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

作者：Tzortzis Ioannis^*; Charalambous Charalambos D; Charalambous Themistoklis; Hadjicostis Christoforos N; Johansson Mikael

来源：IEEE Transactions on Automatic Control, 2017, 62(3): 1030-1045.

DOI：10.1109/TAC.2016.2578299

• Approximating process
• Markov process
• total variation distance
• water-filling

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 optimization 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.

• 出版日期2017-3