Wavelet Packets of Nonstationary Random Processes: Contributing Factors for Stationarity and Decorrelation (original) (raw)
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Some Results on the Wavelet Packet Decomposition of Nonstationary Processes
Eurasip Journal on Advances in Signal Processing, 2002
Wavelet/wavelet packet decomposition has become a very useful tool in describing nonstationary processes. Important examples of nonstationary processes encountered in practice are cyclostationary processes or almost-cyclostationary processes. In this paper, we study the statistical properties of the wavelet packet decomposition of a large class of nonstationary processes, including in particular cyclostationary and almost-cyclostationary processes. We first investigate in a general framework, the existence and some properties of the cumulants of wavelet packet coefficients. We then study more precisely the almost-cyclostationary case, and determine the asymptotic distributions of wavelet packet coefficients. Finally, we particularize some of our results in the cyclostationary case before providing some illustrative simulations.
Wavelets for time series analysis - a survey and new results
Control and Cybernetics, 2005
In the paper we review stochastic properties of wavelet coefficients for time series indexed by continuous or discrete time. The main emphasis is on decorrelation property and its implications for data analysis. Some new properties are developed as the rates of correlation decay for the wavelet coefficients in the case of longrange dependent processes such as the fractional Gaussian noise and the fractional autoregressive integrated moving average processes. It is proved that for such processes the within-scale covariance of the wavelet coefficients at lag k is O(k 2(H−N)−2), where H is the Hurst exponent and N is the number of vanishing moments of the wavelet employed. Some applications of decorrelation property are briefly discussed.
Central Limit Theorems for Wavelet Packet Decompositions of Stationary Random Processes
IEEE Transactions on Signal Processing, 2010
This paper provides central limit theorems for the wavelet packet decomposition of stationary bandlimited random processes. The asymptotic analysis is performed for the sequences of the wavelet packet coefficients returned at the nodes of any given path of the M-band wavelet packet decomposition tree. It is shown that if the input process is centred and strictly stationary, these sequences converge in distribution to white Gaussian processes when the resolution level increases, provided that the decomposition filters satisfy a suitable property of regularity. For any given path, the variance of the limit white Gaussian process directly relates to the value of the input process power spectral density at a specific frequency. Index Terms Wavelet transforms, Band-limited stochastic processes, Spectral analysis. I. INTRODUCTION This paper addresses the statistical properties of the M-Band Discrete Wavelet Packet Transform, hereafter abbreviated as M-DWPT. In [1], asymptotic analysis is given for the correlation structure and the distribution of the M-Band wavelet packet coefficients of stationary random processes. The limit autocorrelation functions and distributions are shown to be the same for every M-DWPT path. This seems to be a paradox because the M-DWPT paths are characterised by several sequences of wavelet filters. Two arbitrary sequences are different, and thus, do not have the same properties. In addition, the
Wavelet Transform Application for/in Non-Stationary Time-Series Analysis: A Review
Applied Sciences, 2019
Non-stationary time series (TS) analysis has gained an explosive interest over the recent decades in different applied sciences. In fact, several decomposition methods were developed in order to extract various components (e.g., seasonal, trend and abrupt components) from the non-stationary TS, which allows for an improved interpretation of the temporal variability. The wavelet transform (WT) has been successfully applied over an extraordinary range of fields in order to decompose the non-stationary TS into time-frequency domain. For this reason, the WT method is briefly introduced and reviewed in this paper. In addition, this latter includes different research and applications of the WT to non-stationary TS in seven different applied sciences fields, namely the geo-sciences and geophysics, remote sensing in vegetation analysis, engineering, hydrology, finance, medicine, and other fields, such as ecology, renewable energy, chemistry and history. Finally, five challenges and future w...
HAL (Le Centre pour la Communication Scientifique Directe), 2006
This paper is a contribution to the analysis of the statistical correlation of the wavelet packet coefficients issued from the decomposition of a random process, stationary in the wide-sense, whose power spectral density is bounded with support in [−π, π]. Consider two quadrature mirror filters (QMF) that depend on a parameter r, such that these filters tend almost everywhere to the Shannon QMF when r increases. The parameter r is called the order of the QMF under consideration. The order of the Daubechies filters (resp. the Battle-Lemarié filters) is the number of vanishing moments of the wavelet function (resp. the spline order of the scaling function). Given any decomposition path in the wavelet packet tree, the wavelet packet coefficients are proved to decorrelate for every packet associated to a large enough resolution level, provided that the QMF order is large enough and above a value that depends on this wavelet packet. Another consequence of our derivation is that, when the coefficients associated to a given wavelet packet are approximately decorrelated, the value of the autocorrelation function of these coefficients at lag 0 is close to the value taken by the power spectral density of the decomposed process at a specific point. This specific point depends on the path followed in the wavelet packet tree to attain the wavelet packet under consideration. Some simulations highlight the good quality of the "whitening" effect that can be obtained in practical cases.
Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significance testing
Physical Review E, 2007
We propose an equivalence class of nonstationary Gaussian stochastic processes defined in the wavelet domain. These processes are characterized by means of wavelet multipliers and exhibit well-defined timedependent spectral properties. They allow one to generate realizations of any wavelet spectrum. Based on this framework, we study the estimation of continuous wavelet spectra, i.e., we calculate variance and bias of arbitrary estimated continuous wavelet spectra. Finally, we develop an areawise significance test for continuous wavelet spectra to overcome the difficulties of multiple testing; it uses basic properties of continuous wavelet transform to decide whether a pointwise significant result is a real feature of the process or indistinguishable from typical stochastic fluctuations. This test is compared to the conventional one in terms of sensitivity and specificity. A software package for continuous wavelet spectral analysis and synthesis is presented.
ó Consider the wavelet packet coefcients issued from the decomposition of a random process stationary in the wide- sense. We address the asymptotic behaviour of the autocorre- lation of these wavelet packet coefcients. In a rst step, we explain why this analysis is more intricate than that already achieved by several authors in the case of the standard discrete orthonormal wavelet decomposition. In a second step, it is shown that the autocorrelation of the wavelet packet coefcients can be rendered arbitrarily small provided that both the decomposition level and the regularity of the quadrature mirror lters are large enough.
Some aspects of wavelet analysis in time series
2000
This thesis consists of two papers dealing with the methodology of wavelet and its application in time series. We give here a brief summary of the contents of the two papers in the thesis. The first paper describes an alternative approach for testing the existence of trend among time series. The test method has been constructed using wavelet analysis which have the ability to decompose a time series into low frequencies (trend) and high frequencies (noise) components. Under the normality assumption the test is distributed as F. However, the distribution of the test is unknown under other conditions, like non-normality. To investigate the properties of the test statistic under