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

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 1/4-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. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

This study investigates the similarities and differences in results when calculating D, H, and RI with different methods, finding that performance varies depending on the dataset used and that the relationship between RI and H and D is non-linear.