The Estimation Of The Hurst Exponent For Fractional Brownian Motion And Its Application

Fractional Brownian motion (FBM) due to self-similarity and long-range correlationin fractal characteristics, and its increment is zero mean and stationary Gaussian process,and is able to describe complex natural phenomena, so the model is widely used in stockmarket options pricing, sea clutter fractal modeling, rotor stability testing, fault detectionsignal, etc., while the Hurst exponent as described self-similarity parameter is important.Estimates of the value of H has been extensively studied, this paper is to summarize someof the existing methods and proposes a new approach and a comparative analysis.Firstly, use DFA series methods (mainly DFA-2and DFA-3) to estimate the H ofFBM, while FBM trajectories is simulated through basic functions in MATLAB which isbased on wavelet decomposition, simulate50samples for each H, use DFA method tocalculate H of the50means, and obtained that when H>0.5, the deviation is relativelysmall, but when the H <0.5, the deviation is significant.For eliminate fluctuations in trend analysis (DFA) method using a polynomial to fitlocal trends defects introduced Empirical Mode Decomposition (EMD) trend (datathrough the remainder after EMD), the use of EMD trend to replace DFA polynomialtrend (tentatively called EMD-DFA), to calculate the H value of FBM. Because ofModal mix problem in EMD, EEMD can solve this problem, an EEMD-DFA methods isoffered.We use EMD-DFA method and EEMD-DFA method to estimate the H value of FBM,and find that the IMF after EMD and EEMD is the same, so there is no problem in mixingmodal wavelet-based simulation of FBM, and these two methods is the same whenestimated H value of FBM.Compare with EMD-DFA, spectral analysis method based on HHT and DFA seriesmethod for calculating the H value of the fractional Brownian motion concluded: When H <0.5, the first two methods of estimation error is small, and estimates are similar; whenH>0.5, estimate error in DFA series method is smaller. However, due to the end effectof EMD in the case of small-scale impact is more serious, so EMD-DFA is suitable fornegative correlation and long data sequences.