ridge_em_svd
Matthew Stephens
2020-06-26
Last updated: 2020-06-26
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Knit directory: misc/analysis/
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| File | Version | Author | Date | Message |
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| Rmd | fab7fde | Matthew Stephens | 2020-06-26 | workflowr::wflow_publish(“ridge_em_svd.Rmd”) |
| html | ae73528 | Matthew Stephens | 2020-06-26 | Build site. |
| Rmd | ced143f | Matthew Stephens | 2020-06-26 | workflowr::wflow_publish(“ridge_em_svd.Rmd”) |
| html | 29360cf | Matthew Stephens | 2020-06-26 | Build site. |
| Rmd | 25cb420 | Matthew Stephens | 2020-06-26 | workflowr::wflow_publish(“ridge_em_svd.Rmd”) |
Introduction
Following up on these EM algorithms to fit Ridge by EB, I look at implementing these kinds of ideas when an SVD for \(X=UDV'\) is available (or, simply by doing SVD of \(X\) as a pre-computation step). Assume we are in the big \(p\) regime, so \(D\) is \(k\) \(k\) with \(k<p\), and \(V'V = I_k\).
The model is: \[Y \sim N(Xb, s^2I_n)\]
Premultiplying by \(U'\) gives: \[U'Y \sim N(DV'b, s^2 I_k)\] which we can write as \[\tilde{Y}_j \sim N(\theta_j, s^2)\] \[\theta_j \sim N(0, s_b^2 d_j^2)\].
And we can solve this by EM, just as before. Of course we can parameterize in various ways.
Here is the EM for the simple parameterization as above:
ridge_indep_em1 = function(y, d2, s2, sb2, niter=10){
k = length(y)
loglik = rep(0,2*niter)
for(i in 1:niter){
prior_var = sb2*d2
data_var = s2
loglik[2*i-1] = sum(dnorm(y,mean=0,sd = sqrt(sb2*d2 + s2),log=TRUE))
# update sb2
post_var = 1/((1/prior_var) + (1/data_var)) #posterior variance of theta
post_mean = post_var * (1/data_var) * y # posterior mean of theta
sb2 = mean((post_mean^2 + post_var)/d2)
loglik[2*i] = sum(dnorm(y,mean=0,sd = sqrt(sb2*d2 + s2),log=TRUE))
# update s2
r = y - post_mean # residuals
s2 = mean(r^2 + post_var)
}
return(list(s2=s2,sb2=sb2,loglik=loglik,postmean = post_mean))
}Scaled parameterization
Here we take the \(s_b\) out of the prior on \(\theta_j\): \[y_j \sim N(s_b \theta_j, s^2)\] \[\theta_j \sim N(0,d_j^2).\]
Note that we could also put the \(d_j\) into the mean of \(y_j\) and have \(\theta_j \sim N(0,1)\) but this ends up leading to exactly the same EM algorithm. (In earlier versions of this document I implemented this, but it turned out to indeed be identical, so I removed it.)
Note also that here I give the option to recompute quantities between updates of sb2 and s2. However, results later generally suggest this is not worth the additional expense.
ridge_indep_em2 = function(y, d2, s2, sb2, niter=10, recompute_between_updates = FALSE){
k = length(y)
loglik = rep(0,2*niter)
for(i in 1:niter){
loglik[2*i-1] = sum(dnorm(y,mean=0,sd = sqrt(sb2*d2 + s2),log=TRUE))
prior_var = d2 # prior variance for theta
data_var = s2/sb2 # variance of y/sb, which has mean theta
post_var = 1/((1/prior_var) + (1/data_var)) #posterior variance of theta
post_mean = post_var * (1/data_var) * (y/sqrt(sb2)) # posterior mean of theta
sb2 = (sum(y*post_mean)/sum(post_mean^2 + post_var))^2
loglik[2*i] = sum(dnorm(y,mean=0,sd = sqrt(sb2*d2 + s2),log=TRUE))
if(recompute_between_updates){
prior_var = d2 # prior variance for theta
data_var = s2/sb2 # variance of y/sb, which has mean theta
post_var = 1/((1/prior_var) + (1/data_var)) #posterior variance of theta
post_mean = post_var * (1/data_var) * (y/sqrt(sb2)) # posterior mean of theta
}
r = y - sqrt(sb2) * post_mean # residuals
s2 = mean(r^2 + sb2 * post_var)
}
return(list(s2=s2,sb2=sb2,loglik=loglik,postmean = post_mean))
}Simple simulation
Here we try a simple simulation to test:
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)
Here I define a function to plot the log-likelihoods:
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)
}
}Run all the methods: the simple parameterization is best here:
X.svd = svd(X)
ytilde = drop(t(X.svd$u) %*% y)
yt.em1 = ridge_indep_em1(ytilde,X.svd$d^2,1,1,100)
yt.em2 = ridge_indep_em2(ytilde,X.svd$d^2,1,1,100)
yt.em2b = ridge_indep_em2(ytilde,X.svd$d^2,1,1,100,recompute_between_updates = TRUE)
plot_loglik(list(yt.em1,yt.em2,yt.em2b))
Try different initializations. Here s2=.1 and sb2=10.
