On an SVD-based algorithm for identifying meta-stable states of Markov chains. (English) Zbl 1321.65056

Tifenbach, Ryan M.

ETNA, Electron. Trans. Numer. Anal. 38, 17-33 (2011).

Summary: A Markov chain is a sequence of random variables \(X=\{x_t\}\) that take on values in a state space \(\mathcal S\).
A meta-stable state with respect to \(X\) is a collection of states \(\mathcal E\subseteq\mathcal S\) such that transitions of the form \(x_t\in\mathcal E\) and \(x_{t+1}\notin\mathcal E\) are exceedingly rare. In [D. Fritzsche et al., ETNA, Electron. Trans. Numer. Anal. 29, 46–69 (2007; Zbl 1171.15008)], an algorithm is presented that attempts to construct the meta-stable states of a given Markov chain. We supplement the discussion contained therein concerning the two main results.

MSC:

65F15	Numerical computation of eigenvalues and eigenvectors of matrices
15A18	Eigenvalues, singular values, and eigenvectors
15B51	Stochastic matrices
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
60J20	Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)

Keywords:

Markov chains; conformation dynamics; singular value decomposition