62H30 Classification and discrimination; cluster analysis [See also 68T10]
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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.
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.
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.
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.
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.
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.
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.
Als Cluster Analyse bezeichnet man den Prozess der Suche und Beschreibung von Gruppen (Clustern) von Objekten, so daß die Objekte innerhalb eines Clusters bezüglich eines gegebenen Maßes maximal homogen sind. Die Homogenität der Objekte hängt dabei direkt oder indirekt von den Ausprägungen ab, die sie für eine Anzahl festgelegter Attribute besitzen. Die Suche nach Clustern läßt sich somit als Optimierungsproblem auffassen, wobei die Anzahl der Cluster vorher bekannt sein muß. Wenn die Anzahl der Objekte und der Attribute groß ist, spricht man von komplexen, hoch-dimensionalen Cluster Problemen. In diesem Fall ist eine direkte Optimierung zu aufwendig, und man benötigt entweder heuristische Optimierungsverfahren oder Methoden zur Reduktion der Komplexität. In der Vergangenheit wurden in der Forschung fast ausschließlich Verfahren für geometrisch basierte Clusterprobleme entwickelt. Bei diesen Problemen lassen sich die Objekte als Punkte in einem von den Attributen aufgespannten metrischen Raum modellieren; das verwendete Homogenitätsmaß basiert auf der geometrischen Distanz der den Objekten zugeordneten Punkte. Insbesondere zur Bestimmung sogenannter metastabiler Cluster sind solche Verfahren aber offensichtlich nicht geeignet, da metastabile Cluster, die z.B. in der Konformationsanalyse von Biomolekülen von zentraler Bedeutung sind, nicht auf einer geometrischen, sondern einer dynamischen Ähnlichkeit beruhen. In der vorliegenden Arbeit wird ein allgemeines Clustermodell vorgeschlagen, das zur Modellierung geometrischer, wie auch dynamischer Clusterprobleme geeignet ist. Es wird eine Methode zur Komplexitätsreduktion von Clusterproblemen vorgestellt, die auf einer zuvor generierten Komprimierung der Objekte innerhalb des Datenraumes basiert. Dabei wird bewiesen, daß eine solche Reduktion die Clusterstruktur nicht zerstört, wenn die Komprimierung fein genug ist. Mittels selbstorganisierter neuronaler Netze lassen sich geeignete Komprimierungen berechnen. Um eine signifikante Komplexitätsreduktion ohne Zerstörung der Clusterstruktur zu erzielen, werden die genannten Methoden in ein mehrstufiges Verfahren eingebettet. Da neben der Identifizierung der Cluster auch deren effiziente Beschreibung notwendig ist, wird ferner eine spezielle Art der Komprimierung vorgestellt, der eine Boxdiskretisierung des Datenraumes zugrunde liegt. Diese ermöglicht die einfache Generierung von regelbasierten Clusterbeschreibungen. Für einen speziellen Typ von Homogenitätsfunktionen, die eine stochastische Eigenschaft besitzen, wird das mehrstufige Clusterverfahren um eine Perroncluster Analyse erweitert. Dadurch wird die Anzahl der Cluster, im Gegensatz zu herkömmlichen Verfahren, nicht mehr als Eingabeparameter benötigt. Mit dem entwickelten Clusterverfahren kann erstmalig eine computergestützte Konformationsanalyse großer, für die Praxis relevanter Biomoleküle durchgeführt werden. Am Beispiel des HIV Protease Inhibitors VX-478 wird dies detailliert beschrieben.
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.
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.