou and Praveen Kumar", series = "Wavelet Analysis and its Applications", volume = "4", publisher = "Academic Press, Inc", year = "1994", address = SD, URL = "http://www.apcatalog.com/cgi-bin/AP?ISBN=0122628500&LOCATION=US&FORM=FORM2", abstract = "Applications of wavelet analysis to the geophysical sciences grew from Jean Morlet's work on seismic signals in the 1980s. Used to detect signals against noise, wavelet analysis excels for transients or for spatially localized phenomena. In this fourth volume in the renown WAVELET ANALYSIS AND ITS APPLICATIONS Series, Efi Foufoula-Georgiou and Praveen Kumar begin with a self-contained overview of the nature, power, and scope of wavelet transforms. The eleven original papers that follow in this edited treatise show how geophysical researchers are using wavelets to analyze such diverse phenomena as intermittent atmospheric turbulence, seafloor bathymetry, marine and other seismic data, and flow in aquifiers. Wavelets in Geophysics will make informative reading for geophysicists seeking an up-to-date account of how these tools are being used as well as for wavelet researchers searching for ideas for applications, or even new points of departure.", } @Book{ant-opp:wavelets, title = "Wavelets and Statistics", booktitle = "Wavelets and Statistics", editor = "Anestis Antoniadis and Georges Oppenheim", series = "Lecture Notes in Statistics", volume = "103", year = "1995", publisher = "Springer-Verlag", address = NY, ISBN = "0-387-94564-4", URL = "http://www.springer-ny.com/catalog/np/aug95np/DATA/0-387-94564-4.html", abstract = "Wavelets theory has found applications in a remarkable diversity of disciplines. The volume presents the proceedings of a conference held at Villard de Lans, France in 1994. Both statistical results and practical contributions were presented. The material is wide in scope and ranges from the development of new tools for nonparametric curve estimation to applied problems such as detection of transients in signal processing and image segmentation.", } @Proceedings{com-gro-tch:wavelets, editor = "Jean-Michel Combes and Alexander Grossman and Philippe Tchamitchian", title = "Wavelets: Time-Frequency Methods and Phase Space", booktitle = "Wavelets: Time-Frequency Methods and Phase Space", series = "Inverse Problems and Theoretical Imaging", publisher = "Springer-Verlag", year = "1989", address = "Berlin", note = "Proceedings of the International Conference, Marseille, France, December 14-18, 1987", } @Book{dag-ste:goodness, title = "Goodness-of-Fit Techniques", booktitle = "Goodness-of-Fit Techniques", editor = "Ralph B. D'Agostino and Michael A. Stephens", volume = "68", series = "STATISTICS: Textbooks and Monographs", publisher = "Marcel Dekker", address = NY, year = "1986", pages = "560", } @Proceedings{lai-uns-ald:wavelet6, title = "Wavelet applications in signal and image processing {VI}", booktitle = "Wavelet applications in signal and image processing {VI}", editor = "Andrew F. Laine and Michael A. Unser and Akram Aldroubi", volume = "3458", year = "1998", pages = "284", series = "Proceedings of SPIE", note = "19-24 July, 1998, San Diego, California", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/3400/3458.html", } @Proceedings{lai-uns:wavelet2, title = "Wavelet applications in signal and image processing {II}", booktitle = "Wavelet applications in signal and image processing {II}", editor = "Andrew F. Laine and Michael A. Unser", volume = "2303", year = "1994", pages = "602", series = "Proceedings of SPIE", note = "24-29 July, 1994, San Diego, California", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/2300/2303.html", } @Proceedings{lai-uns:wavelet3, title = "Wavelet applications in signal and image processing {III}", booktitle = "Wavelet applications in signal and image processing {III}", editor = "Andrew F. Laine and Michael A. Unser and Mladen V. Wickerhauser", volume = "2569", year = "1995", pages = "900", series = "Proceedings of SPIE", note = "12-14 July, 1995, San Diego, California", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/2500/2569.html", } @Proceedings{lai:wavelet1, title = "Mathematical Imaging: Wavelet Applications in Signal and Image Processing", booktitle = "Mathematical Imaging: Wavelet Applications in Signal and Image Processing", editor = "Andrew F. Laine", volume = "2034", series = "Proceedings of the SPIE", year = "1993", note = "11-16 July, 1993, San Diego, California", URL = "http://www.spie.org/web/abstracts/2000/2034.html", } @Book{mul-vid:biwbm, title = "Bayesian Inference in Wavelet-based Models", booktitle = "Bayesian Inference in Wavelet-based Models", editor = "Peter M{\"u}ller and Brani Vidakovic", publisher = "Springer Verlag", address = NY, volume = "141", series = "Lecture Notes in Statistics", year = "1999", pages = "", } @Book{rao-pri-les:applications, title = "Applications of Time Series Analysis in Astronomy and Meteorology", booktitle = "Applications of Time Series Analysis in Astronomy and Meteorology", editor = "T. Subba Rao and M. B. Priestly and O. Lessi", year = "1997", publisher = "Chapman \& Hall", address = "London", ISBN = "0-412-63800-2", abstract = "Statistical techniques, in particular time series techniques, are widely used in astronomy and meteorology. Despite this, until recently there have been few attempts to bring researchers from the fields of statistics, astronomy and meteorology together to discuss and formalize important problems. Applications of Time Series Analysis in Astronomy and Meteorology brings together a series of papers by experts in these fields evenly devoted to the theory and methodology of time series and to its applications to astronomy, meteorology and climatology. The topics covered include detection of periodicities, spectral analysis of unequally spaced data, detection of change points and higher order spectral methods of non-linear and non-Gaussian signals. Estimation of fractal dimension and applications of wavelet methods to astronomy are also considered. In addition, this book includes a floppy disc containing data sets to serve as a benchmark series. Applications of Time Series Analysis in Astronomy and Meteorology is of interest to statisticians, astronomers, meteorologists and climatologists alike.", } @Book{sta-var:methods, title = "Statistical Methods for Physical Science", booktitle = "Statistical Methods for Physical Science", editor = "John L. Stanford and Stephen B. Vardeman", series = "Methods of Experimental Physics", volume = "28", year = "1994", publisher = "Academic Press, Inc.", address = "Boston", keywords = "physical sciences, experiments, statistical methods", } @Proceedings{szu:wavelet1, title = "Wavelet Applications", booktitle = "Wavelet Applications", editor = "Harold H. Szu", volume = "2242", year = "1994", series = "Proceedings of SPIE", pages = "994", note = "4-8, April 1994, Orlando, Florida", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/2200/2242.html", } @Proceedings{szu:wavelet2, title = "Wavelet Applications {II}", booktitle = "Wavelet Applications {II}", editor = "Harold H. Szu", volume = "2491", year = "1995", series = "Proceedings of SPIE", pages = "1183", note = "17-21, April 1995, Orlando, Florida", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/2400/2491.html", } @Proceedings{szu:wavelet3, title = "Wavelet Applications {III}", booktitle = "Wavelet Applications {III}", editor = "Harold H. Szu", volume = "2762", year = "1996", series = "Proceedings of SPIE", pages = "668", note = "8-12 April 1996, Orlando, Florida", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/2700/2762.html", } @Proceedings{szu:wavelet5, title = "Wavelet Applications {V}", booktitle = "Wavelet Applications {V}", editor = "Harold H. Szu", volume = "3391", year = "1998", series = "Proceedings of SPIE", pages = "", note = "14-16 April 1998, Orlando, Florida", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/3300/3391.html", } @Proceedings{szu:wavelet6, title = "Wavelet Applications {VI}", booktitle = "Wavelet Applications {VI}", editor = "Harold H. Szu", volume = "3723", year = "1999", series = "Proceedings of SPIE", pages = "478", note = "4-9 April 1999, Orlando, Florida", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/3700/3723.html", } @Proceedings{uns-ald-lai:wavelet4, title = "Wavelet Applications in Signal and Image Processing {IV}", booktitle = "Wavelet Applications in Signal and Image Processing {IV}", editor = "Michael A. Unser and Akram Aldroubi and Andrew F. Laine", volume = "2825", year = "1996", pages = "1044", series = "Proceedings of SPIE", note = "4-9 August, 1996, Denver, Colorado", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/2800/2825.html", } @Proceedings{uns-ald-lai:wavelet7, title = "Wavelet Applications in Signal and Image Processing {VII}", booktitle = "Wavelet Applications in Signal and Image Processing {VII}", editor = "Michael A. Unser and Akram Aldroubi and Andrew F. Laine", volume = "3813", year = "1999", pages = "", series = "Proceedings of SPIE", note = "?-? October, 1996, Denver, Colorado", keywords = "Image-processing, Signal-processing, Wavelets", URL = "http://www.spie.org/web/abstracts/3800/3813.html", }