Multifractal analyses of row sum signals of elementary cellular automata

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

💡 Research Summary

This paper conducts a systematic comparison of three widely used multifractal analysis techniques—Wavelet Transform Detrended Fluctuation Analysis (WT‑MFDFA), Multifractal Detrended Fluctuation Analysis (MFDFA), and the Wavelet Transform Modulus Maxima method (WTMM)—applied to discrete time series generated by elementary cellular automata (ECA). The authors first validate the methods on synthetic binomial multifractal series, whose analytical scaling exponents h(q) and τ(q) are known. The binomial tests reveal that WT‑MFDFA and MFDFA closely reproduce the theoretical spectra, whereas WTMM tends to over‑estimate the differences between the estimated and exact spectra, especially in the width of the multifractal support.

Having established a baseline, the study proceeds to the main object of interest: the row‑sum signals of two complementary pairs of ECA rules, (90, 165) and (150, 105). For each rule, ten distinct initial conditions are considered, ranging from a single ‘1’ placed at the centre of the first row to a block of ten consecutive ‘1’s covering the central positions. Each simulation runs for 2^16 time steps, producing long time series that capture the evolution of the number of active cells (the row sum) at each step.

All three methods are applied to each series, and three key multifractal descriptors are extracted: (i) the multifractal support (the interval