Maximum likelihood estimation of fractional Brownian motion and Markov noise parameters

Abstract

Abstract : Maximum likelihood estimation and power spectral density analysis are developed as tools for the analysis of stochastic processes. Some useful results from the theory of Markov stochastic processes are then presented followed by the introduction of fractional Brownian motion and fractional Gaussian noise as non-Markov models for systems with power spectral density proportional to fBeta, where -3 < Beta < -1 and -1 < Beta < 1 over all frequencies. Maximum likelihood system identification is applied to estimating the unknown parameters in a Markov model which approximates fractional Brownian motion. The algorithm runs a Kalman filter on states and a maximum likelihood estimator on parameters. Results are presented from estimating trend, white noise, random walk, and exponentially correlated noise parameters from fits to simulated and real test data. Maximum likelihood estimation is applied to the batch estimation of parameters in the non-Markov fractional Brownian motion model. New in this thesis is the use of partial derivatives to minimize the resulting likelihood function, and the capability to estimate the unknown parameters of additional trend and Markov noise processes. Results are presented from fits to computer simulated sample paths.