Fractal Dimension of Self-Affine Signals: Four Methods of Estimation

Abstract

This paper serves as a complementary material to a poster presented at the XXXVI Dynamics Days Europe in Corfu, Greece, on June 6th-10th in 2016. In this study, fractal dimension ($D$) of two types of self-affine signals were estimated with help of four methods of fractal complexity analysis. The methods include the Higuchi method for the fractal dimension computation, the estimation of the spectral decay ($\beta$), the generalized Hurst exponent ($H$), and the detrended fluctuation analysis. For self-affine processes, the next relation between the fractal dimension, Hurst exponent, and spectral decay is valid: $D=2-H=\frac{5-\beta}{2}$. Therefore, the fractal dimension can be get from any of the listed characteristics. The goal of the study is to find out which of the four methods is the most reliable. For this purpose, two types of test data with exactly given fractal dimensions ($D = 1.2, 1.4, 1.5, 1.6, 1.8$) were generated: the graph of the self-affine Weierstrass function and the statistically self-affine fractional Brownian motion. The four methods were tested on the both types of time series. Effect of noise added to data and effect of the length of the data were also investigated. The most biased results were obtained by the spectral method. The Higuchi method and the generalized Hurst exponent were the most successful.