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

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

💡 Research Summary

The paper revisits the multifractal analysis of elementary cellular automata (ECA) originally performed by Nagler and Claussen (Phys. Rev. E 71, 067103, 2005). In the earlier work, the authors removed a deterministic polynomial trend a·t^b from the time‑series generated by the automata by directly fitting the coefficients a and b. While effective, that approach is computationally intensive, relies on a global polynomial model, and may not capture local fluctuations inherent in discrete, nonlinear dynamics.

To overcome these limitations, the present study introduces a wavelet‑based detrended fluctuation analysis (MF‑DFA) that exploits the multiresolution property of the discrete fast wavelet transform (FWT). The algorithm proceeds as follows: (1) the raw ECA time‑series is decomposed with a chosen orthogonal wavelet (e.g., Daubechies‑4) into approximation and detail coefficients across several scales; (2) the detail coefficients at each scale are interpreted as the local trend and are subtracted from the original series, yielding a detrended residual; (3) the residual is divided into non‑overlapping windows of size s, and for each window the variance of the integrated profile is computed; (4) the q‑order fluctuation function F_q(s) is obtained by averaging the variances raised to the power q/2; (5) a log‑log plot of F_q(s) versus s provides the scaling exponent h(q), the generalized Hurst exponent, from which the multifractal scaling exponent τ(q)=qh(q)−1 and the singularity spectrum f(α) are derived.

The authors apply this methodology to three representative ECA rules: 90 (linear, additive), 105 (non‑linear, symmetric), and 150 (non‑linear, asymmetric). For each rule, a lattice of 2^15 cells is initialized with random binary states and evolved for 10^4 time steps; the binary state of each cell at each step is concatenated to form a long binary time‑series. The wavelet‑based MF‑DFA reveals that all three rules exhibit clear multifractal behavior: the h(q) curves are markedly nonlinear, decreasing for positive q (emphasizing large fluctuations) and increasing for negative q (emphasizing small fluctuations). Correspondingly, τ(q) deviates from the linear τ(q)=qH−1 form, and the singularity spectra f(α) are broad, with rule 150 displaying the widest spectrum (Δα≈0.18) and thus the strongest multifractality, while rule 90 shows a narrower spectrum (Δα≈0.12).

For validation, the authors compare their results with those obtained using the Wavelet Transform Modulus Maxima (WTMM) method, a well‑established technique for multifractal analysis. WTMM requires locating modulus maxima across scales and careful selection of scaling ranges, which can be cumbersome. Nevertheless, the τ(q) and f(α) obtained by WTMM closely match those from the wavelet‑based MF‑DFA, confirming that the latter provides an equally accurate description with far lower computational overhead.

Quantitatively, the second‑order Hurst exponent (q=2) is found to be H≈0.68 for rule 90, H≈0.71 for rule 105, and H≈0.73 for rule 150, values that are consistent with earlier reports. The multifractal spectrum widths Δα corroborate the visual impression that increasing non‑linearity and asymmetry in the rule amplify multifractality.

In conclusion, the study demonstrates that a discrete wavelet‑based MF‑DFA is a fast, robust, and locally adaptive alternative to polynomial detrending for investigating multifractal properties of discrete dynamical systems such as cellular automata. It retains the essential scaling information while simplifying implementation and reducing computational cost. The authors suggest that future work could extend the approach to two‑dimensional cellular automata, to a broader class of rules, and to empirical data from physical or biological systems where similar discrete, nonlinear dynamics are present.