Last updated: 2019-10-25
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Unstaged changes:
Modified: analysis/minque.Rmd
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library("ashr")
The idea here is to investigate a parallel approach to updating bhat in multiple regression with ash.
The basic idea is that the optimal b should be the fixed point of the following iterations: 1. r = (Y-Xb) 2. bhat = b + d^{-1}X’r 3. shat = sigma/sqrt(d) 4. b = ash(bhat,shat)
More accurately, I believe that if b is a fixed point of this then it will also be a fixed point of the regular coordinate ascent (at least, something like this should be true.)
To start with we fix g to N(0,1) to keep thing simple….
(Note that the rescaling step was supposed to allow scaling prior and posterior by some constant, but results suggest I may not have the details right here. Ignore this for now.)
mr_ash_parallel_ca_fix = function(X,Y,b_init=NULL,max_iter=100,sigma=1,tol=1e-5,rescale=FALSE){
if(is.null(b_init)){b_init = rep(0,ncol(X))}
b = b_init
d = Matrix::colSums(X * X)
c = 1
for(i in 1:max_iter){
r = Y- X %*% b
bhat = as.vector(b + (1/d)*(t(X) %*% r))
s = sigma/sqrt(d)
bhat.ash = ash(bhat,s,g=normalmix(1,0,c),fixg=TRUE)
bnew = get_pm(bhat.ash)
if(sum((bnew-b)^2)<tol){break;}
b=bnew
if(rescale){
fitted = X %*% b
v = get_psd(bhat.ash)^2
c = sum(fitted*Y)/(sum(fitted^2) + sum(d*v)) # regress Y on fitted values
b = c*b
}
}
print(paste0("niter = ",i))
return(b)
}
mr_ash_ca_fix = function(X,Y,b_init=NULL,max_iter=100,sigma=1,tol=1e-3,rescale=FALSE){
if(is.null(b_init)){b_init = rep(0,ncol(X))}
b = b_init
p = ncol(X)
d = Matrix::colSums(X * X)
c = 1
r = Y - X %*% b
v = rep(0,p) # vector of variances for rescaling
for(i in 1:max_iter){
err = 0
for(j in 1:p){
r = r + b[j]*X[,j]
bhat = (1/d[j])*sum(X[,j]* r)
s = sigma/sqrt(d[j])
bhat.ash = ash(bhat,s,g=normalmix(1,0,c),fixg=TRUE)
bj_new = get_pm(bhat.ash)
v[j] = get_psd(bhat.ash)^2 # store variances
err = err + (b[j]-bj_new)^2
b[j] = bj_new
r = r - b[j]*X[,j] # recompute residuals
}
if(rescale){
fitted = Y - r # so fitted = Xb
c = sum(fitted*Y)/(sum(fitted^2) + sum(d*v)) # regress Y on fitted values
b = c*b
r = Y - c * fitted
}
if(err<tol){break;}
}
print(paste0("niter = ",i))
return(b)
}
ridge = function(X,Y,sigma=1){
p = ncol(X)
S = sigma^2*diag(p) + t(X) %*% X
bhat = solve(S, t(X) %*% Y)
return(bhat)
}
A toy example to check:
set.seed(123)
n= 100
p=10
X = matrix(rnorm(n*p),ncol=p,nrow=n)
btrue = rnorm(p)
Y = X %*% btrue + rnorm(n)
b.pca = mr_ash_parallel_ca_fix(X,Y)
[1] "niter = 10"
b.ca = mr_ash_ca_fix(X,Y)
[1] "niter = 4"
b.ridge = ridge(X,Y)
plot(btrue,b.ca)
points(btrue,b.pca,col=2,pch=2)
points(btrue,b.ridge,col=3,pch=3)
And a sparse example
btrue[1:5]=0
Y = X %*% btrue + rnorm(n)
b.pca = mr_ash_parallel_ca_fix(X,Y)
[1] "niter = 9"
b.ca = mr_ash_ca_fix(X,Y)
[1] "niter = 4"
b.ridge = ridge(X,Y)
plot(btrue,b.ca)
points(btrue,b.pca,col=2,pch=2)
points(btrue,b.ridge,col=3,pch=3)
Now try example with X duplicated. As might have been anticipated, it fails to converge and returns a ridiculous solution that seems to be diverging off to +-infinity.
set.seed(123)
n= 100
p=10
X = matrix(rnorm(n*p),ncol=p,nrow=n)
X = cbind(X ,X)
btrue = rnorm(2*p)
Y = X %*% btrue + rnorm(n)
b.pca = mr_ash_parallel_ca_fix(X,Y)
[1] "niter = 100"
b.ca = mr_ash_ca_fix(X,Y)
[1] "niter = 48"
b.ridge = ridge(X,Y)
plot(btrue,b.ca)
points(btrue,b.pca,col=2,pch=2)
points(btrue,b.ridge,col=3,pch=3)
print(b.pca)
[1] -8.204435e+27 1.079512e+27 4.606275e+27 -7.740797e+27 7.422769e+27
[6] 5.492114e+27 5.070576e+26 -8.111340e+27 2.752684e+27 -5.756308e+27
[11] -8.204435e+27 1.079512e+27 4.606275e+27 -7.740797e+27 7.422769e+27
[16] 5.492114e+27 5.070576e+26 -8.111340e+27 2.752684e+27 -5.756308e+27
Try same thing initializing from truth - it still diverges.
b.pca = mr_ash_parallel_ca_fix(X,Y,b_init = btrue)
[1] "niter = 100"
plot(btrue,b.pca)
Note that the fitted values do not fit Y at all for the parallel case. The others do.
plot(Y,X %*% b.pca)
Version | Author | Date |
---|---|---|
21577d7 | Matthew Stephens | 2019-10-12 |
plot(Y,X %*% b.ca, col=, pch=2, main="fitted values for ridge and CA")
points(Y,X %*% b.ridge, col=3,pch=3)
Version | Author | Date |
---|---|---|
21577d7 | Matthew Stephens | 2019-10-12 |
See if CA matches ridge with more stringent convergence tolerance:
b.ca = mr_ash_ca_fix(X,Y,tol = 1e-8,max_iter=1000)
[1] "niter = 370"
plot(btrue,b.ca)
points(btrue,b.ridge,col=3,pch=3)
Next I tried rescaling the fitted values and prior each iteration by a constant c. This might seem ad hoc, but I think something like this can be justified as scaling both the prior and the posterior approximation (although results later suggest I might have the details wrong…)
In this example, rescaling definitely stabilizes the estimates…
b.pca = mr_ash_parallel_ca_fix(X,Y,b_init = btrue,max_iter = 100,rescale=TRUE)
[1] "niter = 100"
plot(Y,X %*% b.pca)
points(Y,X %*% b.ca, col=2, pch=2)
points(Y,X %*% b.ridge, col=3,pch=3)
Version | Author | Date |
---|---|---|
21577d7 | Matthew Stephens | 2019-10-12 |
I poked around and found that in fact it was flipping between two different solutions. Here I run it for 99 and 98 iterations to illustrate.
b.pca.99= mr_ash_parallel_ca_fix(X,Y,b_init = btrue,max_iter = 99,rescale=TRUE)
[1] "niter = 99"
b.pca.98= mr_ash_parallel_ca_fix(X,Y,b_init = btrue,max_iter = 98,rescale=TRUE)
[1] "niter = 98"
plot(b.pca,b.pca.98)
plot(b.pca,b.pca.99)
This example the X will be highly correlated, but not completely so. It is designed to be challenging but easy to visualize what is going on.
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)
Version | Author | Date |
---|---|---|
21577d7 | Matthew Stephens | 2019-10-12 |
The ridge solution is much better than ca, presumably due to very slow convergence. (The solution looks better with 1000 iterations, but has still not converged; not shown here to keep runtime down…)
bhat_ca = mr_ash_ca_fix(X,Y,max_iter = 100,tol=1-8)
[1] "niter = 100"
bhat_r = ridge(X,Y)
plot(Y)
lines(X %*% bhat_ca,col=2)
lines(X %*% bhat_r, col=3)
Parallel version goes crazy; again rescaling helps stabilize but not really working.
bhat_pca = mr_ash_parallel_ca_fix(X,Y,max_iter= 10)
[1] "niter = 10"
plot(X %*% bhat_pca,col=4)
bhat_pca = mr_ash_parallel_ca_fix(X,Y,max_iter= 100, rescale=TRUE)
[1] "niter = 100"
plot(Y)
lines(X %*% bhat_pca,col=4)
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
other attached packages:
[1] ashr_2.2-38
loaded via a namespace (and not attached):
[1] Rcpp_1.0.2 knitr_1.23 whisker_0.3-2
[4] magrittr_1.5 workflowr_1.4.0 MASS_7.3-51.4
[7] pscl_1.5.2 doParallel_1.0.14 SQUAREM_2017.10-1
[10] lattice_0.20-38 foreach_1.4.7 stringr_1.4.0
[13] tools_3.6.0 parallel_3.6.0 grid_3.6.0
[16] xfun_0.8 git2r_0.26.1 htmltools_0.3.6
[19] iterators_1.0.12 yaml_2.2.0 rprojroot_1.3-2
[22] digest_0.6.20 mixsqp_0.1-97 Matrix_1.2-17
[25] fs_1.3.1 codetools_0.2-16 glue_1.3.1
[28] evaluate_0.14 rmarkdown_1.14 stringi_1.4.3
[31] compiler_3.6.0 backports_1.1.4 truncnorm_1.0-8