Boundary behavior in High Dimension, Low Sample Size asymptotics of PCA
Boundary behavior in High Dimension, Low Sample Size asymptotics of PCA
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DOI：
10.1016/j.jmva.2012.03.005
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发表时间：
2012-08-01
影响因子：
1.6
通讯作者：
Marron, J. S.
中科院分区：
文献类型：
--
作者：
Jung, Sungkyu;Sen, Arusharka;Marron, J. S.
在高维低样本数据（HDLSS）的情况下，当维数d远大于样本量n时，主成分分析（PCA）在统计分析中起着重要的作用。在什么条件下样本主成分分析能很好地反映总体协方差结构？我们回答这个问题在一个相关的渐近背景下，其中d的增长和n是固定的，在一个广义的尖峰协方差模型。具体地说，我们假设最大的人口特征值是d（alpha）的顺序，其中alpha 1。已有的结果给出了样本协方差矩阵特征向量相合和强不相合的条件。在边界的情况下，α = 1，样本PC方向既不一致，也不强烈不一致，我们表明，特征值和特征向量不退化，但有限制的分布。该结果平滑地桥接了由其他两种情况表示的相变，从而给出了HDLSS渐近中样本PCA的极限谱。虽然结果在一般情况下成立，高斯假设下的极限分布进行了更详细的说明。此外，HDLSS数据的几何表示扩展到三个不同的表示，这取决于前几个主成分的方差的大小。(C)2012 Elsevier Inc. All rights reserved.
In High Dimension, Low Sample Size (HDLSS) data situations, where the dimension d is much larger than the sample size n, principal component analysis (PCA) plays an important role in statistical analysis. Under which conditions does the sample PCA well reflect the population covariance structure? We answer this question in a relevant asymptotic context where d grows and n is fixed, under a generalized spiked covariance model. Specifically, we assume the largest population eigenvalues to be of the order d(alpha), where alpha 1. Earlier results show the conditions for consistency and strong inconsistency of eigenvectors of the sample covariance matrix. In the boundary case, alpha = 1, where the sample PC directions are neither consistent nor strongly inconsistent, we show that eigenvalues and eigenvectors do not degenerate but have limiting distributions. The result smoothly bridges the phase transition represented by the other two cases, and thus gives a spectrum of limits for the sample PCA in the HDLSS asymptotics. While the results hold under a general situation, the limiting distributions under Gaussian assumption are illustrated in greater detail. In addition, the geometric representation of HDLSS data is extended to give three different representations, that depend on the magnitude of variances in the first few principal components. (C) 2012 Elsevier Inc. All rights reserved.