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Preprint 15-2006

An SVD approach to identifying meta-stable states of Markov chains

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Author(s) : David Fritzsche , Volker Mehrmann , Daniel Szyld , Elena Virnik

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 15-2006

MSC 2000

15A18 Eigenvalues, singular values, and eigenvectors
15A51 Stochastic matrices

Abstract :
Being one of the key tools in conformation dynamics, the identification of meta-stable states of Markov chains has been subject to extensive research in recent years, especially when the Markov chains represent energy states of biomolecules. Some previous work on this topic involved the computation of the eigenvalue cluster close to one, as well as the corresponding eigenvectors and the stationary probability distribution of the associated stochastic matrix. Later, since the eigenvalue cluster algorithm turned out to be non-robust, an optimisation approach was developed. As a possible less costly alternative, we present an SVD approach to identifying meta-stable states of a stochastic matrix, where we only need the second largest singular vector. We outline some theoretical background and discuss the advantages of this strategy. Some simulated and real numerical examples illustrate the effectiveness of the proposed algorithm.

Keywords : Markov chains, conformation dynamics, singular value decomposition

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