The non-linear relationship between randomness and scaling properties such as fractal dimensions and Hurst exponent in distributed signals

Abstract Fractal-dimensions (D) and Hurst-exponent (H) are often used for determining a randomness (RI) or predictability index in distributed signals, from the linear relationship of RI=1–H=D–1, as H+D=2. This paper investigates the similarities and differences of the results of different methods, when calculating D, H, and RI with the same dataset signals. 8 different methods were tested: Higuchi's (D), Saupe's Variance (H), Dispersional (H), Rescaled-Adjusted-Range R/S (H), Detrended-Fluctuation-Analysis DFA (H), Runs (RI), Persistence-Antipersistence (RI), and ¼-Variance-ratio (RI). These methods were tested with distributed datasets, namely (1) fractional Gaussian noise and its time derivatives, (2) datasets of expected RI, and (3) an EEG signal. All D and H data were converted to RI. For datasets (1), all methods performed equally well for datasets of H=0.5, although the standard deviations of some methods were greater than 0.02. For datasets (2), applied only to RI methods, Runs and Persistence-Antipersistence methods were accurate. All 8 methods performed reasonably well when processing the EEG signal. The relationship between RI and H and D is not a linear one and rather follows the square root of a quadratic function. From this function, however, the actual RI (calculated from RI methods) is not defined if the expected RI (obtained from H and D methods via a linear relationship) equals 1. In this case, the actual RI can be anywhere between 2/3 and 1. Therefore, we suggest, based on the results of this study, that the RI is inaccurately and incompletely determined when using the detour via H & D methods, and that the RI is accurately and directly derived from the Runs or RI p-ap methods, which should be used when the RI, and associated parameters such as persistence, anti-persistence, and predictability are of interest.