Distance-based and RKHS-based dependence metrics in high dimension
Distance-based and RKHS-based dependence metrics in high dimension
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DOI：
10.1214/19-aos1934
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发表时间：
2019-02
期刊：
影响因子：
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通讯作者：
Changbo Zhu;Shu Yao;Xianyang Zhang;Xiaofeng Shao
中科院分区：
文献类型：
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作者：
Changbo Zhu;Shu Yao;Xianyang Zhang;Xiaofeng Shao
本文研究了高维情形下的距离协方差、Hilbert-Schmidt协方差（又称Hilbert-Schmidt独立性准则[格雷顿et al.（2008）]）以及相关的独立性检验。我们证明了两个随机向量之间的样本距离/Hilbert-Schmidt协方差可以近似为分量样本互协方差的平方和，直到一个渐近常数因子，这表明基于距离/Hilbert-Schmidt协方差的检验只能捕获高维线性相关。因此，Szekely和Rizzo（2013）开发的基于距离相关性的独立性t检验显示，当两个随机向量非线性相关但分量不相关时，具有微不足道的限制能力。这种新的和令人惊讶的现象，这似乎是第一次发现，在我们的模拟研究中得到进一步证实。作为一种补救措施，我们提出了测试的基础上聚集的边际样本距离/希尔伯特-施密特协方差，并显示其上级权力行为对他们的联合同行在模拟。我们进一步将基于距离相关的t检验扩展到基于希尔伯特-施密特协方差和边缘距离/希尔伯特-施密特协方差的t检验。提出了一种新的统一的方法来分析学生化样本距离/Hilbert-Schmidt协方差以及在零假设和备择假设下的学生化样本边缘距离协方差。我们的理论和模拟结果揭示了距离/希尔伯特-施密特协方差的限制时，联合使用的高维设置，并建议聚合的边缘距离/希尔伯特-施密特协方差作为一个有用的替代。
In this paper, we study distance covariance, Hilbert-Schmidt covariance (aka Hilbert-Schmidt independence criterion [Gretton et al. (2008)]) and related independence tests under the high dimensional scenario. We show that the sample distance/Hilbert-Schmidt covariance between two random vectors can be approximated by the sum of squared componentwise sample cross-covariances up to an asymptotically constant factor, which indicates that the distance/Hilbert-Schmidt covariance based test can only capture linear dependence in high dimension. As a consequence, the distance correlation based t-test developed by Szekely and Rizzo (2013) for independence is shown to have trivial limiting power when the two random vectors are nonlinearly dependent but component-wisely uncorrelated. This new and surprising phenomenon, which seems to be discovered for the first time, is further confirmed in our simulation study. As a remedy, we propose tests based on an aggregation of marginal sample distance/Hilbert-Schmidt covariances and show their superior power behavior against their joint counterparts in simulations. We further extend the distance correlation based t-test to those based on Hilbert-Schmidt covariance and marginal distance/Hilbert-Schmidt covariance. A novel unified approach is developed to analyze the studentized sample distance/Hilbert-Schmidt covariance as well as the studentized sample marginal distance covariance under both null and alternative hypothesis. Our theoretical and simulation results shed light on the limitation of distance/Hilbert-Schmidt covariance when used jointly in the high dimensional setting and suggest the aggregation of marginal distance/Hilbert-Schmidt covariance as a useful alternative.