A BIC‐based consistent metric between Markovian processes

Abstract

In this paper, we address the problem of deciding if two independent samples coming from discrete Markovian processes are governed by the same stochastic law. We establish a local metric between samples based on the Bayesian information criterion. In addition, we derive the bound that must be used in this metric to take the decision. In the case on which is decided that the laws are not the same, the metric allows to detect the specific elements of the state space where the discrepancies are manifested. We prove that the metric is statistically consistent to detect if the samples follow the same law, tending to zero when the sample sizes increase. Moreover, we show that the metric assumes arbitrarily large values when the sample sizes increase and the stochastic laws are different. This concept is applied to analyze two lines of production of alcohol fuel, described by five variables each. We identify the variables that most contribute to the discrepancy and, using the local nature of the metric, we list the realizations in which the processes behave differently. KEYWORDSBayesian information criterion, Markov processes, proximity between processes, relative entropy 868

Partition Markov Model for Covid-19 Virus

“…We show how algorithms designed for the classic version of Partition Markov Models can have difficulties in recovering the structure investigated here, see Examples 2.2 and 2.3. Under previous determination of the parameters o and G it is possible to adapt all the estimation tools of the usual Partition Markov Models (see [1] and [4]), see Remarks 2.1 and 2.2. This specific structure in the process' partition (see Definition 2.3, Eq.…”

“…We apply the algorithm of [7] in a simulated data from the law given by Table 1, the algorithm is applied in two settings (i) using o = 4, initial set {a, b, c} 4 3). With a sample size n = 5 Â 10 4 , we obtain by (ii) the original partition (Tab.…”

“…We perform simulations of the process following Table 4, with n = 2000. The algorithm of [7] is applied in two settings (i) using o ¼ 4 < log jAj ðnÞ j k À 1 ¼ 5, initial set {a, b, c} 4 and d L as given by Definition 2.2 and, (ii) using o = 0, G = 10, initial set {a, b, c} and d I with the modifications imposed by Remark 2.1. In other words, case (i) reflects the conditions that are generally applied to proceed with the adjustment and determination of the process memory.…”

“…Part Elements 1 (a, aaa), (a, aca), (a, aga), (a, ata), (a, cca), (a, cga), (a, gca), (a, gta), (a, tca), (c, aag), (c, aca), (c, aga), (c, caa), (c, cta), (c, gga), (c, taa), (c, tag), (c, tta), (g, aga), (g, cag), (g, ctc), (g, tag), (g, tca), (g, tta), (t, aag), (t, aca), (t, aga), (t, cag), (t, cca), (t, cga), (t, tca) 2 (a, atg), (a, cgt), (a, ctg), (a, gag), (a, ttg), (c, act), (c, att), (c, tca), (g, aaa), (g, aca), (g, atg), (g, att), (g, cca), (g, ctt), (t, aaa), [4]. It has the same properties; that is to say that i. is a metric and both i. and ii.…”

“…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].…”