PCA CONSISTENCY IN HIGH DIMENSION, LOW SAMPLE SIZE CONTEXT

• 成果类型: Article
• 署名作者: Jung, Sungkyu; Marron, J. S.
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS709
• 发表日期: 2009
• 页码: 4104-4130
• 关键词: geometric representation; Covariance matrices; LARGEST EIGENVALUE

摘要

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace consistency. We show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others, then the corresponding estimated PC directions are consistent or converge to the appropriate subspace (subspace consistency) and most other PC directions are strongly inconsistent. Broad sets of sufficient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations. In preparation for these results, we show that the geometric representation of HDLSS data holds under general conditions, which includes a rho-mixing condition and a broad range of sphericity measures of the covariance matrix.