ridge_em
Matthew Stephens
2020-05-29
Last updated: 2020-06-25
Checks: 7 0
Knit directory: misc/analysis/
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Unstaged changes:
Modified: analysis/ash_delta_operator.Rmd
Modified: analysis/ash_pois_bcell.Rmd
Modified: analysis/lasso_em.Rmd
Modified: analysis/minque.Rmd
Modified: analysis/mr_missing_data.Rmd
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| Rmd | a820c56 | Matthew Stephens | 2020-06-11 | workflowr::wflow_publish(“ridge_em.Rmd”) |
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| Rmd | 4d19a87 | Matthew Stephens | 2020-05-29 | workflowr::wflow_publish(“ridge_em.Rmd”) |
Introduction
Here I am going to experiment with EM algorithm for estimating parameters of ridge regression in different parameterizations.
Initial derivations of EM updates are here. I initially implemented 1,2, and 5 in that document.
A futher derivation for another parameterization is here.
Simple parameterization
\[y \sim N(Xb,s^2)\] \[b \sim N(0,s_b^2I)\]
ridge_em1 = function(y,X, s2,sb2, niter=10){
XtX = t(X) %*% X
Xty = t(X) %*% y
yty = t(y) %*% y
n = length(y)
p = ncol(X)
loglik = rep(0,niter)
for(i in 1:niter){
V = chol2inv(chol(XtX+ diag(s2/sb2,p)))
SigmaY = sb2 *(X %*% t(X)) + diag(s2,n)
loglik[i] = mvtnorm::dmvnorm(as.vector(y),sigma = SigmaY,log=TRUE)
Sigma1 = s2*V # posterior variance of b
mu1 = as.vector(V %*% Xty) # posterior mean of b
s2 = as.vector((yty + sum(diag(XtX %*% (mu1 %*% t(mu1) + Sigma1)))- 2*sum(Xty*mu1))/n)
sb2 = mean(mu1^2+diag(Sigma1))
}
return(list(s2=s2,sb2=sb2,loglik=loglik,postmean=mu1))
}Scaled parameterization
In this parameterization I take the \(s_b\) out of the prior and put it \[y \sim N(s_b Xb,s^2)\] \[b \sim N(0,I)\].
ridge_em2 = function(y,X, s2,sb2, niter=10){
XtX = t(X) %*% X
Xty = t(X) %*% y
yty = t(y) %*% y
n = length(y)
p = ncol(X)
loglik = rep(0,niter)
for(i in 1:niter){
V = chol2inv(chol(XtX+ diag(s2/sb2,p)))
SigmaY = sb2 *(X %*% t(X)) + diag(s2,n)
loglik[i] = mvtnorm::dmvnorm(as.vector(y),sigma = SigmaY,log=TRUE)
Sigma1 = (s2/sb2)*V # posterior variance of b
mu1 = (sqrt(sb2)/s2)*as.vector(Sigma1 %*% Xty) # posterior mean of b
sb2 = (sum(mu1*Xty)/sum(diag(XtX %*% (mu1 %*% t(mu1) + Sigma1))))^2
s2 = as.vector((yty + sb2*sum(diag(XtX %*% (mu1 %*% t(mu1) + Sigma1)))- 2*sqrt(sb2)*sum(Xty*mu1))/n)
}
return(list(s2=s2,sb2=sb2,loglik=loglik,postmean=mu1*sqrt(sb2)))
}A hybrid/redundant parameterization
Motivated by initial observations that 1 and 2 can converge well in different settings I implemented a hybrid of the two:
\[y \sim N(s_b Xb,s^2)\] \[b \sim N(0,\lambda^2).\] Note that there is a redundancy/non-identifiability here as the likelihood depends only on \(s_b^2 \lambda^2\). The hope is to get the best of both worlds…
ridge_em3 = function(y,X, s2, sb2, l2, niter=10){
XtX = t(X) %*% X
Xty = t(X) %*% y
yty = t(y) %*% y
n = length(y)
p = ncol(X)
loglik = rep(0,niter)
for(i in 1:niter){
V = chol2inv(chol(XtX+ diag(s2/(sb2*l2),p)))
SigmaY = l2*sb2 *(X %*% t(X)) + diag(s2,n)
loglik[i] = mvtnorm::dmvnorm(as.vector(y),sigma = SigmaY,log=TRUE)
Sigma1 = (s2/sb2)*V # posterior variance of b
mu1 = (1/sqrt(sb2))*as.vector(V %*% Xty) # posterior mean of b
sb2 = (sum(mu1*Xty)/sum(diag(XtX %*% (mu1 %*% t(mu1) + Sigma1))))^2
s2 = as.vector((yty + sb2*sum(diag(XtX %*% (mu1 %*% t(mu1) + Sigma1)))- 2*sqrt(sb2)*sum(Xty*mu1))/n)
l2 = mean(mu1^2+diag(Sigma1))
}
return(list(s2=s2,sb2=sb2,l2=l2,loglik=loglik,postmean=mu1*sqrt(sb2)))
}Avoiding large 2nd moment computation
The previous parameterizations require the full second moment of \(b\), which is a \(p\) times \(p\) matrix. This can be expensive to compute if \(p\) is big. The following parameterization avoids this.
