Maximum Likelihood Estimation of the Transition Matrix of the Non-homogeneous Markov Chain Terrorist Threat Model

Kód výsledku v IS VaVaI

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Výsledek na webu

<a href="http://www.pdmu.univ.kiev.ua/PDMU_2019W/home.php" target="_blank" >http://www.pdmu.univ.kiev.ua/PDMU_2019W/home.php</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Maximum Likelihood Estimation of the Transition Matrix of the Non-homogeneous Markov Chain Terrorist Threat Model

Popis výsledku v původním jazyce

Markov chain-based models are one of the more popular mathematical tools for modelling time development of risk in various fields of research. The Markov chain is used to govern the change of the risk from one category (state of the chain) to another in discrete time. One particular approach aims to enhance the model by introducing a connection between the Markov chain and some covariates. These covariates describe the development of the environment in which the modelled risk events take place. The abovementioned approach can be seen for example in [1], where a two state Markov chain non-stationary transition probabilities are connected with a set of covariates through a binomial logit generalized linear model (GLM). The estimation of the transition probabilities is carried out within a maximum likelihood framework while utilizing the said connection. Due to the relative simplicity of the model (two state Markov chain) it is possible to obtain the derivatives of the respective log-likelihood function in a closed form. In security research however models with more than two states are needed. Therefore a generalization of the approach to an arbitrary number of states Markov chain model is necessary. This however complicates the computation of the derivatives significantly. In this paper a likelihood function for estimation of the transition probabilities of a Markov chain risk model with dependence on given covariates is presented. An approach allowing to bypass the computation of the derivatives in the process of finding the argument of maximum of the likelihood function is presented. This approach relies on using an optimization algorithm that does not require the knowledge of the derivatives. Computational results obtained through simulations are presented. Panel data approach is considered.

Název v anglickém jazyce

Maximum Likelihood Estimation of the Transition Matrix of the Non-homogeneous Markov Chain Terrorist Threat Model

Popis výsledku anglicky

Markov chain-based models are one of the more popular mathematical tools for modelling time development of risk in various fields of research. The Markov chain is used to govern the change of the risk from one category (state of the chain) to another in discrete time. One particular approach aims to enhance the model by introducing a connection between the Markov chain and some covariates. These covariates describe the development of the environment in which the modelled risk events take place. The abovementioned approach can be seen for example in [1], where a two state Markov chain non-stationary transition probabilities are connected with a set of covariates through a binomial logit generalized linear model (GLM). The estimation of the transition probabilities is carried out within a maximum likelihood framework while utilizing the said connection. Due to the relative simplicity of the model (two state Markov chain) it is possible to obtain the derivatives of the respective log-likelihood function in a closed form. In security research however models with more than two states are needed. Therefore a generalization of the approach to an arbitrary number of states Markov chain model is necessary. This however complicates the computation of the derivatives significantly. In this paper a likelihood function for estimation of the transition probabilities of a Markov chain risk model with dependence on given covariates is presented. An approach allowing to bypass the computation of the derivatives in the process of finding the argument of maximum of the likelihood function is presented. This approach relies on using an optimization algorithm that does not require the knowledge of the derivatives. Computational results obtained through simulations are presented. Panel data approach is considered.

Druh

O - Ostatní výsledky

CEP obor

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OECD FORD obor

10103 - Statistics and probability

Projekt

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Návaznosti

S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů