mr.ash_ridge
Matthew Stephens
2020-05-19
Last updated: 2020-05-20
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Knit directory: misc/analysis/
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library("mr.ash.alpha")Introduction
My idea here is to experiment with a new model \[y=Xb + e\] where \(b_j = (b_1)_j (b_2)_j\). My initial idea was to have elements of \(b_1\) having ridge (normal) prior and elements of \(b_2\) having ash prior. However, it turns out to be interesting, perhaps even more interesting, to use ridge priors for both, and to further extend to \(b_j \prod_k (b_k)_j\).
In any case the motivation is that we can do variational approximation \(q(b_1)\prod q_j(b_2j)\) where \(q(b_1)\) does not factorize - it is a full approximation on all \(p\) (which is tractible due to the normal prior). So it can capture dependence among \(b\) values.
In the non-sparse case we would expect the EB estimation of ash prior to end up fitting a normal, so in that case \(b_j\) will be a product of two normals. So this model does not include ridge regression as a special case.
However, a product of two normals actually has quite an interesting shape:
hist(rnorm(10000)*rnorm(10000),nclass=100)
This is perhaps not a bad “null” non-sparse model in itself.
Fitting
I think we can do the regular variational thing for this model, but for now to get something working quickly I’m going to do a simple 2-stage procedure: fit ridge regression on it’s own to estimate \(b_1\), and then fit ash to \(y=(XB_1) b_2+e\) where \(B_1\) is the diagonal matrix with estimated \(b_1\) on it’s diagonal.
We will try a challenging trend-filtering example (challenging partly as it has highly correlated covariates):
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)
First a function to compute the ridge posterior If \(b_j \sim N(0,s_b^2)\) then \(Y \sim N(0, s^2 I_n + s_b^2(XX'))\).
ridge = function(y,A,prior_variance,prior_mean=rep(0,ncol(A)),residual_variance=1){
n = length(y)
p = ncol(A)
L = chol(t(A) %*% A + (residual_variance/prior_variance)*diag(p))
b = backsolve(L, t(A) %*% y + (residual_variance/prior_variance)*prior_mean, transpose=TRUE)
b = backsolve(L, b)
#b = chol2inv(L) %*% (t(A) %*% y + (residual_variance/prior_variance)*prior_mean)
return(b)
}Plot the ridge fit- looks OK ish.
plot(Y)
b_ridge = ridge(Y,X,10)
lines(X %*% b_ridge)
Apply mr.ash regularly - it does poorly, perhaps (in part) because initialization with glmnet is not very good here.
fit.mrash = mr.ash(X,Y,standardize=FALSE)
plot(Y)
lines(X %*% fit.mrash$beta,col=2)
Try initializing to the ridge fit:
fit.mrash = mr.ash(X,Y,standardize=FALSE,beta.init = b_ridge)
plot(Y)
lines(X %*% fit.mrash$beta,col=3)
Now apply mr.ash with \(X\) scaled by the ridge fit, as suggested above, initializing at 1.
X_scale = t(t(X) * as.vector(b_ridge))
fit.mrash = mr.ash(X_scale,Y,standardize=FALSE,beta.init = rep(1,100),sigma2=1,update.sigma=FALSE)
plot(Y)
lines(X %*% (as.vector(b_ridge)*fit.mrash$beta),col=4)
It didn’t really change much, which is a bit disappointing. Try iterating:
X_scale = t(t(X) * as.vector(fit.mrash$beta))
b_ridge = ridge(Y,X_scale,1,1)
X_scale = t(t(X) * as.vector(b_ridge))
fit.mrash = mr.ash(X_scale,Y,standardize=FALSE,beta.init = rep(1,100))
plot(Y)
lines(X %*% (as.vector(b_ridge)*fit.mrash$beta),col=4)
Iterative ridge regression
So that initial try didn’t work as well as I hoped. There is probably more investigation to be done, but I tried something different, based on \(b_j \prod_k (b_k)_j\) where each \(b_k\) has a ridge prior, so \(b_j\) is a product of gaussians.
Note that product of gaussians becomes sparser and longer tailed the more you use:
hist(rnorm(1000)*rnorm(1000)*rnorm(1000)*rnorm(1000),nclass=100)
So iteration is a way of using ridge regression to induce sparsity…
Here I try iterating 3 times, and you can see the fit becomes gradually “sparser” (sparse solutions are piecewise linear in this basis). (Note: throughout I fix the ridge prior variance to 1, which is not necessarily optimal…)
b_curr = ridge(Y,X,1)
par(mfrow=c(2,3))
plot(Y,main=paste0("iteration ",1))
lines(X %*% b_curr,col=2,lwd=2)
for(i in 2:6){
X_scale = t(t(X) * as.vector(b_curr))
b_ridge = ridge(Y,X_scale,1)
b_curr = b_curr*b_ridge
plot(Y,main=paste0("iteration ",i))
lines(X %*% b_curr,col=2,lwd=2)
}
Try another example
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[20] = .3
btrue[41] = -.4
btrue[60]= .6
Y = X %*% btrue + sd*rnorm(n)b_curr = ridge(Y,X,1)
par(mfrow=c(2,3))
plot(Y,main=paste0("iteration ",1))
lines(X %*% b_curr,col=2,lwd=2)
for(i in 2:6){
X_scale = t(t(X) * as.vector(b_curr))
b_ridge = ridge(Y,X_scale,1)
b_curr = b_curr*b_ridge
plot(Y,main=paste0("iteration ",i))
lines(X %*% b_curr,col=2,lwd=2)
}
A non-sparse example
The above examples are both sparse. We try a non-sparse example here.
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 = rnorm(n,0,0.01)
Y = X %*% btrue + sd*rnorm(n)b_curr = ridge(Y,X,1)
par(mfrow=c(2,3))
plot(Y,main=paste0("iteration ",1))
lines(X %*% b_curr,col=2,lwd=2)
for(i in 2:6){
X_scale = t(t(X) * as.vector(b_curr))
b_ridge = ridge(Y,X_scale,1)
b_curr = b_curr*b_ridge
plot(Y,main=paste0("iteration ",i))
lines(X %*% b_curr,col=2,lwd=2)
}
Summary
I found these results pretty interesting and promising, although very preliminary. Things to do: work out the proper variational approximation, and learn ridge mean and variance by EB. (Note if we allow mean=1 in ridge prior, the it can learn mean=1, variance=0 and the fitting process can stop…)
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
other attached packages:
[1] mr.ash.alpha_0.1-7
loaded via a namespace (and not attached):
[1] workflowr_1.6.1 Rcpp_1.0.4.6 lattice_0.20-40 rprojroot_1.3-2
[5] digest_0.6.25 later_1.0.0 grid_3.6.0 R6_2.4.1
[9] backports_1.1.5 git2r_0.26.1 magrittr_1.5 evaluate_0.14
[13] stringi_1.4.6 rlang_0.4.5 fs_1.3.2 promises_1.1.0
[17] whisker_0.4 Matrix_1.2-18 rmarkdown_2.1 tools_3.6.0
[21] stringr_1.4.0 glue_1.4.0 httpuv_1.5.2 xfun_0.12
[25] yaml_2.2.1 compiler_3.6.0 htmltools_0.4.0 knitr_1.28