The Arnoldi Aggregation for Approximate Transient Distributions of Markov Chains

TL;DR

This paper introduces an Arnoldi iteration-based aggregation method for approximating transient distributions in Markov chains, offering a non-partition-based approach that balances accuracy and computational efficiency.

Contribution

The paper presents a novel Arnoldi aggregation technique for Markov chains, including theoretical guarantees and a practical heuristic for stopping criteria.

Findings

Proven exactness when the invariant Krylov subspace is found.

Demonstrated reduced system size compared to traditional lumping.

Empirical results show competitive accuracy and efficiency.

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.