Partition Markov Model for Multiple Processes
by M. T. A. Cordeiro, Jesús E. García, V.A. González-López and S. L. M. Londoño

In this paper, we analyze the model proposed in [1] in which is considered a set of p independent samples of discrete time Markov chains, over a finite alphabet A and with finite order o.  The model is obtained identifying the states on the state space A^o where two or more samples share the same transition probabilities (see also [3]). This identification establishes a partition on {1,...,p}xA^o the set of samples 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 [1] is given a notion of divergence, derived from the BIC, which serves to identify the proximity/discrepancy between elements of {1,...,p}xA^o (see also [2]). In the present article, we also 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}xA^o. 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.



References:

[1] García Jesús E,  Londoño SLM, (2019, July). Optimal Model for a Set of Markov Processes, in AIP Conference Proceedings. (Vol. 2116: 130002). DOI: 10.1063/1.5114125 

[2] García Jesús E,  Gholizadeh R,  González-López VA, A BIC - based consistent metric between Markovian processes, {\it{Applied Stochastic Models in Business and Industry}} 2018; 34(6): 868-878. DOI: 10.1002/asmb.2346

[3] García Jesús E,  González-López VA, Consistent Estimation of Partition Markov Models, {\it{Entropy }} 2017; {\bf{19}} (4): 160. DOI: 10.3390/e19040160