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Markov chains with memory, tensor formulation, and the dynamics of power iteration

Author

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  • Wu, Sheng-Jhih
  • Chu, Moody T.

Abstract

A Markov chain with memory is no different from the conventional Markov chain on the product state space. Such a Markovianization, however, increases the dimensionality exponentially. Instead, Markov chain with memory can naturally be represented as a tensor, whence the transitions of the state distribution and the memory distribution can be characterized by specially defined tensor products. In this context, the progression of a Markov chain can be interpreted as variants of power-like iterations moving toward the limiting probability distributions. What is not clear is the makeup of the “second dominant eigenvalue” that affects the convergence rate of the iteration, if the method converges at all. Casting the power method as a fixed-point iteration, this paper examines the local behavior of the nonlinear map and identifies the cause of convergence or divergence. As an application, it is found that there exists an open set of irreducible and aperiodic transition probability tensors where the Z-eigenvector type power iteration fails to converge.

Suggested Citation

  • Wu, Sheng-Jhih & Chu, Moody T., 2017. "Markov chains with memory, tensor formulation, and the dynamics of power iteration," Applied Mathematics and Computation, Elsevier, vol. 303(C), pages 226-239.
    • Handle: RePEc:eee:apmaco:v:303:y:2017:i:c:p:226-239
    DOI: 10.1016/j.amc.2017.01.030
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    References listed on IDEAS

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    1. Melnyk, S.S. & Usatenko, O.V. & Yampol'skii, V.A., 2006. "Memory functions of the additive Markov chains: applications to complex dynamic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(2), pages 405-415.
    2. Vladimir Soloviev & Vladimir Saptsin & Dmitry Chabanenko, 2011. "Markov Chains application to the financial-economic time series prediction," Papers 1111.5254, arXiv.org.
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    Cited by:

    1. König, Sandra & Rass, Stefan & Schauer, Stefan, 2019. "Cyber-attack impact estimation for a port," Chapters from the Proceedings of the Hamburg International Conference of Logistics (HICL), in: Jahn, Carlos & Kersten, Wolfgang & Ringle, Christian M. (ed.), Digital Transformation in Maritime and City Logistics: Smart Solutions for Logistics. Proceedings of the Hamburg International Conference of Logistics, volume 28, pages 164-183, Hamburg University of Technology (TUHH), Institute of Business Logistics and General Management.
    2. Mark Kiermayer & Christian Wei{ss}, 2022. "Neural calibration of hidden inhomogeneous Markov chains -- Information decompression in life insurance," Papers 2201.02397, arXiv.org.

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