DIMENSION OF THE SINGULAR SETS OF PLANE-FITTERS

• 成果类型: Article
• 署名作者: ELLIS, SP
• 署名单位: University of Rochester; Rutgers University System; Rutgers University New Brunswick; Rutgers University Biomedical & Health Sciences
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176324532
• 发表日期: 1995
• 页码: 490-501
• 关键词: projection pursuit

摘要

Let n > p > k > O be integers. Let delta be any technique for fitting k-planes to p-variate data sets of size n, for example, linear regression, principal components-or projection pursuit. Let J be the set of data sets which are (1) singularities of delta, that is, near them delta is unstable (for example, collinear data sets are singularities of least squares regression) and (2) nondegenerate, that is, their rank, after centering, is at least k. It is shown that the Hausdorff dimension, dim(H)(J), of J is at least nk + (K + 1)(p - k) - 1. This bound is tight. Under hypotheses satisfied by some projection pursuits (including principal components), dim(H)(J) greater than or equal to np - 2, that is, once singularity is taken into account, only two degrees of freedom remain in the problem. These results have implications for multivariate data description, resistant plane-fitting and jackknifing and bootstrapping plane-fitting.