International Journal of Wavelets, Multiresolution and Information Processing

We consider the testing problem in a fixed-effects functional analysis of variance model. We test the null hypotheses that the functional main effects and the functional interactions are zeros against the composite nonparametric alternative hypotheses that they are separated away from zero in L²-norm and also possess some smoothness properties. We adapt the optimal (minimax) hypothesis testing procedures for testing a zero signal in a Gaussian "signal plus noise" model to derive optimal (minimax) non-adaptive and adaptive hypothesis testing procedures for the functional main effects and the functional interactions. The corresponding tests are based on the empirical wavelet coefficients of the data. Wavelet decompositions allow one to characterize different types of smoothness conditions assumed on the response function by means of its wavelet coefficients for a wide range of function classes. In order to shed some light on the theoretical results obtained, we carry out a simulation study to examine the finite sample performance of the proposed functional hypothesis testing procedures. As an illustration, we also apply these tests to a real-life data example arising from physiology. Concluding remarks and hints for possible extensions of the proposed methodology are also given.

This work analyzes the physical processes occurring in the Mock-up test of the FEBEX I and II projects. FEBEX I and II is an European research project (1996–2004) led by ENRESA, that has financial support from the European Commission. This experiment is based in two large-scale heating tests ("in-situ" test and "Mock-up" test) simulating a radioactive waste repository, and tries to analyze the thermo-hydro-mechanical (THM) processes that could eventually happen in this kind of repositories.

The main objectives of this study have been the following: to identify the physical processes occurring in the Mock-up experiment and to characterize them quantitatively; to understand the nature and consequences of several incidents happening in the Mock-up during the heating phase; and, finally, to analyze the data reliability of the sensors measurements and to predict possible failures.

The analysis techniques used in this work are both statistical (time series correlation and spectral analysis, wavelets analysis, matching pursuit analysis) and non-statistical (spatial distribution analysis of data). These methods aim to establish the existing relationships between several data series registered in the experiment, corresponding to the measured parameters, and to characterize on time and frequency the non-stationary response of the series. A better understanding of the main THM processes affecting the engineered barriers used for the isolation of the radioactive waste is then available with the results obtained from those analyses.

Image restoration problems can naturally be cast as constrained convex programming problems in which the constraints arise from a priori information and the observation of signals physically related to the image to be recovered. In this paper, the focus is placed on the construction of constraints based on wavelet representations. Using a mix of statistical and convex-analytical tools, we propose a general framework to construct wavelet-based constraints. The resulting optimization problem is then solved with a block-iterative parallel algorithm which offers great flexibility in terms of implementation. Numerical results illustrate an application of the proposed framework.

In the first 'Wavelets and Statistics' conference proceedings ¹, our group published 'Wavelab and Reproducible Research', in which we advocated using the internet for publication of software and data so that research results could be duplicated by others. Much has happened in the last decade that bears on the notion of reproducibility, and we will review our experience. We will also describe a new software package BEAMLAB containing routines for multiscale geometric analysis, and describe some of its capabilities.

BEAMLAB makes available, in one package, all the code to reproduce all the figures in our recently published articles on beamlets, curvelets and ridgelets. The interested reader can inspect the source code to see what algorithms were used, and how parameters were set to produce the figures, and will then be able to modify the source codes to produce variations of our results.

Some new examples of numerical studies based on BEAMLAB are provided here.

Deconvolution of a noisy signal in a periodic band-limited wavelet basis exhibits visual artifacts in the neighbourhood of discontinuities. This phenomenon is similar to that appearing in denoising with compactly-supported wavelet transforms and can be reduced by "cycle spinning" as in Coifman and Donoho [3]. In this paper we present an algorithm which "cycle-spins" a periodic band-limited wavelet estimator over all circulant shifts in O(n(log(n))²) steps. Our approach is based on a mathematical idea and takes full advantage of the Fast Fourier Transform. A particular feature of our algorithm is to bounce from the Fourier domain (where deconvolution is performed) to the wavelet domain (where denoising is performed). For both smooth and boxcar convolutions observed in white noise, we illustrate the visual and numerical performances of our algorithm in an extensive simulation study of the estimator recently proposed by Johnstone, Kerkyacharian, Picard, and Raimondo [8]. All figures presented here are reproducible using the software package.

