An entropy-based estimator of the Hurst exponent in fractional Brownian motion

Abstract

The Hurst exponent is a measure of long-term memory of time series and entropy measures the degree of irregularity in them, but there is no specific relationship between these two views of complexity. In addition, classical methods to estimate the Hurst exponent lead to different estimates for this criterion. To fill this gap, in this study, we propose an entropy-based estimator of the Hurst exponent relying on distributive features of the series in fractional Brownian motion (fBm), which is the key model of long-term memory processes. In this regard, we present a relationship between the Hurst exponent and entropy. The entropies used for this purpose are Shannon, Tsallis and Renyi. The result shows that the Hurst exponent can be estimated by a simple linear combination of entropies. In addition, the proposed method has high accuracy and it can be used for any real time series which follows fBm. The findings also indicate that the detection of long-term memory is predictable based on the amount of entropy. We apply the results to a set of real data derived from the forex market and estimate the Hurst exponent and their long-term memory using the proposed method.