Contributions of Wavelet Leaders and Bootstrap to Multifractal Analysis: Images, Estimation Performance, Dependence Structure and Vanishing Moments. Confidence Intervals and Hypothesis Tests.

Scale invariance constitutes a paradigm that is frequently used for the analysis and modeling of empirical data in various applications of very different natures. Multifractal analysis provides a conceptual framework for its theoretical and practical studies. The aim of this thesis is to investigate the benefits of the use of wavelet Leaders, on one hand, and bootstrap methods, for practical multifractal analysis. In the first part of this work, the statistical properties and performance of wavelet Leader based multifractal analysis procedures are studied. It is shown that they compare very favorably to those obtained by wavelet coefficient based ones. Moreover, a practical extension to two dimensional signals (images) is validated. In addition, a number of theoretical questions of fundamental practical importance in applications are investigated: Function space embedding models and minimum regularity, linearization effect, robustness with respect to quantization of the data. The second part of this thesis proposes bootstrap based procedures for statistical inference in multifractal analysis. These procedures are validated by numerical simulations and permit the construction of confidence intervals and hypothesis tests for multifractal attributes, from one single finite length observation of data. This is achieved by an original time-scale block bootstrap approach in the wavelet domain. This work is further completed by the detailed study of the dependence structures of wavelet coefficients and wavelet Leaders. Notably, it is shown that the number of vanishing moments of the analyzing wavelet, which permits to convert long range to weak dependence for fractional Brownian motion, is ineffective for multifractal multiplicative cascades: Increasing the number of vanishing moments still controls the correlation of wavelet coefficients, but has no effect on their long range dependence structure. Finally, the wavelet Leader and bootstrap based multifractal analysis tools are applied to hydrodynamic turbulence data, and to texture image classification.