Wavelet thresholding is an effective method for noise reduction of a wide class of naturally occurring signals. However, bias near to a discontinuity and Gibbs phenomenon are a drawback in wavelet thresholding. The extent to which this is a problem is investigated. The Haar wavelet basis is good at approximating discontinuities, but is bad at approximating other signal artefacts. A method of detecting jumps in a signal is developed that uses non-decimated Haar wavelet coefficients. This is designed to be used in conjunction with most existing thresholding methods.

A detailed simulation study is carried out and results show that when discontinuities are present a substantial reduction in bias can be obtained, leading to a corresponding reduction in mean square error.

Seismic exploration provides information about the ground substructures. Seismic images are generally corrupted by several noise sources. Hence, efficient denoising procedures are required to improve the detection of essential geological information. Wavelet bases provide sparse representation for a wide class of signals and images. This property makes them good candidates for efficient filtering tools, allowing the separation of signal and noise coefficients. Recent works have improved their performance by modelling the intra- and inter-scale coefficient dependencies using hidden Markov models, since image features tend to cluster and persist in the wavelet domain. This work focuses on the use of lapped transforms associated with hidden Markov modelling. Lapped transforms are traditionally viewed as block-transforms, composed of M pass-band filters. Seismic data present oscillatory patterns and lapped transforms oscillatory bases have demonstrated good performances for seismic data compression. A dyadic like representation of lapped transform coefficient is possible, allowing a wavelet-like modelling of coefficients dependencies. We show that the proposed filtering algorithm often outperforms the wavelet performance both objectively (in terms of SNR) and subjectively: lapped transform better preserve the oscillatory features present in seismic data at low to moderate noise levels.

Huang's data-driven technique of Empirical Mode Decomposition (EMD) is applied to the versatile, broadband, model of fractional Gaussian noise (fGn). The experimental spectral analysis and statistical characterization of the obtained modes reveal an equivalent filter bank structure which shares most properties of a wavelet decomposition in the same context, in terms of self-similarity, quasi-decorrelation and variance progression. Furthermore, the spontaneous adaptation of EMD to "natural" dyadic scales is shown, rationalizing the method as an alternative way for estimating the fGn Hurst exponent.

We study the statistical performance of multiresolution-based estimation procedures for the scaling exponents of multifractal processes. These estimators rely on the computation of multiresolution quantities such as wavelet, increment or aggregation coefficients. Estimates are obtained by linear fits performed in log of structure functions of order q versus log of scale plots. Using various and recent types of multiplicative cascades and a large variety of multifractal processes, we study and benchmark, by means of numerical simulations, the statistical performance of these estimation procedures. We show that they all undergo a systematic linearisation effect: for a range of orders q, the estimates account correctly for the scaling exponents; outside that range, the estimates significantly depart from the correct values and systematically behave as linear functions of q. The definition and characterisation of this effect are thoroughly studied. In contradiction with interpretations proposed in the literature, we provide numerical evidence leading to the conclusion that this linearisation effect is neither a finite size effect nor an infiniteness of moments effect, but that its origin should be related to the deep nature of the process itself. We comment on its importance and consequences for the practical analysis of the multifractal properties of empirical data.

We consider linear differential equations with bounded time delay driven by additive white noise. Our aim is the estimation of the maximal delay time from observations of one realisation of the solution process X under nonparametric drift assumptions. In the stationarity case the covariance function has a jump in the third derivative according to the location of the delay time. Based on this result, the delay time estimator is obtained from a singularity detection in the covariance function using a multiresolution framework. It is proved that the estimator attains the rate T^-1/3 for observation times T→∞, which corresponds to change point detection in an ill-posed setting.

The class of locally stationary wavelet processes is a wavelet-based model for covariance nonstationary zero mean time series. This paper presents an algorithm for the pointwise adaptive estimation of their time-varying spectral density. The performance of the procedure is evaluated on simulated and real time series. Two applications of the procedure are also presented and evaluated on real data. The first is a test of local existence for the coefficients of the so-called wavelet periodogram. The second is a new test of covariance stationarity.

We investigate the use of Hilbert wavelet pairs (HWPs) in the non-decimated discrete wavelet transform for the time-varying spectral analysis of multivariate time series. HWPs consist of two high-pass and two low-pass compactly supported filters, such that one high-pass filter is the Hilbert transform (approximately) of the other. Thus, common quantities in the spectral analysis of time series (e.g., power spectrum, coherence, phase) may be estimated in both time and frequency. Compact support of the wavelet filters ensures that the frequency axis will be partitioned dyadically as with the usual discrete wavelet transform. The proposed methodology is used to analyze a bivariate time series of zonal (u) and meridional (v) winds over Truk Island.