Perron cluster

Theorem 1 highlights the strong relation between a decomposition of the state space into metastable subsets and a Perron cluster of dominant eigenvalues close to 1. ft states that the metastability of an arbitrary decomposition d cannot be larger than I-I-A2+... - -Ato, while it is at least 1- -K2 2+- c, which is close to the upper bound whenever the dominant eigenfunctions V2,..., Vm are almost constant on the metastable subsets Pi,..., Dm implying Kj fn 1 and c 0. The term c can be interpreted as a correction that is small whenever a 0 or Kj 1. It is demonstrated in [23] that the lower and upper bounds are sharp and asymptotically exact. 

The parameter fitting step requires the specification of the number of hidden states, which, whenever the hidden states should be metastable states, is in general not apriori known. One policy to overcome this problem is to assume a sufficient large number of hidden states, perform the parameter fitting and conduct a further aggregation of the resulting transition matrix. This can be done by Perron cluster cluster analysis (PCCA), e.g., by the spectral properties of the resulting transition matrix T as proposed in the transfer operator approach (we will illustrate this procedure on an example in the next section), see [11] for details. 

P. Deuflhard and M. Weber (2005) Robust Perron cluster analysis in conformation dynamics. Lin. Alg. Appl. 398, pp. 161-184... 

M. Weber (2004) Improved Perron cluster analysis. ZIB-Report, (Zuse Institute, Berlin, pp. 03-04... 

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