Visualization and Data Mining for High Dimensional Datasets

Abstract

Visualization provides insight through images and can be considered as a collection of application specific mappings: ProblemDomain → VisualRange. For the visualization of multivariate problems a multidimensional system of parallel coordinates (abbr. ∥-coords) is constructed which induces a one-to-one mapping between subsets of N-space and subsets of 2-space. The result is a rigorous methodology for doing and seeing N-dimensional geometry. Starting with an the overview of the mathematical foundations, it is seen that the display of high-dimensional datasets and search for multivariate relations among the variables is transformed into a 2-D pattern recognition problem. This is the basis for the application to Visual Data Mining which is illustrated with a real dataset of VLSI (Very Large Scale Integration — “chip”) production. Then a recent geometric classifier is presented and applied to 3 real datasets. The results compared to those of 23 other classifiers have the least error. The algorithm, has quadratic computational complexity in the size and number of parameters, provides comprehensible and explicit rules, does dimensionality selection — where the minimal set of original variables required to state the rule is found, and orders these variables so as to optimize the clarity of separation between the designated set and its complement. Finally a simple visual economic model of a real country is constructed and analyzed in order to illustrate the special strength of ∥-coords in modeling multivariate relations by means of hypersurfaces.