The Baum Welch-Algorithm for Parameter Estimation of Gaussian Autoregressive Mixture Models
by
Thomas Benesch
Graz University of Technology

A finite discrete Markov chain consists of a finite set of states S={S₁,   ... ,  S_N} and is observed at discrete times t=1,   ... ,  T. We use the variable \xi_t as the state of the Markov chain at discrete time t.

aij(t,  t+1):=I P(\xit+1=Sj|\xit=Si)

is called the transition probabilities from state S_i to S_j at time t. If a_ij(t,  t+1)=a_ij for all t, i.e. the transistion probabilities do not depend on t, we speak of a Markov chain with stationary transition probabilities.

In a Hidden Markov Model, the state \xi_t at time t can not be observed directly, but there exists a set of possible observations V={V₁,   ... ,  V_K} (K < N). \omega_t denotes the observation at time t. Being in state S_i at time t, an observation V_k is made with the probability

bik:=I P(\omegat=Vk|\xit=Si).

In Continuous Density Hidden Markov Models the discrete probability function b_ik is replaced by the continuous density function b_i(X), where X is an element of the continuous observation set that is d-dimensional Euclidean.

In Autoregressive Hidden Markov Model we assume a white noise source with unity variance, i.e. \sigma²=1, followed by an all-pole filter 1/A(z), where

A(z)=1+a1z-1+a2z-2+ ... +apz-p.

In this article we solve the problems of parameter estimation for Autoregressive Hidden Markov Model.

Date received: March 11, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadz-57.