p-exponent and p-leaders, Part I: Negative pointwise regularity

Abstract

Multifractal analysis aims to characterize signals, functions, images or fields, via the fluctuations of their local regularity along time or space, hence capturing crucial features of their temporal/spatial dynamics. Multifractal analysis is becoming a standard tool in signal and image processing, and is nowadays widely used in numerous applications of different natures. Its common formulation relies on the measure of local regularity via the Hölder exponent, by nature restricted to positive values, and thus to locally bounded functions or signals. It is here proposed to base the quantification of local regularity on p-exponents, a novel local regularity measure potentially taking negative values. First, the theoretical properties of p-exponents are studied in detail. Second, wavelet-based multiscale quantities, the p-leaders, are constructed and shown to permit accurate practical estimation of p-exponents. Exploiting the potential dependence with p, it is also shown how the collection of p-exponents enriches the classification of locally singular behaviors in functions, signals or images. The present contribution is complemented by a companion article developing the p-leader based multifractal formalism associated to p-exponents.

Wavelet $p$-Leader Non Gaussian Multiscale Expansions for Heart Rate Variability Analysis in Congestive Heart Failure Patients

“…When based on p-leaders, the regularity at time t is measured by the p-exponent h(t), a generalization of the usual Hölder exponent, cf., [26], [27]. Expressions (3)(4)(5)(6) highlight the strong link between D(h) and the scale invariance properties of X.…”

“…The p-leaders, with p ≥ 0, thus consist of weighted averages of wavelet coefficients existing at all finer scales 2 j ≤ 2 j and within a local temporal neighborhood λ j ,k ⊂ 3λ j,k . For p-leaders, estimation benefits from smaller variance, and both positive and negative statistical moments are well defined and numerically stable (unlike, e.g., those of DWT or DFA coefficients), hence permitting the practical assessment of their scale-by-scale distributions, see [26] for precise definitions and details, beyond the scope of this contribution. Following [27], p = 1 is used in all analyses reported below.…”

“…In this context, elaborating on the concepts of fractal and non linear analyses, the overall goal of the present contribution is to construct flexible non Gaussian multiscale representations that benefit from: i) the use of a large continuum of time scales, yet without requiring a priori exact scale-free dynamics and hence permitting potential departure from multifractal models; ii) the possibility for selecting the (range of) statistical orders for which analysis is to be made, hence clearly departing from the linear, or correlation, or second-order, analysis and its underlying Gaussian assumption. These non Gaussian multiscale representations are embedded within the wavelet p-leader formalism, recently defined in [26] and shown to achieve robust estimation of the statistical characteristics of real-world data [27]. They are presented theoretically and illustrated on synthetic time series in Section II.…”

Multivariate multifractal analysis

“…The support of the multifractal spectrum is {H : E f (H) = ∅} = {H : dim(E f (H)) = −∞}. Different pointwise exponents h f (x 0 ) can be used: The most widespread is the Hölder exponent, cf., e.g., [1] and references therein; one recently studied notion for regularity are the p-exponents h p f (x 0 ), where p > 0 is a parameter, which allow to measure the regularity of non-locally bounded functions, see [2,3]. Two pointwise exponents fitted to a probability measure µ on…”

“…A simple example of this situation is supplied by probability measures for which d λ = µ(3λ) clearly is a multiresolution quantity associated with H µ defined by (3). Multiresolution quantities that yield the Hölder and p-exponents of a function f can be derived from the wavelet decomposition of f , see [1,2,3].…”