Extending multifractal analysis to negative regularity: P-exponents and P-leaders

Scale invariance is a widely used concept to analyze real-world data from many different applications and multifractal analysis has become the standard corresponding signal processing tool. It characterizes data by describing globally and geometrically the fluctuations of local regularity, usually measured by means of the Hölder exponent. A major limitation of the current procedure is that it applies only to locally bounded functions or signals, i.e., to signals with positive regularity. The present contribution proposes to characterize local regularity with a new quantity, the p-exponent, that permits negative regularity in data, a widely observed property in real-world data. Relations to Hölder exponents are detailed and a corresponding p-leader multifractal formalism is devised and shown at work on synthetic multifractal processes, representative of a class of models often used in applications. We formulate a conjecture regarding the equivalence between Hölder and p-exponents for a subclass of processes. Even when Hölder and p-exponents coincide, the p-leader formalism is shown to achieve better estimation performance.