The Arnoldi Aggregation for Approximate Transient Distributions of Markov Chains

Abstract

The paper proposes a new aggregation method, based on the Arnoldi iteration, for computing approximate transient distributions of Markov chains. This aggregation is not partition-based, which means that an aggregate state may represent any portion of any original state, leading to a reduced system which is not a Markov chain. Results on exactness (in case the algorithm finds an invariant Krylov subspace) and minimality of the size of the Arnoldi aggregation are proven. For practical use, a heuristic is proposed for deciding when to stop expanding the state space once a certain accuracy has been reached. Apart from the theory, the paper also includes an extensive empirical section where the new aggregation algorithm is tested on several models and compared to a lumping-based state space reduction scheme.