Combining multivariate Markov chains

Abstract

In this paper we address the problem of modelling multivariate finite order Markov chains, when the dataset is not large enough to apply the usual methodology. The number of parameters needed for a multivariate Markov chain grows exponentially with the process order and dimension of the chain’s alphabet. Usually, when the data set is small, the order of the fitted model is reduced compared to the true process order. In this paper we introduce a strategy to estimate a multivariate process, through this new strategy the estimated order will be greater than the order achieved using standard statistical procedures. We apply the partition Markov models, which is a family of models, where each member is identified by a partition of the state space. The procedure consist in obtaining a partition of the state space that is constructed from a combination of the partitions corresponding to the marginal processes of the multivariate chain, and the partition corresponding to the multivariate Markov chain.