Last updated: 2019-04-18

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Rmd 7010db9 Matthew Stephens 2019-04-18 workflowr::wflow_publish(“analysis/ridge_mle.Rmd”) 
library(mnormt) #for multivariate normal density
library(glmnet)
Loading required package: Matrix
Loading required package: foreach
Loaded glmnet 2.0-16

Introduction

The idea here was to compare estimation of penalty (\(\lambda\)) in ridge regression by two methods: Empirical Bayes and CV (in glmnet)

Model and log-likelihood

We assume linear regression with residual variance 1 (for simplicity): \[Y|b \sim N(Xb, I)\]

Ridge regression assumes a normal prior fo \(b\): \[b \sim N(0, (1/\lambda) I)\] where \(\lambda\) is the prior precision of each \(b_j\).

Note that integrating out \(b\) we get: \[Y | \lambda \sim N(0, (1/\lambda) XX' + I).\]

The following function computes the log-likelihood for log-\(\lambda\) under this model:

loglik_rr = function(log_lambda,Y,X){return(mnormt::dmnorm(t(Y),rep(0,length(Y)),varcov = exp(-log_lambda)*(X %*% t(X)) + diag(rep(1,length(Y))),log=TRUE))}

Set up simulations

Here we simulate \(Y=Xb+e\) where \(b \sim N(0,\sigma=sb)\) (so true precision is \(\lambda=1/sb^2\)). Note that we standardize the columns of \(X\) to have norm 1 (colSums(X^2)=1) because I believe glmnet does this internally and so I think we need this if we want their lambda value to be comparable with the true precision.

simdata = function(n,p,sb){
  X = matrix(rnorm(n*p),ncol=p)
  X = scale(X,center=TRUE,scale=TRUE)
  X = X/sqrt(n-1) # makes colSums = 1
  b = rnorm(p,sd=sb) 
  e = rnorm(n,0,sd=1)
  Y = X %*% b + e
  return(list(Y=Y,X=X,b=b))
}

sb=1 (moderate effect)

set.seed(1)
sb=1
data = simdata(500,100,sb)

Plot log-likelihood for log precision, and true value as vertical line.

l = seq(-5,5,length=20)
ll = rep(0,20)
for(i in 1:length(ll)){ll[i] = loglik_rr(l[i],data$Y,data$X)}
plot(l,ll,type="l")
abline(v=log(1/sb^2))

Now fit ridge regression.

Y.ridge = glmnet(data$X,data$Y,alpha=0)
cv.ridge = cv.glmnet(data$X,data$Y,alpha=0)
plot(cv.ridge)

sb=0.1 (small effect)

Repeat for sb=0.1

set.seed(1)
sb=0.1
data = simdata(500,100,sb)

Plot log-likelihood for log precision, and true value as vertical line.

l = seq(-5,5,length=20)
ll = rep(0,20)
for(i in 1:length(ll)){ll[i] = loglik_rr(l[i],data$Y,data$X)}
plot(l,ll,type="l")
abline(v=log(1/sb^2))

Now fit ridge regression.

Y.ridge = glmnet(data$X,data$Y,alpha=0)
cv.ridge = cv.glmnet(data$X,data$Y,alpha=0)
plot(cv.ridge)

sb=10 (big effect)

set.seed(1)
sb=10
data = simdata(500,100,sb)

Plot log-likelihood for log precision, and true value as vertical line.

l = seq(-5,5,length=20)
ll = rep(0,20)
for(i in 1:length(ll)){ll[i] = loglik_rr(l[i],data$Y,data$X)}
plot(l,ll,type="l")
abline(v=log(1/sb^2))

Now fit ridge regression.

Y.ridge = glmnet(data$X,data$Y,alpha=0)
cv.ridge = cv.glmnet(data$X,data$Y,alpha=0)
plot(cv.ridge)

sb=2 (intermediate-large effect)

set.seed(1)
sb=2
data = simdata(500,100,sb)

Plot log-likelihood for log precision, and true value as vertical line.

l = seq(-5,5,length=20)
ll = rep(0,20)
for(i in 1:length(ll)){ll[i] = loglik_rr(l[i],data$Y,data$X)}
plot(l,ll,type="l")
abline(v=log(1/sb^2))

Now fit ridge regression.

Y.ridge = glmnet(data$X,data$Y,alpha=0)
cv.ridge = cv.glmnet(data$X,data$Y,alpha=0)
plot(cv.ridge)


sessionInfo()
R version 3.5.2 (2018-12-20)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.1

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/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_2.0-16 foreach_1.4.4 Matrix_1.2-15 mnormt_1.5-5 

loaded via a namespace (and not attached):
 [1] workflowr_1.2.0  Rcpp_1.0.0       codetools_0.2-15 lattice_0.20-38 
 [5] digest_0.6.18    rprojroot_1.3-2  grid_3.5.2       backports_1.1.3 
 [9] git2r_0.24.0     magrittr_1.5     evaluate_0.12    stringi_1.2.4   
[13] fs_1.2.6         whisker_0.3-2    rmarkdown_1.11   iterators_1.0.10
[17] tools_3.5.2      stringr_1.3.1    glue_1.3.0       xfun_0.4        
[21] yaml_2.2.0       compiler_3.5.2   htmltools_0.3.6  knitr_1.21