Two-State Imprecise Markov Chains for Statistical Modelling of Two-State Non-Markovian Processes

Matthias Troffaes, Thomas Krak, Henna Bains
Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications, PMLR 103:394-403, 2019.

Abstract

This paper proposes a method for fitting a two-state imprecise Markov chain to time series data from a two-state non-Markovian process. Such non-Markovian processes are common in practical applications. We focus on how to fit modelling parameters based on data from a process where time to transition is not exponentially distributed, thereby violating the Markov assumption. We do so by first fitting a many-state (i.e. having more than two states) Markov chain to the data, through its associated phase-type distribution. Then, we lump the process to a two-state imprecise Markov chain. In practical applications, a two-state imprecise Markov chain might be more convenient than a many-state Markov chain, as we have closed analytic expressions for typical quantities of interest (including the lower and upper expectation of any function of the state at any point in time). A numerical example demonstrates how the entire inference process (fitting and prediction) can be done using Markov chain Monte Carlo, for a given set of prior distributions on the parameters. In particular, we numerically identify the set of posterior densities and posterior lower and upper expectations on all model parameters and predictive quantities. We compare our inferences under a range of sample sizes and model assumptions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v103-troffaes19b, title = {Two-State Imprecise Markov Chains for Statistical Modelling of Two-State Non-Markovian Processes}, author = {Troffaes, Matthias and Krak, Thomas and Bains, Henna}, booktitle = {Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications}, pages = {394--403}, year = {2019}, editor = {De Bock, Jasper and de Campos, Cassio P. and de Cooman, Gert and Quaeghebeur, Erik and Wheeler, Gregory}, volume = {103}, series = {Proceedings of Machine Learning Research}, month = {03--06 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v103/troffaes19b/troffaes19b.pdf}, url = {https://proceedings.mlr.press/v103/troffaes19b.html}, abstract = {This paper proposes a method for fitting a two-state imprecise Markov chain to time series data from a two-state non-Markovian process. Such non-Markovian processes are common in practical applications. We focus on how to fit modelling parameters based on data from a process where time to transition is not exponentially distributed, thereby violating the Markov assumption. We do so by first fitting a many-state (i.e. having more than two states) Markov chain to the data, through its associated phase-type distribution. Then, we lump the process to a two-state imprecise Markov chain. In practical applications, a two-state imprecise Markov chain might be more convenient than a many-state Markov chain, as we have closed analytic expressions for typical quantities of interest (including the lower and upper expectation of any function of the state at any point in time). A numerical example demonstrates how the entire inference process (fitting and prediction) can be done using Markov chain Monte Carlo, for a given set of prior distributions on the parameters. In particular, we numerically identify the set of posterior densities and posterior lower and upper expectations on all model parameters and predictive quantities. We compare our inferences under a range of sample sizes and model assumptions.} }
Endnote
%0 Conference Paper %T Two-State Imprecise Markov Chains for Statistical Modelling of Two-State Non-Markovian Processes %A Matthias Troffaes %A Thomas Krak %A Henna Bains %B Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications %C Proceedings of Machine Learning Research %D 2019 %E Jasper De Bock %E Cassio P. de Campos %E Gert de Cooman %E Erik Quaeghebeur %E Gregory Wheeler %F pmlr-v103-troffaes19b %I PMLR %P 394--403 %U https://proceedings.mlr.press/v103/troffaes19b.html %V 103 %X This paper proposes a method for fitting a two-state imprecise Markov chain to time series data from a two-state non-Markovian process. Such non-Markovian processes are common in practical applications. We focus on how to fit modelling parameters based on data from a process where time to transition is not exponentially distributed, thereby violating the Markov assumption. We do so by first fitting a many-state (i.e. having more than two states) Markov chain to the data, through its associated phase-type distribution. Then, we lump the process to a two-state imprecise Markov chain. In practical applications, a two-state imprecise Markov chain might be more convenient than a many-state Markov chain, as we have closed analytic expressions for typical quantities of interest (including the lower and upper expectation of any function of the state at any point in time). A numerical example demonstrates how the entire inference process (fitting and prediction) can be done using Markov chain Monte Carlo, for a given set of prior distributions on the parameters. In particular, we numerically identify the set of posterior densities and posterior lower and upper expectations on all model parameters and predictive quantities. We compare our inferences under a range of sample sizes and model assumptions.
APA
Troffaes, M., Krak, T. & Bains, H.. (2019). Two-State Imprecise Markov Chains for Statistical Modelling of Two-State Non-Markovian Processes. Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications, in Proceedings of Machine Learning Research 103:394-403 Available from https://proceedings.mlr.press/v103/troffaes19b.html.

Related Material