The Quality of Knockoffs

Introduction

The idea in Knockoff is to generate knockoff variables \(\tilde X\) which 1) keep the relationship of original variables \(X\) 2) are unlike the original variables 3) are null.

• For Fixed Knockoff \[ \begin{array}{c} \tilde X_i^T\tilde X_i = X_i^TX_i \\ \tilde X_i^T\tilde X_j = X_i^T \tilde X_j = \tilde X_i^T X_j = X_i^TX_j \\ \tilde X_i^T X_i: \text{as small as possible} \end{array} \]

• For Model-X Knockoff \[ \begin{array}{c} Var(X_i) = Var(\tilde X_i) \\ cor(\tilde X_i, \tilde X_j) = cor(X_i, \tilde X_j) = cor(\tilde X_i, X_j) = cor(X_i, X_j) \\ cor(\tilde X_i, X_i): \text{as small as possible} \end{array} \]

Session information

sessionInfo()

R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.3

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] compiler_3.4.3  backports_1.1.2 magrittr_1.5    rprojroot_1.3-2
 [5] tools_3.4.3     htmltools_0.3.6 yaml_2.1.16     Rcpp_0.12.15   
 [9] stringi_1.1.6   rmarkdown_1.8   knitr_1.19      git2r_0.21.0   
[13] stringr_1.2.0   digest_0.6.15   evaluate_0.10.1