Multifractal analyses of row sum signals of elementary cellular automata

TL;DR

This study compares multifractal analysis methods on cellular automata signals, finding MFDFA generally more stable and WTMM more sensitive, with results varying based on the specific parameter analyzed.

Contribution

It systematically evaluates the stability and effectiveness of WT-MFDFA, MFDFA, and WTMM methods on cellular automata row sum signals, providing insights into their relative performance.

Findings

MFDFA outperforms WT-MFDFA and WTMM in estimating multifractal support.

WT-MFDFA provides more accurate estimates for Hurst exponent and Holder exponent.

Initial conditions do not significantly affect the multifractal parameters.

Abstract

We first apply the WT-MFDFA, MFDFA, and WTMM multifractal methods to binomial multifractal time series of three different binomial parameters and find that the WTMM method indicates an enhanced difference between the fractal components than the known theoretical result. Next, we make use of the same methods for the time series of the row sum signals of the two complementary ECA pairs of rules (90,165) and (150,105) for ten initial conditions going from a single 1 in the central position up to a set of ten 1's covering the ten central positions in the first row. Since the members of the pairs are actually similar from the statistical point of view, we can check which method is the most stable numerically by recording the differences provided by the methods between the two members of the pairs for various important quantities of the scaling analyses, such as the multifractal support, the most frequent Holder exponent, and the Hurst exponent and considering as the better one the method that provides the minimum differences. According to this criterion, our results show that the MFDFA performs better than WT-MFDFA and WTMM in the case of the multifractal support, while for the other two scaling parameters the WT-MFDFA is the best. The employed set of initial conditions does not generate any specific trend in the values of the multifractal parameters