Fractional Brownian-Motion And Its Fractal Dimension Estimation

A mathematical model of stochastic processes -- fractional Brownian motion -- is addressed. The power-law behaviors of FBM increments are studied in detail for moments, correlation functions, and power spectra. A moment method is proposed to do model testing of fractional Brownian motion. The results of FBM model testing of six simulators show that the covariance matrix transforming algorithm can provide samples with very good approximation of self-affinity. The self-affinity of the FBM samples generated by Fourier transform filtering is not very obvious. The statistical properties of fractal dimension estimation methods are analyzed. The simulation results show that the variance method provides good performance when only estimates of variances with small time lags are used in the least-squares estimation. For the power spectrum method, the bias is not ignorable because of the aliasing and the window effect.