PCA CONSISTENCY IN HIGH DIMENSION, LOW SAMPLE SIZE CONTEXT
PCA CONSISTENCY IN HIGH DIMENSION, LOW SAMPLE SIZE CONTEXT
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DOI：
10.1214/09-aos709
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发表时间：
2009-12-01
影响因子：
4.5
通讯作者：
Marron, J. S.
中科院分区：
文献类型：
--
作者：
Jung, Sungkyu;Marron, J. S.
主成分分析（PCA）是一种重要的降维工具，特别是当维数（或变量数）很高时。渐近研究，其中样本量是固定的，并且维度增长[即，高维度、低样本量（HDLSS）]变得越来越重要。我们研究了主成分（PC）方向的渐近行为。HDLSS渐近性被用来研究一致性、强不一致性和子空间一致性。我们表明，如果人口协方差矩阵的前几个特征值足够大相比，其他的，那么相应的估计PC方向是一致的或收敛到适当的子空间（子空间一致性）和大多数其他PC方向是强烈不一致的。广泛的充分条件，为每一种情况下指定的主要定理给出了一个目录的可能的组合。在准备这些结果，我们表明，HDLSS数据的几何表示在一般条件下，其中包括一个ρ混合条件和广泛的球度的协方差矩阵的措施。
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.