The Computation of Key Properties of Markov Chains via Perturbations

TL;DR

This paper introduces perturbation-based computational methods for efficiently calculating key properties of irreducible Markov chains, including stationary distributions, group inverses, and mean first passage times.

Contribution

It develops six algorithms using perturbation techniques to update matrix solutions, improving computational procedures for Markov chain analysis.

Findings

Six algorithms for Markov chain properties computation

Numerical comparisons demonstrate efficiency

Perturbation methods simplify complex matrix inversions

Abstract

Computational procedures for the stationary probability distribution, the group inverse of the Markovian kernel and the mean first passage times of an irreducible Markov chain, are developed using perturbations. The derivation of these expressions involves the solution of systems of linear equations and, structurally, inevitably the inverses of matrices. By using a perturbation technique, starting from a simple base where no such derivations are formally required, we update a sequence of matrices, formed by linking the solution procedures via generalized matrix inverses and utilising matrix and vector multiplications. Six different algorithms are given, some modifications are discussed, and numerical comparisons made using a test example. The derivations are based upon the ideas outlined in Hunter, J.J., The computation of stationary distributions of Markov chains through perturbations, Journal of Applied Mathematics and Stochastic Analysis, 4, 29-46, (1991).