Jubilee 2019

Multifractal analysis quantifies the fluctuations of regularities in time series or textures through the estimation of their ''multifractal spectrum'' which encapsulates the fractional dimensions of the singularity sets of a function. Wavelet techniques supply robust tools in order to perform multifractal analysis, and they became a standard signal/image processing method for classification or model selection, successfully used in a large variety of applicative contexts. Yet, successes were confined to the analysis of one signal or image at a time (univariate analysis). In view of many modern real-world applications that rely on the joint (multivariate) analysis of collections of signals or images, the need for extensions to multivariate settings became a major challenge.

We will describe the theoretical foundations of multivariate multifractal analysis, which proposes to estimate the multivariate multifractal spectrum of several signals as a new way to reveal the correlations between their singularity sets. We will study the properties and limitations of the most natural extension of the univariate formalism to a multivariate formulation, and explain why, while performing well for some models, this natural extension is not valid in general. Based on the theoretical study of the mechanisms leading to failure, we will show that it is a consequence of the ''large intersection property'' satisfied by the singularity sets of the data. We will illustrate these theoretical results by numerical computations worked out for several classical models of stochastic processes. Finally, we will mention several open problems which are raised by these new techniques. This talk is based on joint works with Patrice Abry, Roberto Leonarduzzi, Stéphane Seuret and Herwig Wendt.