Gradient Filtering Projections for Recovering Structures in Multivariate Data

Abstract

In this paper we introduce several sets of linear projections for low-dimensional projecting multivariate data witch are used for visual revealing structures in them. These projection sets are connected with the gradient filtering process defined by Fukunaga, Hosteller (1975). We define the following projection directions related to the filtering: directions with the extreme change of variance, directions with the extreme relocation of projected data points and the predominant gradient directions.

These projection sets coincide with the principal components set if the density of the data is ellipsoidally symmetric. Two first sets of projections approximate the Rao canonical discriminant projections if the density is a mixture of ellipsoidally symmetric densities with an equal within-covariance matrix.

Extraction of the projections is based on the solution of generalized eigenvector problems. We use “k-nearest neighbours” approch for estimating the density gradient.