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library(glmnet)
Warning: package 'glmnet' was built under R version 3.6.2
Loading required package: Matrix
Loaded glmnet 4.1

Introduction

This is to compare ridge and lasso in “half-dense” simulations with \(n=500, s=p/2\). I compare the cases \(p=1000\) vs \(p=2000\) to see if they behave differently, and confirm that indeed lasso consistently outperforms ridge for \(p=1000\) but ridge is better for \(p=2000\).

p=1000

  set.seed(123)
  n <- 500
  p <- 1000
  p_causal <- p/2 # number of causal variables (simulated effects N(0,1))
  pve <- 0.95
  nrep = 10
  rmse_lasso = rep(0,nrep)
  rmse_ridge = rep(0,nrep)
  
  for(i in 1:nrep){
    sim=list()
    sim$X =  matrix(rnorm(n*p,sd=1),nrow=n)
    B <- rep(0,p)
    causal_variables <- sample(x=(1:p), size=p_causal)
    B[causal_variables] <- rnorm(n=p_causal, mean=0, sd=1)

    sim$B = B
    sim$Y = sim$X %*% sim$B
    sigma2 = ((1-pve)/(pve))*sd(sim$Y)^2
    E = rnorm(n,sd = sqrt(sigma2))
    sim$Y = sim$Y + E
    
    fit_lasso <- cv.glmnet(x=sim$X, y=sim$Y, family="gaussian", alpha=1, standardize=FALSE)  
    
    fit_ridge <- cv.glmnet(x=sim$X, y=sim$Y, family="gaussian", alpha=0, standardize=FALSE)
   
    rmse_lasso[i] = sqrt(mean((sim$B-coef(fit_lasso)[-1])^2))
    rmse_ridge[i] = sqrt(mean((sim$B-coef(fit_ridge)[-1])^2))
  }
  
  plot(rmse_lasso,rmse_ridge, xlim=c(0.5,0.75), ylim=c(0.5,0.75), main="p=1000")
  abline(a=0,b=1)

Version Author Date
83bde66 Matthew Stephens 2021-03-29

p=2000

  set.seed(123)
  n <- 500
  p <- 2000
  p_causal <- p/2 # number of causal variables (simulated effects N(0,1))
  pve <- 0.95
  nrep = 10
  rmse_lasso = rep(0,nrep)
  rmse_ridge = rep(0,nrep)
  
  for(i in 1:nrep){
    sim=list()
    sim$X =  matrix(rnorm(n*p,sd=1),nrow=n)
    B <- rep(0,p)
    causal_variables <- sample(x=(1:p), size=p_causal)
    B[causal_variables] <- rnorm(n=p_causal, mean=0, sd=1)

    sim$B = B
    sim$Y = sim$X %*% sim$B
    sigma2 = ((1-pve)/(pve))*sd(sim$Y)^2
    E = rnorm(n,sd = sqrt(sigma2))
    sim$Y = sim$Y + E
    
    fit_lasso <- cv.glmnet(x=sim$X, y=sim$Y, family="gaussian", alpha=1, standardize=FALSE)  
    
    fit_ridge <- cv.glmnet(x=sim$X, y=sim$Y, family="gaussian", alpha=0, standardize=FALSE)
   
    rmse_lasso[i] = sqrt(mean((sim$B-coef(fit_lasso)[-1])^2))
    rmse_ridge[i] = sqrt(mean((sim$B-coef(fit_ridge)[-1])^2))
  }
  
  plot(rmse_lasso,rmse_ridge, xlim=c(0.5,0.75), ylim=c(0.5,0.75), main="p=2000")
  abline(a=0,b=1)

Version Author Date
83bde66 Matthew Stephens 2021-03-29 

sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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] glmnet_4.1    Matrix_1.2-18

loaded via a namespace (and not attached):
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