Model reduction of nonreversible Markov chains

In many uncertain complex systems it is observed that the system trajectories cluster in several subsets of the state space. In this paper we model the system behavior as a Markov process and consider the problem of finding a low dimensional approximation of the process that captures the clustering phenomena. Furthermore, we concentrate on Markov chain approximations on a finite state space of large dimension. The problem of finding an approximate low dimensional operator is much simpler when the Markov chain is reversible and several solution approaches have been developed for this case. Most of these approaches rely on spectral properties of the Markov chain. In this paper we consider the general nonreversible case. Our approach is based on a reversibilization procedure, spectral methods for the identification of the dominant components and constrained projection of the original system onto the low dimensional space.