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- Hidden Markov Model (1)
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In this paper we analyze decompositions of reversible nearly uncoupled
Markov chains into rapidly mixing subchains. We state upper
bounds on the 2nd eigenvalue for restriction and stochastic complementation
chains of reversible Markov chains, as well as a relation between
them. We illustrate the obtained bounds analytically for bunkbed
graphs, and furthermore apply them to restricted Markov chains that
arise when analyzing conformation dynamics of a small biomolecule.
We shortly review the uncoupling-coupling method, a Markov chain
Monte Carlo based approach to compute statistical properties of systems like
medium-sized biomolecules. This technique has recently been proposed for the efficient computation of biomolecular conformations. One crucial step of UC is the
decomposition of reversible nearly uncoupled Markov chains into rapidly mixing
subchains. We show how the underlying scheme of uncoupling-coupling can also be
applied to stochastic differential equations where it can be translated into a domain
decomposition technique for partial differential equations.
We report on a novel approach to the automatic identification
of metastable states from long term simulation of complex
molecular systems. The new approach is based on a hierarchical concept
of metastability: metastable states are understood as subsets of
state or configuration space from which the dynamics exits only very rarely;
subsets with the smallest exit probabilities are of most interest, their
further decomposition then may reveal subsets from which exiting
is less but comparably difficult for the system under investigation.
The article gives a survey of the theoretical foundation of
the approach and its algorithmic realization that generalizes
the well-known concept of Hidden Markov Models.
The performance of the resulting algorithm are illustrated by
application to a 100 ns simulation of penta-alanine with explicit water.
We demonstrate the resulting metastable states allow to
reveal the conformation dynamics of the moelcule.