Last updated: 2020-06-26

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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 \(\theta_j\): \[y_j \sim N(s_b \theta_j, s^2)\] \[\theta_j \sim N(0,d_j^2).\]

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))
}

Scaled parameterization, part II

We could also do the same with \(d_j\) resulting in this variation: \[y_j \sim N(s_b d_j \theta_j, s^2)\] \[\theta_j \sim N(0,1).\]

ridge_indep_em2b = 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 = 1 # prior variance for theta
    data_var = s2/(sb2*d2) # variance of y/(sb d), 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)*sqrt(d2)) # posterior mean of theta
    
    sb2 = (sum(y*post_mean*sqrt(d2))/sum(d2*(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 = 1 # prior variance for theta
      data_var = s2/(sb2*d2) # variance of y/(sb d), 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)*sqrt(d2)) # posterior mean of theta
    }
    
    r = y - sqrt(d2) * sqrt(sb2) * post_mean # residuals
    s2 = mean(r^2 + sb2 * d2 * 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)

Version Author Date
29360cf Matthew Stephens 2020-06-26 
X.svd = svd(X)
ytilde = drop(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$d^2,1,1,100)
yt.em2b = ridge_indep_em2b(ytilde,X.svd$d^2,1,1,100)


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)
  }

}

plot_loglik(list(yt.em1,yt.em2,yt.em2b))

Version Author Date
29360cf Matthew Stephens 2020-06-26

No signal

This simulation has no signal (b=0):

btrue = rep(0,n)
y = X %*% btrue + sd*rnorm(n)

X.svd = svd(X)
ytilde = drop(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$d^2,1,1,100)
yt.em2b = ridge_indep_em2b(ytilde,X.svd$d^2,1,1,100)


plot_loglik(list(yt.em1,yt.em2,yt.em2b))

Version Author Date
29360cf 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)

X.svd = svd(X)
ytilde = drop(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_em2b(ytilde,X.svd$d^2,1,1,100)

plot_loglik(list(yt.em1,yt.em2,yt.em2b))

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)

X.svd = svd(X)
ytilde = drop(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_em2b(ytilde,X.svd$d^2,1,1,100)

plot_loglik(list(yt.em1,yt.em2,yt.em2b))

plot_loglik(list(yt.em1,yt.em2))


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    rprojroot_1.3-2 digest_0.6.25  
 [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   
[21] xfun_0.12       yaml_2.2.1      compiler_3.6.0  htmltools_0.4.0
[25] knitr_1.28