Application of a Time-Scale Local Hurst Exponent analysis to time series

Abstract

This paper introduces a method to perform a Time-Scale Local Hurst Exponent (TS-LHE) analysis for time series. The traditional Hurst exponent methods usually analyze time series as a whole, providing a single value that characterizes their global behavior. In contrast, the methods based on the Local Hurst Exponent allow the evaluation of the fractal structure of a time series on local events. However, a critical parameter in these methods is the selection of scale. Here, a TS-LHE method is presented, based on a systematic implementation of the rescaled-range (R/S) method, in a set of sliding windows of different sizes. This method allows calculating instantaneous values of Local Hurst Exponents at different scales, associating them with individual samples of a time series. This paper is organized as follows: first, an overview of the TS-LHE is provided; then, a proof-of-concept of this analysis is presented, considering (a) different fractional Brownian motion series, (b) a synthetic seismic signal under different noise conditions, and (c) a group of real seismic traces. Finally, the obtained results show that the TS-LHE analysis is particularly sensitive to sudden behavior changes of the time series, such as frequency or phase variations. This sensitivity is independent of the amplitude of the data, and thus, it can be used to identify pattern changes as well as long- and short-range correlations within a time series.