Semi-Parametric Reduction of dimensionality
Elmar Diederichs · Universitätsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver · 2009
Concerning the analysis of large molecular systems increasing amounts of simulation data and growing dimensionality have led to the demand of data- driven approaches to extract physically interpretable information from large data sets. Hence a mapping to a low dimensional manifold, representing the essential degrees of freedom of a molecular system is sought. A general obstacle to such an analysis is the curse of dimensionality. This thesis is motivated by the fact that most dimension reduction methods are either not reliable in dimensionality regimes of realistic biomolecular systems or restricted to data sets with special features. On the one hand the aim is to develop an unsupervised linear feature extraction method, that allows to extract any multimodal distributed component to a given high dimensional data density. On the other hand the development of a geometric approach to the analysis of the large scale dynamical behavior of biological active molecules is intended. To this end a very general semi-parametric framework for unsupervised feature extraction based on weak structural assumptions on the data density is introduced. We discuss and develop different iterative and non-iterative approaches to semi-parametric dimension reduction allowing for identifying a low-dimensional non-Gaussian component of the whole distribution in a structure adaptive way. The main difference between the approaches discussed consist in the reconstruction of the low dimensional, non-Gaussian target space of the method on focus. We discuss methods based on Principle Component Analysis (PCA), convex projection and semi-definite programming. It turns out that the choice of the optimization problem to be solved in order to reconstruct the target space from some estimators is decisive for the statistical sensitivity of the method to a variety of non-Gaussian components. Currently the best alternative is Sparse NonGaussian Component Analysis based on semidefinite programming. Combining this linear projective method with the so called dip index specialized on the detection of multimodality, we come up with NonGaussian Cluster Analysis (NCA). It is demonstrated that NCA used as a preprocessing step to the metastablility analysis of biomolecules is superior to comparable dimension reduction methods. Combining NCA with the state-of- the-art approach of Hidden Markov Models to metastablility analysis, results in an almost geometrical approach to high dimensional analysis of metastablility as requested.