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Estimation and Modelling Repeated Patterns in High Order Markov Chains with the Mixture Transition Distribution Model

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  • Adrian Raftery
  • Simon Tavaré

Abstract

The mixture transition distribution (MTD) model was introduced by Raftery as a parsimonious model for high order Markov chains. It is flexible, can represent a wide range of dependence patterns, can be physically motivated, fits data well and is in several ways a discrete‐valued analogue for the class of autoregressive time series models. However, estimation has presented difficulties because the parameter space is highly non‐convex, being defined by a large number of non‐linear constraints. Here we propose a computational algorithm for maximum likelihood estimation which is based on a way of reducing the large number of constraints. This also allows more structured versions of the model, e.g. those involving structural zeros, to be fitted quite easily. A way of fitting the model by using GLIM is also discussed. The algorithm is applied to a sequence of wind directions, and also to two sequences of deoxyribonucleic acid bases from introns from mouse genes. In each case, the MTD model fits better than the conventional Markov chain model, and for the wind data it provides superior out‐of‐sample predictions. A modification of the model to represent repeated patterns is proposed and a very parsimonious version of this modified model is successfully applied to data representing bird songs.

Suggested Citation

  • Adrian Raftery & Simon Tavaré, 1994. "Estimation and Modelling Repeated Patterns in High Order Markov Chains with the Mixture Transition Distribution Model," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 43(1), pages 179-199, March.
    • Handle: RePEc:bla:jorssc:v:43:y:1994:i:1:p:179-199
    DOI: 10.2307/2986120
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    1. Arapis, Anastasios N. & Makri, Frosso S. & Psillakis, Zaharias M., 2016. "On the length and the position of the minimum sequence containing all runs of ones in a Markovian binary sequence," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 45-54.
    2. Kharin, Yuriy & Voloshko, Valeriy, 2021. "Robust estimation for Binomial conditionally nonlinear autoregressive time series based on multivariate conditional frequencies," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    3. Ioannis Kontoyiannis & Lambros Mertzanis & Athina Panotopoulou & Ioannis Papageorgiou & Maria Skoularidou, 2022. "Bayesian context trees: Modelling and exact inference for discrete time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1287-1323, September.
    4. Topuz, Kazim & Urban, Timothy L. & Yildirim, Mehmet B., 2024. "A Markovian score model for evaluating provider performance for continuity of care—An explainable analytics approach," European Journal of Operational Research, Elsevier, vol. 317(2), pages 341-351.
    5. Jonsson, Robert, 2011. "A Markov Chain Model for Analysing the Progression of Patient’s Health States," Research Reports 2011:6, University of Gothenburg, Statistical Research Unit, School of Business, Economics and Law.
    6. P.-C. G. Vassiliou & T. P. Moysiadis, 2010. "$\boldsymbol{\mathcal{G}-}$ Inhomogeneous Markov Systems of High Order," Methodology and Computing in Applied Probability, Springer, vol. 12(2), pages 271-292, June.
    7. Riccardo De Blasis & Giovanni Batista Masala & Filippo Petroni, 2021. "A Multivariate High-Order Markov Model for the Income Estimation of a Wind Farm," Energies, MDPI, vol. 14(2), pages 1-16, January.
    8. Bruno Damásio & João Nicolau, 2020. "Time Inhomogeneous Multivariate Markov Chains: Detecting and Testing Multiple Structural Breaks Occurring at Unknown," Working Papers REM 2020/0136, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.
    9. Ankinakatte, Smitha & Edwards, David, 2015. "Modelling discrete longitudinal data using acyclic probabilistic finite automata," Computational Statistics & Data Analysis, Elsevier, vol. 88(C), pages 40-52.
    10. Li, Li & Yan, Xihong & Zhang, Xinzhen, 2022. "An SDP relaxation method for perron pairs of a nonnegative tensor," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    11. Francesco Bartolucci & Alessio Farcomeni, 2010. "A note on the mixture transition distribution and hidden Markov models," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(2), pages 132-138, March.
    12. James, Marilyn, 1999. "Towards an integrated needs and outcome framework," Health Policy, Elsevier, vol. 46(3), pages 165-177, March.
    13. Wai Ki Ching & Eric S. Fung & Michael K. Ng, 2004. "Higher‐order Markov chain models for categorical data sequences," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(4), pages 557-574, June.
    14. Damásio, Bruno & Nicolau, João, 2024. "Time inhomogeneous multivariate Markov chains: Detecting and testing multiple structural breaks occurring at unknown dates," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    15. Nikolaos Stavropoulos & Alexandra Papadopoulou & Pavlos Kolias, 2021. "Evaluating the Efficiency of Off-Ball Screens in Elite Basketball Teams via Second-Order Markov Modelling," Mathematics, MDPI, vol. 9(16), pages 1-13, August.
    16. Fokianos, Konstantinos & Fried, Roland & Kharin, Yuriy & Voloshko, Valeriy, 2022. "Statistical analysis of multivariate discrete-valued time series," Journal of Multivariate Analysis, Elsevier, vol. 188(C).

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