Last updated: 2018-05-12

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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 \(n = 3000\), \(p = 1000\), 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 \(3.5\).
Loading required package: foreach
Loading required package: iterators
Loading required package: parallel
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 <- svd(X)$u
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk

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

X <- matrix(rnorm(n * p), n , p)
X <- svd(X)$u
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk

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

X <- matrix(rnorm(n * p), n , p)
X <- svd(X)$u
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk

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Fix-\(X\) Knockoffs

Orthogonal design

\(X\) has random orthonormal columns

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
0.0979 0.1082 0.0863 0.7339 0.7329 0.6939 0.7339 0.7329 0.6939 NA NA NA
0.0847 0.0921 0.0795 0.3836 0.3867 0.3691 0.7791 0.7799 0.7555 0.1858 0.1901 0.176
0.0756 0.0817 0.0727 0.3294 0.3340 0.3195 0.8116 0.8122 0.7932 0.2088 0.2145 0.201

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Independent design

\(X_{n \times p}\) has independent columns simulated from \(N(0, 1)\) and then normalized to have \(\|X_j\|_2^2 \equiv 1\).

X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk
X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk
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
0.0950 0.0713 0.0483 0.4344 0.5399 0.4135 0.4344 0.5399 0.4135 NA NA NA
0.0828 0.0587 0.0397 0.2180 0.2106 0.1584 0.4789 0.4820 0.3699 0.0876 0.0749 0.0527
0.0747 0.0456 0.0325 0.1838 0.1613 0.1243 0.5135 0.4372 0.3432 0.1013 0.0923 0.0696

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Local correlation design

\(X_{n \times p}\) has correlation \(\Sigma_{ij} = \rho^{|i - j|}\). Each row is independently \(N(0, \Sigma)\) and then normalized to have \(\|X_j\|_2^2 \equiv 1\).

rho <- 0.25
Sigma <- toeplitz(rho^(0 : (p - 1)))
X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
X <- X %*% chol(Sigma)
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk
X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
X <- X %*% chol(Sigma)
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk
X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
X <- X %*% chol(Sigma)
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
0.0957 0.0563 0.0344 0.3367 0.4985 0.3484 0.3367 0.4985 0.3484 NA NA NA
0.0824 0.0407 0.0249 0.1710 0.1656 0.1070 0.3820 0.3620 0.2348 0.0655 0.0673 0.0431
0.0738 0.0258 0.0170 0.1441 0.1393 0.0988 0.4137 0.4040 0.2893 0.0767 0.0731 0.0511

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Model-\(X\) Knockoffs

Independent design

\(X_{n \times p}\) has independent columns simulated from \(N(0, (1/\sqrt n)^2)\) so they are roughly normalized.

Loaded glmnet 2.0-13
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
0.0903 0.1164 0.0935 0.4312 0.6632 0.6124 0.4312 0.6632 0.6124 NA NA NA
0.0782 0.1150 0.0929 0.2213 0.3417 0.3161 0.4808 0.6924 0.6526 0.0916 0.1664 0.1478
0.0693 0.0923 0.0806 0.1803 0.2731 0.2573 0.5156 0.6962 0.6718 0.0965 0.1674 0.1536

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Local correlation design

\(X_{n \times p}\) has correlation \(\Sigma_{ij} = \rho^{|i - j|}\). Each row is independently \(N(0, \frac1n\Sigma)\).

rho <- 0.5
Sigma <- toeplitz(rho^(0 : (p - 1)))
Sigma.5 <- chol(Sigma)
Cov.X <- Sigma / n
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
0.1199 0.1230 0.0946 0.1268 0.6036 0.5196 0.1268 0.6036 0.5196 NA NA NA
0.0840 0.0971 0.0800 0.0635 0.2759 0.2469 0.1504 0.5784 0.5246 0.0200 0.1246 0.1081
0.1562 0.0671 0.0467 0.0944 0.1253 0.0953 0.2850 0.3564 0.2782 0.0467 0.0676 0.0496

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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 10.13.4

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] parallel  stats     graphics  grDevices utils     datasets  methods  
[8] base     

other attached packages:
[1] knitr_1.20      lattice_0.20-35 doMC_1.3.5      iterators_1.0.9
[5] foreach_1.4.4   ggplot2_2.2.1   reshape2_1.4.3  Matrix_1.2-12  
[9] knockoff_0.3.0 

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.16      compiler_3.4.3    pillar_1.0.1     
 [4] git2r_0.21.0      plyr_1.8.4        highr_0.6        
 [7] workflowr_1.0.1   R.methodsS3_1.7.1 R.utils_2.6.0    
[10] tools_3.4.3       digest_0.6.15     evaluate_0.10.1  
[13] tibble_1.4.1      gtable_0.2.0      rlang_0.1.6      
[16] yaml_2.1.18       stringr_1.3.0     rprojroot_1.3-2  
[19] grid_3.4.3        rmarkdown_1.9     magrittr_1.5     
[22] whisker_0.3-2     backports_1.1.2   scales_0.5.0     
[25] codetools_0.2-15  htmltools_0.3.6   colorspace_1.3-2 
[28] labeling_0.3      stringi_1.1.6     lazyeval_0.2.1   
[31] munsell_0.4.3     R.oo_1.21.0

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