Revisiting scaling, multifractal, and multiplicative cascades with the wavelet leader lens

摘要

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.