Partition Markov model for multiple processes

• Published in: Mathematical methods in the applied sciences 2020-09, Vol.43 (13), p.7677-7691
• Main Authors: Cordeiro, Marcos Tadeu A., García, Jesús Enrique, González‐López, Verónica Andrea, Mercado Londoño, Sergio Luis
• Format: Article
• Language: English
• Subjects: Bayesian information criterion; Markov chains; metric between stochastic processes; parsimonious model; Partitions (mathematics); stochastic processes; Transition probabilities
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.6079

In this paper, we analyze the model proposed in García and Londoño1 in which a set of p‐independent sequences of discrete time Markov chains is considered, over a finite alphabet A and with finite order o. The model is obtained identifying the states on the state space Ao where two or more sequences share the same transition probabilities (see also García and González‐López2). This identification establishes a partition on {1,…,p}×Ao, the set of sequences, and the state space. We show that by means of the Bayesian information criterion (BIC), the partition can be estimated eventually almost surely. Also, in García and Londoño,1 it is given a notion of divergence, derived from the BIC, which serves to identify the proximity/discrepancy between elements of {1,…,p}×Ao (see also García et al3). In the present article, we prove that this notion is a metric in the space where the model is built and that it is statistically consistent to determine proximity/discrepancy between the elements of the space {1,…,p}×Ao. We apply the notions discussed here for the construction of a parsimonious model that represents the common stochastic structure of 153 complete genomic Zika sequences, coming from tropical and subtropical regions.