Paulo Gonçalves

Empirical mode decomposition as a filter bank

Empirical mode decomposition (EMD) has recently been pioneered by Huang et al. for adaptively representing nonstationary signals as sums of zero-mean amplitude modulation frequency modulation components. In order to better understand the way EMD behaves in stochastic situations involving broadband noise, we report here on numerical experiments based on fractional Gaussian noise. In such a case, it turns out that EMD acts essentially as a dyadic filter bank resembling those involved in wavelet decompositions. It is also pointed out that the hierarchy of the extracted modes may be similarly exploited for getting access to the Hurst exponent.

Empirical Mode Decompositions as data-driven wavelet-like expansions

Huangs 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.

Wavelets, spectrum analysis and 1/f processes

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.

Computational methods for hidden Markov tree models-an application to wavelet trees

The hidden Markov tree models were introduced by Crouse et al. in 1998 for modeling nonindependent, non-Gaussian wavelet transform coefficients. In their paper, they developed the equivalent of the forward-backward algorithm for hidden Markov tree models and called it the upward-downward algorithm. This algorithm is subject to the same numerical limitations as the forward-backward algorithm for hidden Markov chains (HMCs). In this paper, adapting the ideas of Devijver from 1985, we propose a new upward-downward algorithm, which is a true smoothing algorithm and is immune to numerical underflow. Furthermore, we propose a Viterbi-like algorithm for global restoration of the hidden state tree. The contribution of those algorithms as diagnosis tools is illustrated through the modeling of statistical dependencies between wavelet coefficients with a special emphasis on local regularity changes.

Pseudo affine Wigner distributions: definition and kernel formulation

We introduce a new set of tools for time-varying spectral analysis: the pseudo affine Wigner distributions. Based on the affine Wigner distributions of J. and P. Bertrand (1992), these new time-scale distributions support efficient online operation at the same computational cost as the continuous wavelet transform. Moreover, they take advantage of the proportional bandwidth smoothing inherent in the sliding structure of their implementation to suppress cumbersome interference components. To formalize their place within the echelon of the affine class of time-scale distributions (TSDs), we introduce and study an alternative set of generators for this class.

Empirical mode decomposition, fractional Gaussian noise and Hurst exponent estimation

Huangs data-driven technique of empirical mode decomposition (EMD) is applied to the versatile, broadband, model of fractional Gaussian noise (fGn). The spectral analysis and statistical characterization of the obtained modes reveal an equivalent filter bank structure together with gamma distributed variances, both sharing some properties with wavelet decompositions. These common features are then used to mimic wavelet based techniques aimed at estimating the Hurst exponent.

A simple statistical analysis of wavelet-based multifractal spectrum estimation

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. 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.

Multifractal analysis of fetal heart rate variability in fetuses with and without severe acidosis during labor.

We performed multifractal analysis of fetal heart rate (FHR) variability in fetuses with and without acidosis during labor. Multifractal analysis was performed on fetal electrocardiograms in 10-minute sliding windows within the last 2 hours before delivery in 45 term fetuses divided in three groups according to umbilical arterial pH and FHR pattern: group A had pH ≥7.30 and normal FHR, group B had pH ≥7.30 and intermediate or abnormal FHR, and group C had acidosis (pH ≤7.05) and intermediate or abnormal FHR. Six multifractal parameters were compared using Wilcoxon rank sum test. Multifractal parameters were significantly different between the three groups in the last 10 minutes before delivery (P <0.05). Two parameters (H(min), zeta(2)) exhibited a significant difference 70 minutes before delivery, and one parameter (C(2)) was different 10 minutes before birth (P <0.05). Multifractal parameters were significantly different in acidotic and nonacidotic fetuses, independently from FHR pattern.

Adaptive diffusion as a versatile tool for time-frequency and time-scale representations processing: a review

Inspired by the work on image processing by Perona and Malik, diffusion-based models were first investigated by Goncalve/spl grave/s and Payot to improve the readability of Cohen class time-frequency representations. They rely on signal-dependent partial differential equations that yield adaptive smoothed representations with sharpened time-frequency components. Here, we demonstrate the versatility and utility of this family of methods, and we propose new adaptive diffusion processes to locally control both the orientation and the strength of smoothing. Our approach is an improvement on previous works as it provides a unified framework not only for the Cohen class but for the affine class as well. The latter is of particular interest because, except for some special techniques such as the reassignment method, no signal-dependent smoothing technique exists to process bilinear time-scale distributions, nor even a transposition of the adaptive optimal-kernel method proposed by Baraniuk and Jones.

Multiple-window wavelet transform and local scaling exponent estimation

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 satisfy 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.