Estimating the Eigenvalue Error of Markov State Models

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 (MSMs) 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. MSMs have been used for a number of applications, including molecular dynamics (cf. [F. Noé et al., Proc. Natl. Acad. Sci. USA, 106 (2009), pp. 19011–19016]), for more than a decade. The rigorous and fully general (no zero temperature limit or comparable restrictions) analysis of their approximation quality, however, has only recently begun. Our first article on this topics [M. Sarich, F. Noé, and Ch. Schütte, Multiscale Model. Simul., 8 (2010), pp. 1154–1177] introduces an error bound for the difference in propagation of probability densities between the MSM and the original process on long timescales. Herein we provide upper bounds for the error in the eigenvalues between the MSM and the original process, which means that we analyze how well the longest timescales in the original process are approximated by the MSM. Our findings are illustrated by numerical experiments.