yt.em1 = ridge_indep_em1(ytilde,X.svd$d^2,.1,10,100)
yt.em2 = ridge_indep_em2(ytilde,X.svd$d^2,.1,10,100)
yt.em2b = ridge_indep_em2(ytilde,X.svd$d^2,.1,10,100,recompute_between_updates = TRUE)
plot_loglik(list(yt.em1,yt.em2,yt.em2b))
Here s2=10 and sb2=.1.
yt.em1 = ridge_indep_em1(ytilde,X.svd$d^2,10,.1,100)
yt.em2 = ridge_indep_em2(ytilde,X.svd$d^2,10,.1,100)
yt.em2b = ridge_indep_em2(ytilde,X.svd$d^2,10,.1,100,recompute_between_updates = TRUE)
plot_loglik(list(yt.em1,yt.em2,yt.em2b))
| Version | Author | Date |
|---|---|---|
| ae73528 | Matthew Stephens | 2020-06-26 |
No signal
This simulation has no signal (b=0). Methods are similar here. (Note that the green line requires approximately twice as much computation per iteration…)
btrue = rep(0,n)
y = X %*% btrue + sd*rnorm(n)
X.svd = svd(X)
ytilde = drop(t(X.svd$u) %*% y)
yt.em1 = ridge_indep_em1(ytilde,X.svd$d^2,1,1,100)
yt.em2 = ridge_indep_em2(ytilde,X.svd$d^2,1,1,100)
yt.em2b = ridge_indep_em2(ytilde,X.svd$d^2,1,1,100, recompute_between_updates = TRUE)
plot_loglik(list(yt.em1,yt.em2,yt.em2b))
| Version | Author | Date |
|---|---|---|
| ae73528 | Matthew Stephens | 2020-06-26 |
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)
| Version | Author | Date |
|---|---|---|
| ae73528 | Matthew Stephens | 2020-06-26 |
Run the methods: there is a clear advantage of simple parameterization.
X.svd = svd(X)
ytilde = drop(t(X.svd$u) %*% y)
yt.em1 = ridge_indep_em1(ytilde,X.svd$d^2,1,1,100)
yt.em2 = ridge_indep_em2(ytilde,X.svd$d^2,1,1,100)
yt.em2b = ridge_indep_em2(ytilde,X.svd$d^2,1,1,100, recompute_between_updates = TRUE)
plot_loglik(list(yt.em1,yt.em2,yt.em2b))
| Version | Author | Date |
|---|---|---|
| ae73528 | Matthew Stephens | 2020-06-26 |
No signal case
Try no signal case
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)
Run the EM: there is a clear advantage of scaled parameterization; not much benefit of recomputing between updates.
X.svd = svd(X)
ytilde = drop(t(X.svd$u) %*% y)
yt.em1 = ridge_indep_em1(ytilde,X.svd$d^2,1,1,100)
yt.em2 = ridge_indep_em2(ytilde,X.svd$d^2,1,1,100)
yt.em2b = ridge_indep_em2(ytilde,X.svd$d^2,1,1,100, recompute_between_updates = TRUE)
plot_loglik(list(yt.em1,yt.em2,yt.em2b))
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):
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[5] later_1.0.0 R6_2.4.1 backports_1.1.5 git2r_0.26.1
[9] magrittr_1.5 evaluate_0.14 stringi_1.4.6 rlang_0.4.5
[13] fs_1.3.2 promises_1.1.0 whisker_0.4 rmarkdown_2.1
[17] tools_3.6.0 stringr_1.4.0 glue_1.4.0 httpuv_1.5.2
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[25] knitr_1.28