TY - GEN A1 - Galliat, Tobias A1 - Deuflhard, Peter A1 - Roitzsch, Rainer A1 - Cordes, Frank T1 - Automatic Identification of Metastable Conformations via Self-Organized Neural Networks N2 - As has been shown recently, the identification of metastable chemical conformations leads to a Perron cluster eigenvalue problem for a reversible Markov operator. Naive discretization of this operator would suffer from combinatorial explosion. As a first remedy, a pre-identification of essential degrees of freedom out of the set of torsion angles had been applied up to now. The present paper suggests a different approach based on neural networks: its idea is to discretize the Markov operator via self-organizing (box) maps. The thus obtained box discretization then serves as a prerequisite for the subsequent Perron cluster analysis. Moreover, this approach also permits exploitation of additional structure within embedded simulations. As it turns out, the new method is fully automatic and efficient also in the treatment of biomolecules. This is exemplified by numerical results. T3 - ZIB-Report - 00-51 KW - biochemical conformations KW - cluster analysis KW - Molecular Dynamics KW - Monte-Carlo methods KW - operator discretization KW - Perron cluster Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6197 ER - TY - GEN A1 - Deuflhard, Peter A1 - Weber, Marcus T1 - Robust Perron Cluster Analysis in Conformation Dynamics N2 - The key to molecular conformation dynamics is the direct identification of metastable conformations, which are almost invariant sets of molecular dynamical systems. Once some reversible Markov operator has been discretized, a generalized symmetric stochastic matrix arises. This matrix can be treated by Perron cluster analysis, a rather recent method involving a Perron cluster eigenproblem. The paper presents an improved Perron cluster analysis algorithm, which is more robust than earlier suggestions. Numerical examples are included. T3 - ZIB-Report - 03-19 KW - Markov chains KW - cluster algorithms KW - Perron cluster analysis KW - conformation dynamics Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7415 ER - TY - GEN A1 - Weber, Marcus A1 - Walter, Lionel A1 - Kube, Susanna A1 - Deuflhard, Peter T1 - Stable computation of probability densities for metastable dynamical systems N2 - Whenever the invariant stationary density of metastable dynamical systems decomposes into almost invariant partial densities, its computation as eigenvector of some transition probability matrix is an ill-conditioned problem. In order to avoid this computational difficulty, we suggest to apply an aggregation/disaggregation method which only addresses wellconditioned sub-problems and thus results in a stable algorithm. In contrast to existing methods, the aggregation step is done via a sampling algorithm which covers only small patches of the sampling space. Finally, the theoretical analysis is illustrated by two biomolecular examples. T3 - ZIB-Report - 06-39 KW - dynamical systems KW - metastability KW - molecular conformations KW - cluster analysis KW - sampling KW - aggregation/disaggregation KW - domain decomposition Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9331 ER - TY - GEN A1 - Galliat, Tobias A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter T1 - Self-Organizing Maps Combined with Eigenmode Analysis for Automated Cluster Identification N2 - One of the important tasks in Data Mining is automated cluster analysis. Self-Organizing Maps (SOMs) introduced by {\sc Kohonen} are, in principle, a powerful tool for this task. Up to now, however, its cluster identification part is still open to personal bias. The present paper suggests a new approach towards automated cluster identification based on a combination of SOMs with an eigenmode analysis that has recently been developed by {\sc Deuflhard et al.} in the context of molecular conformational dynamics. Details of the algorithm are worked out. Numerical examples from Data Mining and Molecular Dynamics are included. T3 - ZIB-Report - SC-99-38 KW - Self-Organizing Maps KW - cluster analysis Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4279 ER - TY - GEN A1 - Galliat, Tobias A1 - Deuflhard, Peter T1 - Adaptive hierarchical cluster analysis by Self-Organizing Box Maps N2 - The present paper aims at an extension of {\sc Kohonen's} Self-Organizing Map (SOM) algorithm to be called Self-Organizing Box Map (SOBM) algorithm; it generates box codebooks in lieu of point codebooks. Box codebooks just like point codebooks indirectly define a Voronoi tessellation of the input space, so that each codebook vector represents a unique set of points. Each box codebook vector comprises a multi-dimensional interval that approximates the related partition of the Voronoi tessellation. Upon using the automated cluster identification method that has recently been developed by the authors, the codebook vectors can be grouped in such a way that each group represents a point cluster in the input space. Since the clustering usually depends on the size of the SOM, one cannot be sure, whether the clustering comes out to be optimal. Refinement of part of the identified clusters would often improve the results. This paper presents the concept of an adaptive multilevel cluster algorithm that performs such refinements automatically. Moreover the paper introduces a concept of essential dimensions and suggests a method for their identification based on our herein suggested box codebooks. Applications of the algorithm to molecular dynamics will be described in a forthcoming paper. T3 - ZIB-Report - 00-13 KW - Self-Organizing Maps KW - cluster analysis KW - Voronoi tessellation KW - feature extraction KW - essential dimensions KW - multilevel methods Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5815 ER -