Consistent Estimation of Partition Markov Models

Garćıa, Jesús E.; González‐López, V. A.

doi:10.3390/e19040160

Cited by 30 publications

(31 citation statements)

References 13 publications

(18 reference statements)

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“…A well‐established methodology for model selection is based on the BIC (see the work of Schwarz). For the Markovian processes, the BIC is consistent (see for example the work of Csiszár and Shields, Csiszár and Talata, and García and González‐López). We will formulate those questions from the BIC‐based perspective.…”

Section: Preliminariesmentioning

confidence: 64%

“…The BIC has already been used to obtain consistent methods for the selection of models in Markovian processes. See for example the works of Csiszár and Talata and García and González‐López . This paper contributes to the theory of showing that the BIC can also be used to consistently decide whether two independent samples of stochastic processes can be considered as coming from the same process, ie, it allows identifying if such samples are governed by the same law of probability.…”

Section: Discussionmentioning

confidence: 97%

“…Proof Following similar calculations of theorem in the work of García and González‐López . For (i), define

a_{i} = N_{n_{i}} false(s, a false)

and

b_{i} = N_{n_{i}} false(s false), i = 1, 2 .

Consider the log‐sum inequality, ie,

a_{1} \ln false(\frac{a_{1}}{b_{1}} false) + a_{2} \ln false(\frac{a_{2}}{b_{2}} false) \geq \sum_{s = normal1, normal2} a_{s} \ln ((), \frac{\sum_{s = normal1, normal2} a_{s}}{\sum_{s = normal1, normal2} b_{s}}),

with equality

\Leftrightarrow \frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}}

, and the result follows.…”

Section: Theoretical Properties Of Dmentioning

confidence: 98%

“…In general, given a specific model and a sample

x_{1}^{n},

the BIC value (see the work of Schwarz) can be written as the log‐maximum likelihood for the model minus a penalization term depending on the number of parameters of the model and the size n of the sample

x_{1}^{n}

. Under this perspective, the BIC associated with an order o Markov chain and the sample

x_{1}^{n}

is given by

B I C ((), x_{1}^{n}) = M L ((), scriptS, x_{1}^{n}) - \frac{false(false| A false| - 1 false)}{α} false| scriptS false| \ln false(n false) .

This criterion can be modified for different families of Markov models, see for instance the expression of the BIC in the work of Csiszár and Talata for context tree models and in the work of García and González‐López for partition Markov models.…”

Section: Preliminariesmentioning

confidence: 99%

“…This criterion was experimentally used in the work of García et al, and it is built through the Bayesian information criterion (BIC) (see the work of Schwarz). The BIC was used to obtain consistent estimates in several Markov models, as is the case of context trees (Csiszár and Talata) and partition Markov models (García and González‐López). In this paper, we prove that it is possible to use the BIC to derive a local similarity measure for deciding whether or not two independent samples can be considered as coming from the same local stochastic law.…”

Section: Introductionmentioning

confidence: 99%

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A BIC‐based consistent metric between Markovian processes

Garćıa

Gholizadeh

González‐López

2018

Appl Stoch Models Bus & Ind

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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

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Section: Preliminariesmentioning

confidence: 64%

Section: Discussionmentioning

confidence: 97%

“…Proof Following similar calculations of theorem in the work of García and González‐López . For (i), define

a_{i} = N_{n_{i}} false(s, a false)

and

b_{i} = N_{n_{i}} false(s false), i = 1, 2 .

Consider the log‐sum inequality, ie,

a_{1} \ln false(\frac{a_{1}}{b_{1}} false) + a_{2} \ln false(\frac{a_{2}}{b_{2}} false) \geq \sum_{s = normal1, normal2} a_{s} \ln ((), \frac{\sum_{s = normal1, normal2} a_{s}}{\sum_{s = normal1, normal2} b_{s}}),

with equality

\Leftrightarrow \frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}}

, and the result follows.…”

Section: Theoretical Properties Of Dmentioning

confidence: 98%

“…In general, given a specific model and a sample

x_{1}^{n},

x_{1}^{n}

. Under this perspective, the BIC associated with an order o Markov chain and the sample

x_{1}^{n}

is given by

B I C ((), x_{1}^{n}) = M L ((), scriptS, x_{1}^{n}) - \frac{false(false| A false| - 1 false)}{α} false| scriptS false| \ln false(n false) .

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A BIC‐based consistent metric between Markovian processes

Garćıa

Gholizadeh

González‐López

2018

Appl Stoch Models Bus & Ind

View full text Add to dashboard Cite

show abstract

Comparison of Stochastic Processes

Garćıa¹,

Gholizadeh²,

González‐López³

2020

Data Analysis and Applications 4

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Sample selection procedure in daily trading volume processes

Fernández¹,

Garćıa

Gholizadeh

et al. 2019

Math Methods in App Sciences

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In this paper, we propose a procedure of selecting samples from a set of samples coming from Markovian processes of finite order and finite alphabet. Under the assumption of the existence of a law that prevails in at least q% of the samples of the collection, we show that the procedure allows to identify samples governed by the predominant law. The approach is based on a local metric between samples, which tends to zero when we compare samples of identical law and tends to infinity when comparing samples with different laws. The local metric allows to define a criterion which takes arbitrarily large values when the previous assumption about the existence of a predominant law does not hold. By means of this procedure, we map similarities and dissimilarities of some Brazilian stocks' daily trading volume dynamic.

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Consistent Estimation of Partition Markov Models

Cited by 30 publications

References 13 publications

A BIC‐based consistent metric between Markovian processes

A BIC‐based consistent metric between Markovian processes

Comparison of Stochastic Processes

Sample selection procedure in daily trading volume processes

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