In the recent past years, scaling, random multiplicative cascades, multifractal stochastic processes became common paradigms used to analyse a large variety of different empirical times series characterised by scale invariance phenomena or properties.
Scale invariance implies that no characteristic scale can be identified in data or equivalently that all scales are equally
important. It also means that all scales are in relation ones with the others, hence the connection to multiplicative cascades, which, by construction, tie together a wide range of scales. Data with scale invariance are also often characterised by a high irregularity of their sample path. This variability is usually accounted for by Multifractal analysis. Hence, in applications, the three notions, scaling, multiplicative cascade and multifractal are often used ones for the others and even confusingly mixed up. These assimilations, that turned out to be fruitful in the early stages of the study of scaling, are now often responsible for misleading analysis and erroneous conclusions. Wavelet coefficients have long been used with relevance to analyse scaling. However, very recently, it has been shown that the analysis of multifractal properties can be significantly improved both conceptually and practically by the use of quantities referred to as wavelet leaders. The goals of this article are to introduce the wavelet leader based multifractal analysis, to detail its qualities and to show how it enables an insightful visit of the relationships between scaling, multifractal and multiplicative cascades.
In this paper, we consider the problem of detecting changes in dynamical systems from the analysis of the signals they produce. A notion of continuous multiresolution entropy is introduced, which combine advantages stemming from both classical entropy and wavelet analysis. The relevance of the approach, together with its robustness in the presence of moderate noise, is supported by numerical investigations.
Reassignment is a technique which consists in moving the computed value of a time-frequency or time-scale energy distribution to a different location in the plane, so as to increase its readability. In the case of scalograms (squared modulus of wavelet transforms), a general form is given for the reassignment operators and their properties are discussed with respect to the chosen wavelet. Characterization of local singularities after reassignment is investigated by simulation and some examples (from mathematics and physics) are presented in order to support the usefulness of the approach. Since reassigning a scalogram amounts to compute two extra wavelet transforms, it is finally shown how this can be achieved in a fast and efficient way within a multiresolution framework.
Often, the Discrete Wavelet Transform is performed and implemented with the Daubechies wavelets, the Battle-Lemarie wavelets or the splines wavelets whereas in continuous time wavelet decomposition a much larger variety of mother wavelets are used. Maintaining the dyadic time-frequency sampling and the recursive pyramidal computational structure, we present various methods to obtain any chosen analyzing wavelet (psi) w, with some desired shape and properties and which is associated with a semi-orthogonal multiresolution analysis or to a pair of bi-orthogonal multiresolutions. We explain in details how to design one's own wavelet, starting from any given Multiresolution Analysis or any pair of bi-orthogonal multiresolutions. We also explicitly derive, in a very general oblique (or bi-orthogonal) framework, the formulae of the filter bank structure that implements the designed wavelet. We illustrate these wavelet design, techniques with examples that we have programmed with Matlab routines, available upon request.
Conference Committee Involvement (7)
Wavelet Applications in Industrial Processing VII
18 January 2010 | San Jose, California, United States
Wavelet Applications in Industrial Processing VI
21 January 2009 | San Jose, California, United States
Wavelet Applications in Industrial Processing V
11 September 2007 | Boston, MA, United States
Wavelet Applications in Industrial Processing IV
2 October 2006 | Boston, Massachusetts, United States
Wavelet Applications in Industrial Processing III
24 October 2005 | Boston, MA, United States
Wavelet Applications in Industrial Processing II
27 October 2004 | Philadelphia, Pennsylvania, United States
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