@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{Weber2003, author = {Weber, Marcus}, title = {Clustering by using a simplex structure}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7782}, number = {04-03}, year = {2003}, abstract = {In this paper we interpret clustering as a mapping of data into a simplex. If the data itself has simplicial struture this mapping becomes linear. Spectral analysis is an often used tool for clustering data. We will show that corresponding singular vectors or eigenvectors comprise simplicial structure. Therefore they lead to a cluster algorithm, which consists of a simple linear mapping. An example for this kind of algorithms is the Perron cluster analysis (PCCA). We have applied it in practice to identify metastable sets of molecular dynamical systems. In contrast to other algorithms, this kind of approach provides an a priori criterion to determine the number of clusters. In this paper we extend the ideas to more general problems like clustering of bipartite graphs.}, language = {en} } @misc{WeberRungsarityotinSchliep2004, author = {Weber, Marcus and Rungsarityotin, Wasinee and Schliep, Alexander}, title = {Perron Cluster Analysis and Its Connection to Graph Partitioning for Noisy Data}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8140}, number = {04-39}, year = {2004}, abstract = {The problem of clustering data can be formulated as a graph partitioning problem. Spectral methods for obtaining optimal solutions have reveceived a lot of attention recently. We describe Perron Cluster Cluster Analysis (PCCA) and, for the first time, establish a connection to spectral graph partitioning. We show that in our approach a clustering can be efficiently computed using a simple linear map of the eigenvector data. To deal with the prevalent problem of noisy and possibly overlapping data we introduce the min Chi indicator which helps in selecting the number of clusters and confirming the existence of a partition of the data. This gives a non-probabilistic alternative to statistical mixture-models. We close with showing favorable results on the analysis of gene expressi on data for two different cancer types.}, language = {en} } @misc{HuisingaBestCordesetal.1998, author = {Huisinga, Wilhelm and Best, Christoph and Cordes, Frank and Roitzsch, Rainer and Sch{\"u}tte, Christof}, title = {From Simulation Data to Conformational Ensembles: Structure and Dynamics based Methods}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3797}, number = {SC-98-36}, year = {1998}, abstract = {Statistical methods for analyzing large data sets of molecular configurations within the chemical concept of molecular conformations are described. The strategies are based on dependencies between configurations of a molecular ensemble; the article concentrates on dependencies induces by a) correlations between the molecular degrees of freedom, b) geometrical similarities of configurations, and c) dynamical relations between subsets of configurations. The statistical technique realizing aspect a) is based on an approach suggested by {\sc Amadei et al.} (Proteins, 17 (1993)). It allows to identify essential degrees of freedom of a molecular system and is extended in order to determine single configurations as representatives for the crucial features related to these essential degrees of freedom. Aspects b) and c) are based on statistical cluster methods. They lead to a decomposition of the available simulation data into {\em conformational ensembles} or {\em subsets} with the property that all configurations in one of these subsets share a common chemical property. In contrast to the restriction to single representative conformations, conformational ensembles include information about, e.g., structural flexibility or dynamical connectivity. The conceptual similarities and differences of the three approaches are discussed in detail and are illustrated by application to simulation data originating from a hybrid Monte Carlo sampling of a triribonucleotide.}, language = {en} } @misc{CordesWeberSchmidtEhrenberg2002, author = {Cordes, Frank and Weber, Marcus and Schmidt-Ehrenberg, Johannes}, title = {Metastable Conformations via successive Perron-Cluster Cluster Analysis of dihedrals}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7074}, number = {02-40}, year = {2002}, abstract = {Decomposition of the high dimensional conformational space of bio-molecules into metastable subsets is used for data reduction of long molecular trajectories in order to facilitate chemical analysis and to improve convergence of simulations within these subsets. The metastability is identified by the Perron-cluster cluster analysis of a Markov process that generates the thermodynamic distribution. A necessary prerequisite of this analysis is the discretization of the conformational space. A combinatorial approach via discretization of each degree of freedom will end in the so called ''curse of dimension''. In the following paper we analyze Hybrid Monte Carlo simulations of small, drug-like biomolecules and focus on the dihedral degrees of freedom as indicators of conformational changes. To avoid the ''curse of dimension'', the projection of the underlying Markov operator on each dihedral is analyzed according to its metastability. In each decomposition step of a recursive procedure, those significant dihedrals, which indicate high metastability, are used for further decomposition. The procedure is introduced as part of a hierarchical protocol of simulations at different temperatures. The convergence of simulations within metastable subsets is used as an ''a posteriori'' criterion for a successful identification of metastability. All results are presented with the visualization program AmiraMol.}, 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{DeuflhardWeber2003, author = {Deuflhard, Peter and Weber, Marcus}, title = {Robust Perron Cluster Analysis in Conformation Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7415}, number = {03-19}, year = {2003}, abstract = {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.}, language = {en} } @misc{WeberMeyer2005, author = {Weber, Marcus and Meyer, Holger}, title = {ZIBgridfree - Adaptive Conformation Analysis with qualified Support of Transition States and Thermodynamic Weights}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8514}, number = {05-17}, year = {2005}, abstract = {This paper introduces a new algorithm of conformational analysis based on mesh-free methods as described in [M. Weber. Mehless methods in Conformation Dynamics.(2005)]. The adaptive decomposition of the conformational space by softly limiting functions avoids trapping effects and allows adaptive refinement strategies. These properties of the algorithm makes ZIBgridfree particularly suitable for the complete exploration of high-dimensional conformational space. The adaptive control of the algorithm benefits from the tight integration of molecular simulation and conformational analysis. An emphasized part of the analysis is the Robust Perron Cluster Analysis (PCCA+) based on the work of Peter Deuflhard and Marcus Weber. PCCA+ supports an almost-characteristic cluster definition with an outstanding mapping of transition states. The outcome is expressed by the metastable sets of conformations, their thermodynamic weights and flexibility.}, 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} }