| Projection Pursuit (PP) technology is a key for exploring the "non-linear" structure of high dimensional data. PP algorithms make it possible to find interesting directions so that the structure in data can be seen clearly after projection on these directions. It is useful to have a significance test to help us decide whether apparent structure is real or just the effect of noise. Monte Carlo methods can be helpful to achieve this goal; but unfortunately in this case, they are difficult to use and computationally expensive.;In addition to its contribution to the problem of projection pursuit, this work gives a general two terms approximation for the tail probability of the extreme of a class of differentiable Gaussian random fields and illustrates some of the potential for using Weyl's formula in Probability and Statistics. In the particular case of Friedman's PP index, the matrix of the metric tensor turns out to be diagonal and hence it is possible to calculate Weyl's second coefficient, which involves the total scalar curvature of the manifold, and in general is extremely complicated. The numerical results show that this term can improve the quality of the approximation enormously, and that use of only the first term, which involves the comparatively easily computed volume of the manifold, is inadequate.;My work is to look for analytical methods for calculating the P-value associated with various projection pursuit indices, especially, the index suggested by J. Friedman (1987). In this talk, under the null hypothesis that the data are an independent, identically distributed, sample from a p-dimensional normal population, a theoretical approximation formula for the P-value is derived using Weyl's formula (1939) for the volume of a tube about a manifold imbedded in the unit sphere in Euclidean space. Weyl's formula involves some complicated constants, for which applicable formulas and a table of numerical values are given. The result of Monte Carlo simulations is compared with our analytical result. The comparisons show that special care is needed when the number of dimensions is large. |
