p-exponent and p-leaders, Part II: Multifractal Analysis. Relations to Detrended Fluctuation Analysis

TL;DR

This paper introduces p-exponents and p-leaders as novel tools for multifractal analysis, extending the traditional framework by allowing negative regularity measures and improving the estimation of multifractal properties in real data.

Contribution

It proposes a new multifractal formalism based on p-exponents and p-leaders, with theoretical foundations and practical advantages over existing methods, including connections to detrended fluctuation analysis.

Findings

p-exponents can take negative values, unlike H"older exponents.

p-leaders enable accurate multifractal spectrum estimation.

The formalism outperforms previous methods on synthetic data.

Abstract

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 H\"older 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 H\"older exponent with a collection of novel exponents for measuring local regularity, the $p$-exponents. One of the major virtues of $p$-exponents is that they can potentially take negative values. The corresponding wavelet-based multiscale quantities, the $p$-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 $p$-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 $p$-exponents and the rich classification of local singularities it permits.