ECA: High-Dimensional Elliptical Component Analysis in Non-Gaussian Distributions
ECA: High-Dimensional Elliptical Component Analysis in Non-Gaussian Distributions
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DOI：
10.1080/01621459.2016.1246366
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发表时间：
2013-10
影响因子：
3.7
通讯作者：
Fang Han;Han Liu
中科院分区：
文献类型：
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作者：
Fang Han;Han Liu
摘要我们提出了一个强大的替代主成分分析（PCA）-椭圆成分分析（ECA）-用于分析高维，椭圆分布的数据。ECA估计椭圆数据的协方差矩阵的特征空间。为了科普重尾椭圆分布，利用多元秩统计量。在模型级，我们考虑两种设置：要么协方差矩阵的前导特征向量是非稀疏的，要么它们是稀疏的。方法上，我们提出了非稀疏和稀疏设置的ECA程序。从理论上讲，我们提供了非渐近和渐近分析量化的ECA的理论性能。在非稀疏设置，我们表明，ECA的性能是高度相关的协方差矩阵的有效秩。在稀疏设置，结果是双重的：（i）我们表明，稀疏ECA估计的组合程序的基础上达到最佳的收敛速度;（ii）根据最近的一些发展，估计稀疏领先的特征向量，我们表明，计算效率稀疏ECA估计达到最佳的收敛速度下的次优缩放。本文的补充材料可在网上查阅。
ABSTRACT We present a robust alternative to principal component analysis (PCA)—called elliptical component analysis (ECA)—for analyzing high-dimensional, elliptically distributed data. ECA estimates the eigenspace of the covariance matrix of the elliptical data. To cope with heavy-tailed elliptical distributions, a multivariate rank statistic is exploited. At the model-level, we consider two settings: either that the leading eigenvectors of the covariance matrix are nonsparse or that they are sparse. Methodologically, we propose ECA procedures for both nonsparse and sparse settings. Theoretically, we provide both nonasymptotic and asymptotic analyses quantifying the theoretical performances of ECA. In the nonsparse setting, we show that ECA’s performance is highly related to the effective rank of the covariance matrix. In the sparse setting, the results are twofold: (i) we show that the sparse ECA estimator based on a combinatoric program attains the optimal rate of convergence; (ii) based on some recent developments in estimating sparse leading eigenvectors, we show that a computationally efficient sparse ECA estimator attains the optimal rate of convergence under a suboptimal scaling. Supplementary materials for this article are available online.