Partition Markov model for multiple processes

Abstract

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.

Partition Markov Model for Covid-19 Virus

“…As is usual in DNA sequences [2], the transition probabilities are moderate (in this case 0.44, see Tab. 8).…”

“…8). Note that there is a predilection of the process to choose as the next state a or t. Sequences of other viruses lead to other predilections, see for example [2], in which the Zika process is modeled, revealing a predilection for the states a or g. We observe that under Definition 2.3 and without imposing the partition structure, the total number of parameters to be estimated is unfeasible. For example, with o = 3 and G = 9 we have |A  A o |  (|A| À 1) = 768 parameters to estimate, and when using the strategy given by equation 2and Remark 2.1, it is necessary to estimate jI j  (|A| À 1) = 39 parameters.…”

“…By construction, this is a parsimonious model, since it reduces the number of probabilities to estimate by identifying equivalent states (found in the same part). PMM models have already shown sufficient flexibility in the field of genomic structure modeling, for several purposes, such as determining similarities between Zika's genomic sequences and modeling the genomic Zika's profile [2]. This family of models has also been used to define the genomic profile of the Epstein-Barr virus [3].…”

“…The consistent estimation of a PMM [1], is achieved by the Bayesian Information Criterion, BIC, which has led to the definition of a BIC-based metric, see also [4]. The metric has allowed the use of the dynamics of the PMM models for other open problems; it has been efficiently applied in subjects such as (1) the comparison between Dengue's genomic sequences of different origins [5] and (2) to compare genomic sequences of Burkitt lymphoma/leukemia [6]. Since a PMM is defined by a partition on the state space of the process, we understand that the partition's structure could be the key to modeling certain phenomena.…”

An Efficient Coding Technique for Stochastic Processes

“…One aspect that could be considered in the compression process of many sequences of the same virus is to use a model that determines the parts considering a generalization of the Definition 8, such as the notion introduced in [ 16 ]. The notion introduced in [ 16 ] proposes to determine the parts using any sequence and any state, joining in the same part sequences and states if they share the transition probabilities. With such a model in hand, the bits used for the description of the model would be informed only once, remaining only to inform the coding of the sequences.…”