@misc{WeberWalterKubeetal.2006, author = {Weber, Marcus and Walter, Lionel and Kube, Susanna and Deuflhard, Peter}, title = {Stable computation of probability densities for metastable dynamical systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9331}, number = {06-39}, year = {2006}, abstract = {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.}, language = {en} } @misc{WeberKubeRiemeretal.2006, author = {Weber, Marcus and Kube, Susanna and Riemer, Alexander and Bujotzek, Alexander}, title = {Efficient Sampling of the Stationary Distribution of Metastable Dynamical Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9467}, number = {07-03}, year = {2006}, abstract = {In this article we aim at an efficient sampling of the stationary distribution of dynamical systems in the presence of metastabilities. In the past decade many sophisticated algorithms have been inven ted in this field. We do not want to simply add a further one. We address the problem that one has applied a sampling algorithm for a dynamical system many times. This leads to different samplings which more or less represent the stationary distribution partially very well, but which are still far away from ergodicity or from the global stationary distribution. We will show how these samplings can be joined together in order to get one global sampling of the stationary distribution.}, language = {en} } @misc{Weber2003, author = {Weber, Marcus}, title = {Improved Perron Cluster Analysis}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7260}, number = {03-04}, year = {2003}, abstract = {The problem of clustering data can often be transformed into the problem of finding a hidden block diagonal structure in a stochastic matrix. Deuflhard et al. have proposed an algorithm that state s the number \$k\$ of clusters and uses the sign structure of \$k\$ eigenvectors of the stochastic matrix to solve the cluster problem. Recently Weber and Galliat discovered that this system of eigenvectors can easily be transformed into a system of \$k\$ membership functions or soft characteristic functions describing the clusters. In this article we explain the corresponding cluster algorithm and point out the underlying theory. By means of numerical examples we explain how the grade of membership can be interpreted.}, language = {en} } @misc{Fischer2000, author = {Fischer, Alexander}, title = {An Uncoupling-Coupling Technique for Markov Chain Monte Carlo Methods}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5720}, number = {00-04}, year = {2000}, abstract = {Uncoupling-coupling Monte Carlo (UCMC) combines uncoupling techniques for finite Markov chains with Markov chain Monte Carlo methodology. By determining almost invariant sets of the associated Markov operator, the Monte Carlo sampling splits by a hierarchical annealing process into the essential regions of the state space; therefore UCMC aims at avoiding the typical metastable behavior of Monte Carlo techniques. From the viewpoint of Monte Carlo, a slowly converging long-time Markov chain is replaced by a limited number of rapidly mixing short-time ones. The correct weighting factors for the various Markov chains are obtained via a coupling matrix, that connects the samplings from the different almost invariant sets. The underlying mathematical structure of this approach is given by a general examination of the uncoupling-coupling procedure. Furthermore, the overall algorithmic scheme of UCMC is applied to the \$n\$-pentane molecule, a well-known example from molecular dynamics.}, language = {en} } @misc{GalliatHuisingaDeuflhard1999, author = {Galliat, Tobias and Huisinga, Wilhelm and Deuflhard, Peter}, title = {Self-Organizing Maps Combined with Eigenmode Analysis for Automated Cluster Identification}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4279}, number = {SC-99-38}, year = {1999}, abstract = {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.}, language = {en} } @misc{Galliat1999, author = {Galliat, Tobias}, title = {Clustering Data of Different Information Levels}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4319}, number = {SC-99-42}, year = {1999}, abstract = {For using Data Mining, especially cluster analysis, one needs measures to determine the similarity or distance between data objects. In many application fields the data objects can have different information levels. In this case the widely used euclidean distance is an inappropriate measure. The present paper describes a concept how to use data of different information levels in cluster analysis and suggests an appropriate similarity measure. An example from practice is included, that shows the usefulness of the concept and the measure in combination with {\sc Kohonens} Self-Organizing Map algorithm, a well-known and powerful tool for cluster analysis.}, language = {en} } @misc{GalliatDeuflhard2000, author = {Galliat, Tobias and Deuflhard, Peter}, title = {Adaptive hierarchical cluster analysis by Self-Organizing Box Maps}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5815}, number = {00-13}, year = {2000}, abstract = {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.}, language = {en} } @misc{GalliatDeuflhardRoitzschetal.2000, author = {Galliat, Tobias and Deuflhard, Peter and Roitzsch, Rainer and Cordes, Frank}, title = {Automatic Identification of Metastable Conformations via Self-Organized Neural Networks}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6197}, number = {00-51}, year = {2000}, abstract = {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.}, language = {en} } @misc{WeberGalliat2002, author = {Weber, Marcus and Galliat, Tobias}, title = {Characterization of Transition States in Conformational Dynamics using Fuzzy Sets}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6791}, number = {02-12}, year = {2002}, abstract = {Recently, a novel approach for the analysis of molecular dynamics on the basis of a transfer operator has been introduced. Therein conformations are considered to be disjoint metastable clusters within position space of a molecule. These clusters are defined by almost invariant characteristic functions that can be computed via {\em Perron Cluster} analysis. The present paper suggests to replace crisp clusters with {\em fuzzy} clusters, i.e. to replace characteristic functions with membership functions. This allows a more sufficient characterization of transiton states between different confor conformations and therefore leads to a better understanding of molecular dynamics. Fur thermore, an indicator for the uniqueness of metastable fuzzy clusters and a fast algorithm for the computation of these clusters are described. Numerical examples are included.}, language = {en} }