Last updated: 2018-02-08

Code version: 1a2b7bf

Introduction

In the Knockoff paper simulations, \(\beta\)’s are either \(0\) or \(A\). Here we are replicating the results, and investigating how well Knockoff deal with small signals.

In the following simulations, we always have \(n = 3000\), \(p = 1000\), \(X_{n \times p}\) has independent columns simulated from \(N(0, 1)\) and then normalized to have \(\|X_j\|_2^2 \equiv 1\). For a certain \(\beta\), \(Y_n \sim N(X_{n\times p}\beta_p, I_n)\). Out of \(p = 1000\) \(\beta_j\)’s, here are three scenarios.

  • Scenario 1: \(950\) are 0, and the rest \(50\) are \(A = 3.5\). (replicating a data point on Fig 3 of the Knockoff paper)
  • Scenario 2: \(850\) are 0, \(50\) are \(A = 3.5\) as large signals, and the rest \(100\) are small signals uniformly from \(0\) to \(3.5\).
  • Scenario 3: \(750\) are 0, \(50\) are \(A = 3.5\) as large signals, and the rest \(200\) are small signals uniformly from \(0\) to \(3.5\).
n <- 3000
p <- 1000
k <- 50
q <- 0.1
A <- 3.5

Scenario 1: 50 large signals, no small signals, 950 zeroes.

X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk

Scenario 2: 50 large signals, 100 small signals, 850 zeroes.

X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk

Scenario 3: 50 large signals, 200 small signals, 750 zeroes.

X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk

FDP.BH FDP.Knockoff FDP.Knockoff.Plus Power.BH Power.Knockoff Power.Knockoff.Plus Power.Large.BH Power.Large.Knockoff Power.Large.Knockoff.Plus Power.Small.BH Power.Small.Knockoff Power.Small.Knockoff.Plus
0.0936 0.0677 0.0431 0.4277 0.5400 0.4076 0.4277 0.5400 0.4076 NA NA NA
0.0824 0.0477 0.0390 0.3678 0.3523 0.3106 0.5760 0.5849 0.5222 0.1756 0.1538 0.1319
0.0589 0.0350 0.0292 0.3607 0.2416 0.2108 0.6638 0.5659 0.5068 0.2254 0.1309 0.1110

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 knitr_1.19   

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.14     magrittr_1.5     munsell_0.4.3    colorspace_1.3-2
 [5] rlang_0.1.6      stringr_1.2.0    highr_0.6        plyr_1.8.4      
 [9] tools_3.4.3      grid_3.4.3       gtable_0.2.0     git2r_0.21.0    
[13] htmltools_0.3.6  yaml_2.1.16      lazyeval_0.2.1   rprojroot_1.3-2 
[17] digest_0.6.14    tibble_1.4.1     evaluate_0.10.1  rmarkdown_1.8   
[21] labeling_0.3     stringi_1.1.6    compiler_3.4.3   pillar_1.0.1    
[25] scales_0.5.0     backports_1.1.2

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