Statistical inference and distribution theory in exploratory projection pursuit

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The exploratory projection pursuit (EPP) is a method for extracting non-linearity or non-normality from multidimensional data through maximizing the projection pursuit index and then projecting data into low-dimensional subspaces. Because of recent developments' of computers and techniques of optimization, EPP has been developing in the viewpoint of algorithm, but not yet fully developed in the viewpoint of statistical inference. The aims of this research is to make it practicable.The first step toward the practical use is to obtain the distribution of the maximum of projection pursuit index. In the first year, we characterized the limiting distribution of the maximum of the moment index (Jones and Sibson, 1987, J.Roy. Statist. Soc. Ser. A) as the maximum of a Gaussian random field, when the sample size goes to infinity. Then we derived an approximate upper tail probability of the maximum by an integral-geometric approach called the tube method or the Euler characteristic method.In these methods, the index set of the random field is regarded as a Riemannian submanifold endowed with the metric induced by it correlation structure. The approximate upper tail probability is derived as geometric characteristics of the manifold.In the second year, we evaluated the error term (remainder term) of the resulting formula. The error term is determined by so-called critical radius, which is a measure of convexity of submanifold. We conjectured the value of the critical radius through a large scale numerical experiments. However, we have not yet given any mathematical proof that the suggested value is truly the critical radius.These results were presented at the 3rd International Conference on Multiple Comparisons (MCP2002), Bethesda, Maryland, USA, August 2002.