Visualization and knowledge discovery for high dimensional data

The goal here is to present a multidimensional visualization methodology and its applications to Visual and Automatic Knowledge Discovery in a coherent paper. Visualization provides insight through images and can be considered as a collection of application specific mappings: Problem Domain --> Visual Range. 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. We start with an overview of the mathematical foundations where it is seen that from the display of high-dimensional datasets the 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 Knowledge Discovery which is illustrated in the second part with real dataset of VLSI 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.