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The development of wavelet theory has in recent years spawned applications in signal processing, in fast algorithms for integral transforms, and in image and function representation methods. This last application has stimulated interest in…
The notion of wavelets is defined. It is briefly described {\it what} are wavelets, {\it how} to use them, {\it when} we do need them, {\it why} they are preferred and {\it where} they have been applied. Then one proceeds to the…
Wavelet thresholding generally assumes independent, identically distributed normal errors when estimating functions in a nonparametric regression setting. VisuShrink and SureShrink are just two of the many common thresholding methods based…
This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous…
We consider the problem of constructing nonparametric undirected graphical models for high-dimensional functional data. Most existing statistical methods in this context assume either a Gaussian distribution on the vertices or linear…
In this study, we perform some analysis for the probability distributions in the space of frequency and time variables. However, in the domain of high frequencies, it behaves in such a way as the highly non-linear dynamics. The wavelet…
Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive…
We present a general nonparametric approach for testing whether a statistical parameter defined through conditional distributions is constant across the conditioning variables. Such hypotheses arise naturally in problems such as assessing…
A very important property of a statistical distribution is to know whether it obeys Gaussian statistics or not. On the one hand, it is of paramount importance in the context of CMB anisotropy studies, since deviations from a Gaussian…
This paper addresses the problem of regularity properties of functions represented as an expansion in a wavelet basis with random coefficients in terms of finiteness of their Besov norm with probability 1. Such representations are used to…
We obtain a general solution for the probability density function of wave intensities in non-stationary Wave Turbulence. The solution is expressed in terms of the wave action spectrum evolving according the the wave-kinetic equation. We…
Functional data analysis is ubiquitous in most areas of sciences and engineering. Several paradigms are proposed to deal with the dimensionality problem which is inherent to this type of data. Sparseness, penalization, thresholding, among…
We propose a new wavelet-based method for density estimation when the data are size-biased. More specifically, we consider a power of the density of interest, where this power exceeds 1/2. Warped wavelet bases are employed, where warping is…
The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets…
Wavelets provide the flexibility to analyse stochastic processes at different scales. Here, we apply them to multivariate point processes as a means of detecting and analysing unknown non-stationarity, both within and across data streams.…
Objective detection of specific patterns in statistical distributions, like groupings or gaps or abrupt transitions between different subsets, is a task with a rich range of applications in astronomy: Milky Way stellar population analysis,…
We study non-parametric regression estimates for random fields. The data satisfies certain strong mixing conditions and is defined on the regular $N$-dimensional lattice structure. We show consistency and obtain rates of convergence. The…
I discuss approaches to optimally remove noise from images. A generalization of Wiener filtering to Non-Gaussian distributions and wavelets is described, as well as an approach to measure the errors in the reconstructed images. We argue…
We propose a wavelet-based technique for the nonparametric estimation of functions contaminated with noise whose mean and variance are linked via a possibly unknown variance function. Our method, termed the data-driven wavelet-Fisz…
In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a nonparametric estimator of the…