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Statistical inference and distribution theory in exploratory projection pursuit

探索性投影追踪中的统计推断和分布理论

基本信息

批准号：
13680380
负责人：
KURIKI Satoshi
金额：
$ 1.41万
依托单位：
The Institute of Statistical Mathematics
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13680380/
关键词：
projection pursuit index Gaussian random field tail probability

项目摘要

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.

探索性投影寻踪(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分布，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在算法方面有了很大的发展，但在统计推理方面还没有完全发展起来。这项研究的目的是使其具有实用性。走向实际使用的第一步是获得投影寻踪指数的最大值的分布。在第一年，我们刻画了矩指数的最大值的极限分布(Jones和Sibson，1987，J.Roy)。统计学家。SoC。先生。A)当样本大小为无穷大时，作为高斯随机场的最大值。然后，我们利用一种称为筒子法或欧拉特征线法的积分几何方法，将随机场的指标集看作具有由其相关结构诱导的度量的黎曼子流形，得到了一个近似的上尾概率。根据流形的几何特征，推导出近似的上尾概率。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，临界半径是子流形凸性的量度。通过大规模的数值实验，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来证明建议的值确实是临界半径。这些结果已于2002年8月在美国马里兰州贝塞斯达举行的第三届国际多重比较会议(MCP2002)上公布。