pp-exponent and pp-leaders, Part II: Multifractal analysis. Relations to detrended fluctuation analysis

• We propose to base multifractal analysis on p-exponents and p-leaders.
• This tool can directly be used with non locally bounded data (negative regularity).
• Estimation performance improves as the parameter p is decreased.
• We provide theoretical and practical connections with Multifractal DFA.

Multifractal analysis studies signals, functions, images or fields via the fluctuations of their local regularity along time or space, which capture crucial features of their temporal/spatial dynamics. It has become a standard signal and image processing tool and is commonly used in numerous applications of different natures. In its common formulation, it relies on the Hölder exponent as a measure of local regularity, which is by nature restricted to positive values and can hence be used for locally bounded functions only. In this contribution, it is proposed to replace the Hölder exponent with a collection of novel exponents for measuring local regularity, the pp-exponents. One of the major virtues of pp-exponents is that they can potentially take negative values. The corresponding wavelet-based multiscale quantities, the pp-leaders, are constructed and shown to permit the definition of a new multifractal formalism, yielding an accurate practical estimation of the multifractal properties of real-world data. Moreover, theoretical and practical connections to and comparisons against another multifractal formalism, referred to as multifractal detrended fluctuation analysis, are achieved. The performance of the proposed pp-leader multifractal formalism is studied and compared to previous formalisms using synthetic multifractal signals and images, illustrating its theoretical and practical benefits. The present contribution is complemented by a companion article studying in depth the theoretical properties of pp-exponents and the rich classification of local singularities it permits.