ridge_trend_filter
Matthew Stephens
2019-10-12
Last updated: 2019-10-12
Checks: 7 0
Knit directory: misc/analysis/
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Modified: analysis/mr_ash_sca.Rmd
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Introduction
One of the key challenges we face is convergence of VB approaches to local optima. In some applications – particularly non-parametric regression – we have found that estimating the residual variance accurately first can be helpful/necessary. For example, in susie_trend_filter we use the MAD estimate to get a good estimate of the residual variance first.
The idea here is to look at whether ridge regression could be useful for estimating the residual variance in a trend filtering application. Note that ridge regression (not necessarily mle - usually methods of moments) is widely used in heritability estimation, and has some theoretical support as well as empirically performing well. I’m also curious how this relates to using the MAD estimate, and whether the MAD estimate idea can be extended to other non-trend-filtering applications.
First a function to compute the ridge log-likelihood. If \(b_j \sim N(0,s_b^2)\) then \(Y \sim N(0, s^2 I_n + s_b^2(XX'))\).
ridge_log_lik = function(X,Y,s,sb){
p = ncol(X)
n = nrow(X)
S = s^2 * diag(n) + sb^2*(X %*% t(X))
mvtnorm::dmvnorm(as.vector(Y),sigma=S,log=TRUE)
}Simulate some data from a trend filter model:
set.seed(100)
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 + rnorm(n)
plot(Y)
lines(X %*% btrue)
xx = 20
s = seq(0,2,length=xx)
sb = seq(0.01,0.4,length=xx)
ll = rep(0,xx)
for(j in 1:length(sb)){
for(i in 1:length(s)){
ll[i] = ridge_log_lik(X,Y,s=s[i],sb=sb[j])
}
plot(s,ll,ylim=c(max(ll)-10,max(ll)),type="l",main=paste0("sb=",sb[j]))
}
Look at the inverse of covariance matrix S (here use small s, so approximately s=XX’)
s = 0.01
sb = 1
S = s^2 * diag(n) + sb^2*(X %*% t(X))
invS = solve(S)
plot(invS[1,])
sessionInfo()R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.4
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_1.4.0 Rcpp_1.0.2 mvtnorm_1.0-11 digest_0.6.20
[5] rprojroot_1.3-2 backports_1.1.4 git2r_0.26.1 magrittr_1.5
[9] evaluate_0.14 stringi_1.4.3 fs_1.3.1 whisker_0.3-2
[13] rmarkdown_1.14 tools_3.6.0 stringr_1.4.0 glue_1.3.1
[17] xfun_0.8 yaml_2.2.0 compiler_3.6.0 htmltools_0.3.6
[21] knitr_1.23