A Characterization of Categorical Markov Chains

Abstract

We prove a representation theorem for {X_t } (t denotes time), an rth order categorical Markov chain. We prove that the conditional probability P(X_t |X _t-1, …, X_t-r ) can be written as a linear combination of the monomials of past process responses X _t-1,…,X_t-r . Simulations show that the “partial likelihood estimation” and the representation together give us satisfactory results. We also check the performance of “BIC” criterion for selecting optimal models and find that to be quite satisfactory. An advantage of this model over existing models is its capacity to admit covariates as linear terms by extension. For example, we can add some seasonal processes to get a non-stationary chain for daily precipitation values.

Key Words:

• Categorical stochastic processes
• Markov chains
• Characterization
• Estimation
• BIC