Multidimensional Relative Projection Pursuit

Abstract

We propose a new multidimensional projection index for relative projection pursuit (RPP; Mizuta, 2002). RPP is a dimension reduction method that is an extension of conventional projection pursuit (Friedman and Tukey, 1974). Conventional projection pursuit finds 'interesting' structures which differ from the normal distribution. RPP finds structures that differ from a reference data set predefined by the user as having 'uninteresting' structure. We have already proposed a one-dimensional projection index for RPP, the area index, which measures the difference between target data and reference data as a degree of 'Interestingness'. However, it cannot be applied when a user wants to reduce high dimensional data into spaces of more than one dimension. Therefore, we extend the area index so that it can be applied even when the target data set is projected into multidimensional space. In addition, we develop a new index for RPP, which is based on the Hall index (Hall, 1989), called the Hall type relative projection index.We demonstrate the effectiveness of multidimensional RPP using artificial and actual data. In the numerical example with artificial data, it is shown that with the Hall type relative projection index we can detect more 'interesting' multidimensional spaces than that with Area index. When we apply multidimensional RPP to actual data, we can obtain 'interesting' structures of high dimensional data that cannot be derived using conventional projection pursuit.