\[y \sim N(sXb, s^2 I)\]
\[b \sim N(0,s_b^2I)\]
(Note that for simplicity I still do compute the \(p \times p\) matrix, as for now it is the easiest way to implement the ridge regression).
dot = function(x,y){sum(x*y)}
ridge_em4 = function(y, X, s2, sb2, niter=10){
XtX = t(X) %*% X
Xty = t(X) %*% y
yty = t(y) %*% y
n = length(y)
p = ncol(X)
loglik = rep(0,niter)
for(i in 1:niter){
SigmaY = s2*sb2 *(X %*% t(X)) + diag(s2,n)
loglik[i] = mvtnorm::dmvnorm(as.vector(y),sigma = SigmaY,log=TRUE)
Sigma1 = chol2inv(chol(XtX + diag(1/sb2,p))) # posterior variance of b
mu1 = (1/sqrt(s2))*as.vector(Sigma1 %*% Xty) # posterior mean of b
sb2 = mean(mu1^2+diag(Sigma1))
yhat = X %*% mu1
s2 = drop((0.5/n)* (sqrt(dot(y,yhat)^2 + 4*n*yty) - dot(y,yhat)))^2
}
return(list(s2=s2,sb2=sb2,loglik=loglik,postmean=mu1*sqrt(s2)))
}Simple Simulations
This is a simple simulation with independent design matrix.
High signal:
This simulation has high signal:
set.seed(100)
sd = 1
n = 100
p = n
X = matrix(rnorm(n*p),ncol=n)
btrue = rnorm(n)
y = X %*% btrue + sd*rnorm(n)
plot(X %*% btrue, y)
y.em1 = ridge_em1(y,X,1,1,100)
y.em2 = ridge_em2(y,X,1,1,100)
y.em3 = ridge_em3(y,X,1,1,1,100)
y.em4 = ridge_em4(y,X,1,1,100)
plot_loglik = function(res){
maxloglik = max(res[[1]]$loglik)
minloglik = min(res[[1]]$loglik)
maxlen =length(res[[1]]$loglik)
for(i in 2:length(res)){
maxloglik = max(c(maxloglik,res[[i]]$loglik))
minloglik = min(c(minloglik,res[[i]]$loglik))
maxlen= max(maxlen, length(res[[i]]$loglik))
}
plot(res[[1]]$loglik,type="n",ylim=c(minloglik,maxloglik),xlim=c(0,maxlen),ylab="log-likelihood",
xlab="iteration")
for(i in 1:length(res)){
lines(res[[i]]$loglik,col=i,lwd=2)
}
}
res = list(y.em1,y.em2,y.em3,y.em4)
plot_loglik(res)
No signal
This simulation has no signal (b=0):
btrue = rep(0,n)
y = X %*% btrue + sd*rnorm(n)
y.em1 = ridge_em1(y,X,1,1,100)
y.em2 = ridge_em2(y,X,1,1,100)
y.em3 = ridge_em3(y,X,1,1,1,100)
y.em4 = ridge_em4(y,X,1,1,100)
plot_loglik(list(y.em1,y.em2,y.em3,y.em4))
Trendfiltering Simulations
This is more challenging example (in that the design matrix is correlated)
High Signal
set.seed(100)
sd = 1
n = 100
p = n
X = matrix(0,nrow=n,ncol=n)
for(i in 1:n){
X[i:n,i] = 1:(n-i+1)
}
btrue = rep(0,n)
btrue[40] = 8
btrue[41] = -8
y = X %*% btrue + sd*rnorm(n)
plot(y)
lines(X %*% btrue)
y.em1 = ridge_em1(y,X,1,1,100)
lines(X %*% y.em1$postmean,col=1,lwd=2)
y.em2 = ridge_em2(y,X,1,1,100)
lines(X %*% y.em2$postmean,col=2,lwd=2)
y.em3 = ridge_em3(y,X,1,1,1,100)
lines(X %*% y.em3$postmean,col=3,lwd=2)
y.em4 = ridge_em4(y,X,1,1,100)
lines(X %*% y.em4$postmean,col=4,lwd=2)
