Approximation of Markov Processes by Lower Dimensional Processes via Total Variation Metrics

Ioannis Tzortzis, Charalambos D. Charalambous, Themistoklis Charalambous, Christoforos N. Hadjicostis, Mikael Johansson

Research output: Contribution to journalArticleScientificpeer-review


The aim of this paper is to approximate a Finite-State Markov (FSM) process by another process defined on a lower dimensional state space, called the approximating process, with respect to a total variation distance fidelity criterion. The approximation problem is formulated as an optimization problem using two different approaches. The first approach is based on approximating the transition probability matrix of the FSM process by a lower-dimensional transition probability matrix, resulting in an approximating process which is a Finite-State Hidden Markov (FSHM) process. The second approach is based on approximating the invariant probability vector of the original FSM process by another invariant probability vector defined on a lower-dimensional state space. Going a step further, a method is proposed based on optimizing a Kullback-Leibler divergence to approximate the FSHM processes by FSM processes. The solutions of these optimisation problems are described by optimal partition functions which aggregate the states of the FSM process via a corresponding water-filling solution, resulting in lower-dimensional approximating processes which are FSHM or FSM processes. Throughout the paper, the theoretical results are justified by illustrative examples that demonstrate our proposed methodology.
Original languageEnglish
Pages (from-to)1030-1045
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number3
Publication statusPublished - 2017
MoE publication typeA1 Journal article-refereed


  • Markov process
  • approximating process
  • total variation distance

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