Hurst exponent estimation using natural visibility graph embedding in Fisher–Shannon plane

Abstract

In this article, two important stochastic processes, namely fractional Brownian motions (fBm) and fractional Gaussian noises (fGn) are analyzed, within a Fisher–Shannon framework. These processes are well suited for the realistic modeling of phenomena occurring across various domains in science and engineering. An unique feature that characterizes both fBm and fGn, is the Hurst parameter $H$, that measures the long/short range dependence of such stochastic processes. In this paper, we show that these processes, from which we extract the degree distribution of the associated natural visibility graph (NVG), can be located in an informational plane, defined by normalized Shannon entropy $S$ and Fisher information measure $F$, in order to estimate their Hurst exponents. The aim of this work is to map signals onto this informational plane, in which a reference backbone is built using generated fBm and fGn processes with known Hurst exponents. To show the effectiveness of the developed graphical estimator, some real-world data are analyzed, and it found that the $H$ estimated by our method are quite comparable to those obtained from four well-known estimators of the literature. Besides, estimation of $H$ parameter is very fast and requires a reduced number of samples of the input signal. Using the constructed reference backbone in the Fisher–Shannon plane, the associated $H$ exponent can be easily estimated by a simple orthogonal projection of the point $(S,F)$ extracted from the truncated degree distribution of the considered signal NVG representation.