Computing the multifractal spectrum from time series: An algorithmic approach

TL;DR

This paper introduces a new automated algorithmic scheme for accurately computing the multifractal spectrum from time series data, improving upon existing methods by capturing the full spectrum and applicable to noisy real-world data.

Contribution

The paper presents a novel algorithm that efficiently computes the complete multifractal spectrum from time series, adaptable to high dimensions and noisy data, with potential diagnostic applications.

Findings

Successfully tested on logistic attractor with known spectrum

Effectively applied to higher-dimensional and noisy real-world data

Preliminary results suggest parameters as diagnostic measures

Abstract

We show that the existing methods for computing the f(\alpha) spectrum from a time series can be improved by using a new algorithmic scheme. The scheme relies on the basic idea that the smooth convex profile of a typical f(\alpha) spectrum can be fitted with an analytic function involving a set of four independent parameters. While the standard existing schemes [16, 18] generally compute only an incomplete f(\alpha) spectrum (usually the top portion), we show that this can be overcome by an algorithmic approach which is automated to compute the Dq and f(\alpha) spectrum from a time series for any embedding dimension. The scheme is first tested with the logistic attractor with known f(\alpha) curve and subsequently applied to higher dimensional cases. We also show that the scheme can be effectively adapted for analysing practcal time series involving noise, with examples from two widely different real world systems. Moreover, some preliminary results indicating that the set of four independant parameters may be used as diagnostic measures is also included.