Intrinsic Dimensionality Estimation in Visualizing Toxicity Data

Over the years, a number of dimensionality reduction techniques have been proposed and used in chemo informatics to perform nonlinear mappings. Nevertheless, data visualization techniques can be efficiently applied for dimensionality reduction mainly in a case if the data are not really high-dimensional and can be represented as a nonlinear low-dimensional manifold when it is possible to reduce dimensionality without significant information loss. In this study several intrinsic dimensionality estimation approaches have been investigated: the Geodesic Minimum Spanning Tree, the Eigen value-based and the Maximum Likelihood Estimators. Their performance has been compared for visualizing toxicity data in different descriptor spaces. INTRODUCTION Over the years, a number of dimensionality reduction techniques have been proposed and used in chemo informatics to perform nonlinear mappings. Nevertheless, data visualization techniques can be efficiently applied for dimensionality reduction mainly in a case if the data are not really high-dimensional and can be represented as a nonlinear low-dimensional manifold when it is possible to reduce dimensionality without significant information loss [1]. In this study several intrinsic dimensionality estimation [2] approaches have been investigated: the Geodesic Minimum Spanning Tree [3], the Eigen value-based [4,5] and the Maximum Likelihood Estimators [1]. Their performance has been compared for visualizing toxicity data in different descriptor spaces. The obtained values of data intrinsic dimensionality (ID) were compared with the quantitative results of data visualization for two applied dimensionality reduction approaches: Diffusion maps and Isomap. MATERIALS AND METHODS For intrinsic dimensionality estimation and dimensionality reduction the implementations provided by Matlab Toolbox for Dimensionality Reduction (v 0.7.1b) [6] were used. Intrinsic dimensionality estimators The intrinsic dimensionality of the data can be defined as the minimal number of variables needed to describe the data x. The intrinsic dimensionality estimators can be related to two main categories: the eigen value or projection methods and the geometric methods. Eigen value methods are based on principal component analysis (PCA) [7]. PCA projects the data along the directions of maximal variance. It computes eigen values and eigenvectors of the covariance matrix of data. Intrinsic Dimensionality (ID) is defined by the number of eigen values that exceed a predefined value of threshold. The geometric methods are mostly based on fractal dimensions or nearest neighbor distances. In this study, the Geodesic Minimum Spanning Tree [3] and Maximum Likelihood Estimator [1] were used as representatives of second group of methods. In Geodesic Minimum Spanning Tree (GMST) several steps are considered. First, a complete graph based on geodesic distances between all pairs of data points is built. A minimal spanning graph, or the GMST, is obtained by the reduction of the initial graph to a subgraph, in which every data point xi is connected to its k nearest neighbors. The intrinsic dimension is estimated from the GMST length functional L: