Statistical inference and distribution theory in exploratory projection pursuit

探索性投影寻踪（exploratory projection pursuit， EPP）是一种通过最大化投影寻踪指数，从多维数据中提取非线性或非正态性，然后将数据投影到低维子空间的方法。由于近年来计算机和优化技术的发展，EPP在算法的角度上得到了发展，但在统计推理的角度上还没有得到充分的发展。这项研究的目的是使其可行。实现实际应用的第一步是求出投影跟踪指标最大值的分布。在第一年，我们描述了矩指数最大值的极限分布(Jones and Sibson, 1987； J.Roy；中央集权。Soc。爵士。A)为高斯随机场的最大值，当样本量趋于无穷时。然后，我们通过一种称为管法或欧拉特征法的积分几何方法推导出最大值的近似上尾概率。在这些方法中，将随机场的指标集看作一个黎曼子流形，并赋予其相关结构所产生的度量。近似的上尾概率推导为流形的几何特征。第二年，我们对所得公式的误差项（余项）进行了评估。误差项由所谓的临界半径决定，它是子流形凹凸度的度量。通过大规模数值实验，推测了临界半径的取值。然而，我们还没有给出任何数学证明，证明所建议的值确实是临界半径。这些结果于2002年8月在美国马里兰州贝塞斯达举行的第三届多重比较国际会议（MCP2002）上发表。

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.