Hidden Markov Processes: The Complete Realization Problem
M. Vidyasagar · Princeton University Press eBooks · 2014
This chapter considers hidden Markov processes (HMPs), focusing on the so-called complete realization problem. It is quite easy to prove a universal necessary condition for the given process to have a hidden Markov model (HMM). However, this condition is not sufficient in general. In principle, one can derive a “necessary and sufficient condition,” but the “necessary and sufficient condition” is virtually a restatement of the problem to be solved and does not shed any insight into the solution. The chapter first introduces a very useful matrix known as the “Hankel” matrix before discussing the nonsufficiency of the finite Hankel rank condition, an abstract necessary and sufficient condition, and the existence of regular quasi-realizations. It also describes the spectral properties of alpha-mixing processes and goes on to analyze ultra-mixing processes and a sufficient condition for the existence of HMMs.