Look at the likelihoods:
plot_loglik(list(y.em1,y.em2,y.em3,y.em4))
Run the second one longer and check it:
y.em2 = ridge_em2(y,X,1,1,1000)
plot_loglik(list(y.em1,y.em2,y.em3,y.em4))
y.em1$sb2[1] 0.02203472y.em2$sb2[1] 0.02305466y.em3$sb2*y.em3$l2[1] 0.02189412y.em4$sb2 * y.em4$s2[1] 0.02435217y.em1$s2[1] 1.612878y.em2$s2[1] 1.606894y.em3$s2[1] 1.613795y.em4$s2[1] 1.566927Different initializations
Try starting \(s\) in wrong place
y.em1 = ridge_em1(y,X,10,1,100)
y.em2 = ridge_em2(y,X,10,1,100)
y.em3 = ridge_em3(y,X,10,1,1,100)
y.em4 = ridge_em4(y,X,10,1,100)
plot_loglik(list(y.em1,y.em2,y.em3,y.em4))
Try starting \(s2\) in wrong place
y.em1 = ridge_em1(y,X,1,10,100)
y.em2 = ridge_em2(y,X,1,10,100)
y.em3 = ridge_em3(y,X,1,10,10,100)
y.em4 = ridge_em4(y,X,1,10,100)
plot_loglik(list(y.em1,y.em2,y.em3,y.em4))
| Version | Author | Date |
|---|---|---|
| 547645e | Matthew Stephens | 2020-05-29 |
Try starting both in wrong place. Interestingly in this example em4 seems to converge to a local optimum?
y.em1 = ridge_em1(y,X,.1,10,100)
y.em2 = ridge_em2(y,X,.1,10,100)
y.em3 = ridge_em3(y,X,.1,10,10,100)
y.em4 = ridge_em4(y,X,.1,10,100)
plot_loglik(list(y.em1,y.em2,y.em3,y.em4))
y.em4$s2[1] 0.1075621y.em1$s2[1] 1.609084No signal case
Try no signal case – the convergence issues are reversed!
sd = 1
n = 100
p = n
X = matrix(0,nrow=n,ncol=n)
for(i in 1:n){
X[i:n,i] = 1:(n-i+1)
}
btrue = rep(0,n)
y = X %*% btrue + sd*rnorm(n)
plot(y)
lines(X %*% btrue)
y.em1 = ridge_em1(y,X,1,1,100)
lines(X %*% y.em1$postmean,col=1,lwd=2)
y.em2 = ridge_em2(y,X,1,1,100)
lines(X %*% y.em2$postmean,col=2,lwd=2)
y.em3 = ridge_em3(y,X,1,1,1,100)
lines(X %*% y.em3$postmean,col=3,lwd=2)
y.em4 = ridge_em4(y,X,1,1,100)
lines(X %*% y.em4$postmean,col=4,lwd=2)
The EM2 and EM3 converge faster here:
plot_loglik(list(y.em1,y.em2,y.em3,y.em4))
Try starting the expanded algorithm from very large lambda… it still seems to work.
y.em3b = ridge_em3(y,X,1,1,100,100)
plot_loglik(list(y.em1,y.em2,y.em3b,y.em4))
The next step seems to be to combine the expanded idea with algorithm em4….
sessionInfo()R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6
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
loaded via a namespace (and not attached):
[1] workflowr_1.6.1 Rcpp_1.0.4.6 mvtnorm_1.1-1 rprojroot_1.3-2
[5] digest_0.6.25 later_1.0.0 R6_2.4.1 backports_1.1.5
[9] git2r_0.26.1 magrittr_1.5 evaluate_0.14 stringi_1.4.6
[13] rlang_0.4.5 fs_1.3.2 promises_1.1.0 whisker_0.4
[17] rmarkdown_2.1 tools_3.6.0 stringr_1.4.0 glue_1.4.0
[21] httpuv_1.5.2 xfun_0.12 yaml_2.2.1 compiler_3.6.0
[25] htmltools_0.4.0 knitr_1.28