Multifractal properties of elementary cellular automata in a discrete wavelet approach of MF-DFA

TL;DR

This paper applies a wavelet-based multifractal detrended fluctuation analysis to elementary cellular automata time series, confirming their multifractal nature more efficiently than previous polynomial fitting methods.

Contribution

It introduces a wavelet-based MF-DFA method for analyzing ECA time series, improving trend removal speed and accuracy over traditional polynomial fitting.

Findings

Confirmed multifractal behavior of ECA rules 90, 105, and 150

Demonstrated the efficiency of wavelet-based trend filtering

Provided scaling parameters for the analyzed ECA rules

Abstract

In 2005, Nagler and Claussen (Phys. Rev. E 71 (2005) 067103) investigated the time series of the elementary cellular automata (ECA) for possible (multi)fractal behavior. They eliminated the polynomial background at^b through the direct fitting of the polynomial coefficients a and b. We here reconsider their work eliminating the polynomial trend by means of the multifractal-based detrended fluctuation analysis (MF-DFA) in which the wavelet multiresolution property is employed to filter out the trend in a more speedy way than the direct polynomial fitting and also with respect to the wavelet transform modulus maxima (WTMM) procedure. In the algorithm, the discrete fast wavelet transform is used to calculate the trend as a local feature that enters the so-called details signal. We illustrate our result for three representative ECA rules: 90, 105, and 150. We confirm their multifractal behavior and provide our results for the scaling parameters