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60
Wavelet Thresholding via a Bayesian Approach
 J. R. STATIST. SOC. B
, 1996
"... We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. ..."
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Cited by 204 (27 self)
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We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might b...
The Discrete Wavelet Transform in S
 Journal of Computational and Graphical Statistics
, 1996
"... The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called wavethresh that works within the statistical language S to perform one and twodimensional discrete wavelet transforms. The transforms and their inverses can be computed using any par ..."
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Cited by 81 (24 self)
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The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called wavethresh that works within the statistical language S to perform one and twodimensional 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 twodimensional 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 objectorientated functionality of S makes wavethresh easy to use. We provide a tutorial introduction to wavelets and the 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 ...
Wavelet shrinkage using crossvalidation
, 1996
"... Wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics. This article concentrates on the estimation of functions and images from noisy data using wavelet shrinkage. A modified form of twofold crossvalidation is introduced to ..."
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Cited by 77 (13 self)
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Wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics. This article concentrates on the estimation of functions and images from noisy data using wavelet shrinkage. A modified form of twofold crossvalidation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of two. The crossvalidation algorithm is then extended to data sets of any length and to multidimensional data sets. The algorithms are compared to established threshold choosers using simulation. An application to a real data set arising from anaesthesia is presented.
Short Wavelets and Matrix Dilation Equations
, 1995
"... Scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter (in a twoband orthogonal filter bank). For "multifilters" those coefficients are matrices. This gives a new block structure for the filter bank, and leads to multiple scaling functions and w ..."
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Cited by 69 (10 self)
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Scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter (in a twoband orthogonal filter bank). For "multifilters" those coefficients are matrices. This gives a new block structure for the filter bank, and leads to multiple scaling functions and wavelets. Geronimo, Hardin, and Massopust constructed two scaling functions that have extra properties not previously achieved. The functions \Phi 1 and \Phi 2 are symmetric (linear phase) and they have short support (two intervals or less), while their translates form an orthogonal family. For any single function \Phi, apart from Haar's piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2 by 2 matrix coefficients while retaining orthogonality. This note derives the properties of \Phi 1 and \Phi 2 from the matrix dilation equation that they satisfy. Then our main step is to construct associated wavelets: two wavelets for two scaling functions....
Vector Quantization of Image Subbands: A Survey
 IEEE Transactions on Image Processing
, 1996
"... Subband and wavelet decompositions are powerful tools in image coding, because of their decorrelating effects on image pixels, the concentration of energy in a few coefficients, their multirate/multiresolution framework, and their frequency splitting which allows for efficient coding matched to the ..."
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Cited by 53 (4 self)
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Subband and wavelet decompositions are powerful tools in image coding, because of their decorrelating effects on image pixels, the concentration of energy in a few coefficients, their multirate/multiresolution framework, and their frequency splitting which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system. Vector quantization provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter and intraband correlation as well as of the more flexible partitions of higher dimensional vector spaces. Since 1988 a growing body of research has examined the use of vector quantization for subband/wavelet transform coefficients. We present a survey of these methods. 1 Introduction Image compression maps an original image into a bit stream suitable for communication over or storage in a digital medium. The number of bits required to represent the coded image should b...
Wavelet Processes and Adaptive Estimation of the Evolutionary Wavelet Spectrum
, 1998
"... This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power va ..."
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Cited by 48 (28 self)
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This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable timelocalized autocovariance. We illustrate our theory with a pedagogical example based on discrete nondecimated Haar wavelets and also a real medical time series example.
Wavelet Analysis and Its Statistical Applications
, 1999
"... In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this ..."
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Cited by 44 (9 self)
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In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this article is intended to give a relatively accessible introduction to standard wavelet analysis and to provide an up to date review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be e ective, rather than to researchers already familiar with the eld. Given that objective, we do not emphasise mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in...
On estimation of the wavelet variance
 Biometrika
, 1995
"... The wavelet variance provides a scalebased decomposition of the process variance for a time series or random field. It has seen increasing use in geophysics, astronomy, genetics, hydrology, medical imaging, oceanography, soil science, signal processing and texture analysis. In practice, however, da ..."
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Cited by 34 (4 self)
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The wavelet variance provides a scalebased decomposition of the process variance for a time series or random field. It has seen increasing use in geophysics, astronomy, genetics, hydrology, medical imaging, oceanography, soil science, signal processing and texture analysis. In practice, however, data collected in the form of a time series or random field often suffer from various types of contamination. We discuss the difficulties and limitations of existing contamination models (pure replacement models, additive outliers, level shift models and innovation outliers that hide themselves in the original time series) for robust nonparametric estimates of secondorder statistics. We then introduce a new model based upon the idea of scalebased multiplicative contamination. This model supposes that contamination can occur and affect data at certain scales and thus arises naturally in multiscale processes and in the wavelet variance context. For this new contamination model, we develop a full Mestimation theory for the wavelet variance and derive its large sample theory when the underlying time series or random field is Gaussian. Our approach treats the wavelet variance as a scale parameter and offers protection against contamination that operates additively on the log of squared wavelet coefficients and acts independently at different scales.
Wavelets through a Looking Glass. The World of the Spectrum
, 2001
"... harmonic analysis and wavelets in R n , The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, 1999) (L.W. Baggett and D.R. Larson, eds.), Contemp. Math., vol. 247, American Mathematical Society, Providence, 1999, pp. 1727. 56 References [BBC+95] A. Barenco, C.H. Bennett, R ..."
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Cited by 30 (20 self)
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harmonic analysis and wavelets in R n , The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, 1999) (L.W. Baggett and D.R. Larson, eds.), Contemp. Math., vol. 247, American Mathematical Society, Providence, 1999, pp. 1727. 56 References [BBC+95] A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J.A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A 52 (1995), 34573467. [BBGK71] V. Bargmann, P. Butera, L. Girardello, and J.R. Klauder, On the completeness of the coherent states, Rep. Mathematical Phys. 2 (1971), 221228. [BDMT98] G.P. Berman, G.D. Doolen, R. Mainieri, and V.I. Tsifrinovich,
A survey on wavelet applications in data mining
 SIGKDD Explor. Newsl
"... Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a highlevel datamining framework tha ..."
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Cited by 30 (3 self)
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Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a highlevel datamining framework that reduces the overall process into smaller components. Then applications of wavelets for each component are reviewd. The paper concludes by discussing the impact of wavelets on data mining research and outlining potential future research directions and applications. 1.