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Markov chain order estimation with conditional mutual information

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  • Papapetrou, M.
  • Kugiumtzis, D.

Abstract

We introduce the Conditional Mutual Information (CMI) for the estimation of the Markov chain order. For a Markov chain of K symbols, we define CMI of order m, Ic(m), as the mutual information of two variables in the chain being m time steps apart, conditioning on the intermediate variables of the chain. We find approximate analytic significance limits based on the estimation bias of CMI and develop a randomization significance test of Ic(m), where the randomized symbol sequences are formed by random permutation of the components of the original symbol sequence. The significance test is applied for increasing m and the Markov chain order is estimated by the last order for which the null hypothesis is rejected. We present the appropriateness of CMI-testing on Monte Carlo simulations and compare it to the Akaike and Bayesian information criteria, the maximal fluctuation method (Peres–Shields estimator) and a likelihood ratio test for increasing orders using ϕ-divergence. The order criterion of CMI-testing turns out to be superior for orders larger than one, but its effectiveness for large orders depends on data availability. In view of the results from the simulations, we interpret the estimated orders by the CMI-testing and the other criteria on genes and intergenic regions of DNA chains.

Suggested Citation

  • Papapetrou, M. & Kugiumtzis, D., 2013. "Markov chain order estimation with conditional mutual information," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(7), pages 1593-1601.
    • Handle: RePEc:eee:phsmap:v:392:y:2013:i:7:p:1593-1601
    DOI: 10.1016/j.physa.2012.12.017
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    References listed on IDEAS

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    1. Buldyrev, S.V. & Dokholyan, N.V. & Goldberger, A.L. & Havlin, S. & Peng, C.-K. & Stanley, H.E. & Viswanathan, G.M., 1998. "Analysis of DNA sequences using methods of statistical physics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 249(1), pages 430-438.
    2. Kugiumtzis, D. & Provata, A., 2004. "Statistical analysis of gene and intergenic DNA sequences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 342(3), pages 623-638.
    3. L. Zhao & C. Dorea & C. Gonçalves, 2001. "On Determination of the Order of a Markov Chain," Statistical Inference for Stochastic Processes, Springer, vol. 4(3), pages 273-282, October.
    4. M. L. Menéndez & L. Pardo & M. C. Pardo & K. Zografos, 2011. "Testing the Order of Markov Dependence in DNA Sequences," Methodology and Computing in Applied Probability, Springer, vol. 13(1), pages 59-74, March.
    5. Dalevi Daniel & Dubhashi Devdatt & Hermansson Malte, 2006. "A New Order Estimator for Fixed and Variable Length Markov Models with Applications to DNA Sequence Similarity," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 5(1), pages 1-26, March.
    6. Menendez, M.L. & Pardo, J.A. & Pardo, L. & Zografos, K., 2006. "On tests of independence based on minimum [phi]-divergence estimator with constraints: An application to modeling DNA," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 1100-1118, November.
    7. M. Menéndez & J. Pardo & L. Pardo, 2001. "Csiszar’s ϕ-divergences for testing the order in a Markov chain," Statistical Papers, Springer, vol. 42(3), pages 313-328, July.
    8. Kugiumtzis Dimitris, 2008. "Evaluation of Surrogate and Bootstrap Tests for Nonlinearity in Time Series," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 12(1), pages 1-26, March.
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    Cited by:

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