Some Results on the Wavelet Packet Decomposition of Nonstationary Processes

Touati, Sami; Pesquet, Jean-Christophe

doi:10.1155/s1110865702207040

Cited by 11 publications

(7 citation statements)

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“…In particular, it seems relevant to study to what extent the results of this paper extend to nonstationary or almost cyclostationary processes and relate to properties such as those given in [32].…”

Section: Resultsmentioning

confidence: 99%

On the statistical decorrelation of the wavelet packet coefficients of a band-limited wide-sense stationary random process

2007

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This paper is a contribution to the analysis of the statistical correlation of the wavelet packet coefficients resulting 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 with 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 with 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.

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Section: Resultsmentioning

confidence: 99%

On the statistical decorrelation of the wavelet packet coefficients of a band-limited wide-sense stationary random process

2007

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“…The wavelet packet coefficients of subspaces W j,l ; j = 1, 2, ..., J; l = 0, 1, ..., 2 j − 1 are approximately distributed as N (0, σ 2 ) [7]. As the level of decomposition increases, the distribution function converges to the real normal distribution [8].…”

Section: Wavelet Packet Decompositionmentioning

confidence: 99%

“…The detail coefficients can be modeled as a zero-mean Gaussian random process [7]. The problem of anomaly detection in the wavelet packet domain can be formulated as hypothesis testing with the absence of anomaly as the null hypothesis and the presence of the anomaly as the alternative hypothesis.…”

Section: Anomaly Detectionmentioning

confidence: 99%

A multi-scale energy detector for anomaly detection in dynamic networks

Mahyari

Aviyente

2013

2013 Asilomar Conference on Signals, Systems and Computers

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Complex networks have attracted a lot of attention for representing relational data, where the weights of the edges, where edges' weights show the strength of relationships. Complex networks have found numerous applications in social and biological sciences. Studying the topology of these networks is important for a better understanding of the underlying systems and data. Currently, most of the network analysis tools are limited to static networks. However, most networks of interest have edges or relationships that vary across time. Therefore, there is a need to develop methods for the study of these dynamic networks. In this paper, we introduce another aspect of threat detection, which is identifying abrupt changes in edges' weights over time. Wavelet decomposition method is used to separate the transient activity from the stationary activity in the edges. A hypothesis testing is proposed for the wavelet coefficients to detect any anomalous edges. Finally, the time points where the anomalous activity occurs are identified through the ratio of the energy of the anomalous to normal edges.

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“…This fact further emphasizes the interest in studying more precisely the dual-tree wavelet decomposition of a white noise. Note also that, by calculating higher order cumulants of the dual-tree wavelet coefficients and using techniques as in [18], [40], it could be proved that, for all…”

Section: Propositionmentioning

confidence: 99%

Noise Covariance Properties in Dual-Tree Wavelet Decompositions

Chaux

Pesquet

Duval

2007

IEEE Trans. Inform. Theory

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Abstract-Dual-tree wavelet decompositions have recently gained much popularity, mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy. When the decomposition of a random process is performed -which occurs in particular when an additive noise is corrupting the signal to be analyzed -it is useful to characterize the statistical properties of the dual-tree wavelet coefficients of this process. As dual-tree decompositions constitute overcomplete frame expansions, correlation structures are introduced among the coefficients, even when a white noise is analyzed. In this paper, we show that it is possible to provide an accurate description of the covariance properties of the dual-tree coefficients of a wide-sense stationary process. The expressions of the (cross-) covariance sequences of the coefficients are derived in the one and two-dimensional cases. Asymptotic results are also provided, allowing to predict the behaviour of the second-order moments for large lag values or at coarse resolution. In addition, the crosscorrelations between the primal and dual wavelets, which play a primary role in our theoretical analysis, are calculated for a number of classical wavelet families. Simulation results are finally provided to validate these results.

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Some Results on the Wavelet Packet Decomposition of Nonstationary Processes

Cited by 11 publications

References 11 publications

On the statistical decorrelation of the wavelet packet coefficients of a band-limited wide-sense stationary random process

On the statistical decorrelation of the wavelet packet coefficients of a band-limited wide-sense stationary random process

A multi-scale energy detector for anomaly detection in dynamic networks

Noise Covariance Properties in Dual-Tree Wavelet Decompositions

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