Estimating the eigenvalue error of Markov State Models

Abstract

We consider a continuous-time, ergodic Markov process on a large continuous or discrete state space. The process is assumed to exhibit a number of metastable sets. Markov state models (MSM) are designed to represent the effective dynamics of such a process by a Markov chain that jumps between the metastable sets with the transition rates of the original process. MSM are used for a number of applications, including molecular dynamics (cf. Noe et al, PNAS(106) 2009)[1], since more than a decade. The rigorous and fully general (no zero temperature limit or comparable restrictions) analysis of their approximation quality, however, has only been started recently. Our first article on this topics (Sarich et al, MMS(8) 2010)[2] introduces an error bound for the difference in propagation of probability densities between the MSM and the original process on long time scales. Herein we provide upper bounds for the error in the eigenvalues between the MSM and the original process which means that we analyse how well the longest timescales in the original process are approximated by the MSM. Our findings are illustrated by numerical experiments.