Multifractal Analysis Based on p-Exponents and Lacunarity Exponents

Abstract

Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the Hölder exponent, is not feasible. We present a multifractal analysis based on another quantity, the p-exponent, which can take arbitrarily large negative values. We investigate some mathematical properties of this exponent, and show how it allows us to model the idea of "lacunarity" of a singularity at a point. We finally adapt the wavelet based multifractal analysis in this setting, and we give applications to a simple mathematical model of multifractal processes: Lacunary wavelet series.

An integrate-and-fire model to generate spike trains with long-range dependence

“…More generally, fractional Brownian motion provides a good, mathematically tractable, model of so-called 1/f noise, see e.g. (Abry et al, 1995). 1/f noise has been successfully applied to describe many phenomena, from heartbeat (Peng et al, 1993) to internet traffic (Willinger et al, 1997), including neuronal fractal dynamics (Lowen et al, 2001;Sobie et al, 2011).…”

“…The second approach to model LRD is an Integrateand-Fire model with 1/f noise proposed by Sobie et al (2011), strongly related to our model. The link between fractional Brownian motion and 1/f noise is explained in (Abry et al, 1995), although there is no definition of 1/f noise as clear and universally accepted as the definition of fBm can be. Besides, an advantage of using fBm is that it can be exactly simulated, which ensures that all frequencies are present in its spectrum and that LRD holds, while a simulated 1/f noise is an approximate 1/f noise with limited bandwidth.…”

Multifractal analysis of the Brjuno function

“…In contradistinction with the Hölder case, few p-spectrums have been determined: Let us mention the characteristic functions of some fractal sets [21] and random wavelet series [2]; generic results (in the Baire and prevalence settings) for functions in a Sobolev space were obtained by A. Fraysse [16]; recently, 2-exponents of trigonometric series which are not locally bounded were obtained by S. Seuret and A. Ubis [33].…”

“…We can assume, by extracting a subsequence if necessary, that for all n ≥ 1, x 0 − p n /q n ≤ 1/q τ n , and we pick for r the sequence of convergents r n = p n /q n , ρ n = 1/q τ n , and h n = 1/q τ n (log q n ) 2 . We obtain that…”