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

"In 2005, Nagler and Claussen (Phys. Rev. E 71, 067103 (2005)) investigated the time series of the elementary cellular automata (ECA) for possible (multi)fractal behavior. They eliminated the polynomial background atb through the direct ?t-ting of the polynomial coe?cients a and b. We here reconsider their work eliminating the polynomial trend by means of the multifractal-based detrended ?uctuation analysis (MF-DFA) in which the wavelet multiresolution property is employed to ?lter out the trend in a more speedy way than the direct polynomial ?tting 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 con?rm their multifractal behavior and provide our results for the scaling parameters.