Comparison with Knockoff: Start from \(\Sigma_{\hat\beta}\)

Introduction

Similar to previous simulations, only that the design matrix \(X\) is simulated such that \(\Sigma_{\hat\beta} = \sigma_e^2(X^TX)^{-1}\) has non negligible off-diagnoal correlations.

\(d\) and \(\Sigma_{\hat\beta}\)

Let \(\Sigma_{\hat\beta} / \sigma_e^2 = B_{p \times d} \cdot B_{p \times d}^T + I\), where \(B_{i, j} \stackrel{\text{iid}}{\sim} N(0, 1)\). Then rescale the matrix such that the mean of its diagnal \(= 1\). Generate \(X_{n \times p}\) such that \((X^TX)^{-1} = \Sigma_{\hat\beta} / \sigma_e^2\).

Recall that the random coefficient of the second order of Gaussian deviative with the empirical distribution of correlated null \(W_2\) has the property \[ Var(W_2) = \bar{\rho_{ij}^2} \] We take a look at how \(d\) is related to this quantity.

\(\eta \in \{0.5, 0.6, 0.7, 0.8, 0.9\}\), \(\sigma_\beta / \sigma_e = 3\), \(d \sim Unif\{1, 2, \cdots, 50\}\)

\(\eta \in \{0.75, 0.80, 0.85, 0.90, 0.95\}\), \(\sigma_\beta / \sigma_e = 3\), \(d \sim Unif\{1, 2, \cdots, 20\}\)

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 10.13.2

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     

other attached packages:
 [1] ggplot2_2.2.1     reshape2_1.4.3    knockoff_0.3.0   
 [4] qvalue_2.10.0     ashr_2.2-2        Rmosek_8.0.69    
 [7] PolynomF_1.0-1    CVXR_0.94-4       REBayes_1.2      
[10] Matrix_1.2-12     SQUAREM_2017.10-1 EQL_1.0-0        
[13] ttutils_1.0-1    

loaded via a namespace (and not attached):
 [1] gmp_0.5-13.1      Rcpp_0.12.14      pillar_1.0.1     
 [4] plyr_1.8.4        compiler_3.4.3    git2r_0.21.0     
 [7] R.methodsS3_1.7.1 R.utils_2.6.0     iterators_1.0.9  
[10] tools_3.4.3       digest_0.6.14     bit_1.1-12       
[13] tibble_1.4.1      gtable_0.2.0      evaluate_0.10.1  
[16] lattice_0.20-35   rlang_0.1.6       foreach_1.4.4    
[19] yaml_2.1.16       parallel_3.4.3    Rmpfr_0.6-1      
[22] ECOSolveR_0.3-2   stringr_1.2.0     knitr_1.18       
[25] rprojroot_1.3-2   bit64_0.9-7       grid_3.4.3       
[28] R6_2.2.2          rmarkdown_1.8     magrittr_1.5     
[31] splines_3.4.3     scales_0.5.0      MASS_7.3-47      
[34] backports_1.1.2   codetools_0.2-15  htmltools_0.3.6  
[37] scs_1.1-1         colorspace_1.3-2  labeling_0.3     
[40] stringi_1.1.6     lazyeval_0.2.1    munsell_0.4.3    
[43] doParallel_1.0.11 pscl_1.5.2        truncnorm_1.0-7  
[46] R.oo_1.21.0