Fractional Brownian motion and its applications
Description
This dissertation addresses a new mathematical model, fractional Brownian motion. A review of the concepts and properties of fractional Brownian motion is presented in detail as the groundwork of the whole thesis. The algorithms of FBM simulation and fractal dimension estimation are summarized. A model-based approach is developed to generate the samples of fractional Brownian motion recursively. The results of FBM model testing of seven simulators show that the covariance matrix transformation method and the model-based approach can provide FBM samples with very good approximation of self-affinity. The samples generated by the Fourier transform filtering method are not very self-affine. It has been found that the absolute displacement of a fractional Brownian motion has a power-law dependence on the time resolution. With use of the principal that the spectrum of a product of a random signal and a deterministic signal is equal to the convolution of their power spectra, the spectrum of the finite-duration discrete FBM is analyzed. Both theoretical and experimental results show that the rectangular window effect is not ignorable when the spectrum method is used for fractal dimension estimation. A comparison of the simulation results with an asymptotic Cramer-Rao Bound is presented and shows that the variance method and the displacement method provide better performance than the spectrum method. The results of applications show that FBM is a useful mathematical model for time series analysis and image analysis. The S&P stock price index shows strong self-affinity. The fractal-based cell classification proves that fractal dimension is a useful feature for pattern recognition. Based on the analysis of a sample terrain map, the moments of the elevation increments of a mountain area have a power-law dependence on the distance