Abstract: Estimating of the Hurst exponent for experimental data plays a very important role in the research of processes which show properties of self-similarity. There are many methods for estimating the Hurst exponent using time series. The aim of this research is to carry out the comparative analysis of the statistical properties of the Hurst exponent estimators obtained by different methods using model stationary and nonstationary fractal time series. In this paper the most commonly used methods for estimating the Hurst exponents are examined. There are: /RS-analysis, variance-time analysis, detrended fluctuation analysis (DFA) and wavelet-based estimation. The fractal Brownian motion that is constructed using biorthogonal wavelets have been chosen as a model random process which exhibit fractal properties. In this paper, the results of a numerical experiment are represented where the fractal Brown motion was modelled for the specified values of the exponent H. The values of the Hurst exponent for the model realizations were varied within the whole interval of possible values 0 < H < 1.The lengths of the realizations were defined as 500, 1000, 2000 and 4000 values. For the nonstationary case model time series are presented by the sum of fractional noise and the trend component, which are a polynomial in varying degrees, irrational, transcendental and periodic functions. The estimates of H were calculated for each generated time series using the methods mentioned above. Samples of the exponent H estimates were obtained for each value of H and their statistical characteristics were researched. The results of the analysis have shown that the estimates of the Hurst exponent, which were obtained for the stationary realisations using the considered methods, are biased normal random variables. For each method the bias depends on the true value of the degrees self-similarity of the process and length of time series. Those estimates which are obtained by the DFA method and the wavelet transformation have the minimal bias. Standard deviations of the estimates depending on the estimation method and decrease, while the length of the series increases. Those estimates which are obtained by using the wavelet analysis have the minimal standard deviation. In the case of a nonstationary time series, represented by a trend and additive fractal noise, more accurate evaluation is obtained using the DFA method. This method allows estimating the Hurst exponent for experimental data with trend components of virtually any kind. The greatest difficulty in estimating, presents a series with a periodic trend component. It is desirable in addition to investigate the spectrum of the wavelet energy, which is demonstrated in the structure of the time series. In the presence of a slight trend, the wavelet-estimation is quite effective.

Keywords: Hurst exponent, estimate of the Hurst exponent, self-similar stochastic process, nonstationary time series, methods for estimating the Hurst exponent.

ACM Classification Keywords: G.3 Probability and statistics - Time series analysis, Stochastic processes, G.1 Numerical analysis, G.1.2 Approximation - Wavelets and fractals.

Link:

COMPARATIVE ANALYSIS FOR ESTIMATING OF THE HURST EXPONET FOR STATIONARY AND NONSTATIONARY TIME SERIES

Ludmila Kirichenko, Tamara Radivilova, Zhanna Deineko

http://foibg.com/ijitk/ijitk-vol05/ijitk05-4-p05.pdf