Power analysis of knockoff filters for correlated designs
Power analysis of knockoff filters for correlated designs
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发表时间：
2019-10
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通讯作者：
Jingbo Liu;P. Rigollet
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作者：
Jingbo Liu;P. Rigollet
Barber和Candes 2016年引入的仿制品过滤器是一个优雅的框架，用于控制变量选择中的错误发现率。虽然实证结果表明，这种方法并不太保守，但没有结论性的理论结果，其权力。当预测因子是独立同分布时。高斯，已知随着信噪比趋于无穷大，敲除滤波器在可以使FDR同时变为0并且功率变为1的意义上是一致的。在这项工作中，我们研究的情况下，预测有一个一般的协方差矩阵$\Sigma$。我们引入一个简单的功能称为有效信号不足（ESD）的协方差矩阵$\Sigma$，预测各种变量选择方法的一致性。特别是，ESD揭示了精度矩阵$\Sigma^{-1}$的结构在一致性中起着核心作用，因此，预测器的条件独立结构也是如此。为了利用这种连接，我们引入了条件独立敲除，这是一个简单的过程，能够与更复杂的敲除过滤器竞争，并且当预测器服从高斯树图形模型时（或者当图形足够稀疏时）定义。我们的理论结果得到了合成数据的数值证据的支持。
The knockoff filter introduced by Barber and Candes 2016 is an elegant framework for controlling the false discovery rate in variable selection. While empirical results indicate that this methodology is not too conservative, there is no conclusive theoretical result on its power. When the predictors are i.i.d. Gaussian, it is known that as the signal to noise ratio tend to infinity, the knockoff filter is consistent in the sense that one can make FDR go to 0 and power go to 1 simultaneously. In this work we study the case where the predictors have a general covariance matrix $\Sigma$. We introduce a simple functional called effective signal deficiency (ESD) of the covariance matrix $\Sigma$ that predicts consistency of various variable selection methods. In particular, ESD reveals that the structure of the precision matrix $\Sigma^{-1}$ plays a central role in consistency and therefore, so does the conditional independence structure of the predictors. To leverage this connection, we introduce Conditional Independence knockoff, a simple procedure that is able to compete with the more sophisticated knockoff filters and that is defined when the predictors obey a Gaussian tree graphical models (or when the graph is sufficiently sparse). Our theoretical results are supported by numerical evidence on synthetic data.