Estimate V – fewer iterations
Yuxin Zou
2018-10-22
Last updated: 2020-12-21
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Knit directory: mash_application/analysis/
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library(mashr)Loading required package: ashrlibrary(ggplot2)
source('../code/generateDataV.R')penalty <- function(prior, pi_s){
subset <- (prior != 1.0)
sum((prior-1)[subset]*log(pi_s[subset]))
}
mixture.MV <- function(mash.data, Ulist, init_V=diag(ncol(mash.data$Bhat)), max_iter = 500, tol=1e-5, prior = c('nullbiased', 'uniform'), cor = TRUE, track_fit = FALSE){
prior <- match.arg(prior)
tracking = list()
m.model = fit_mash_V(mash.data, Ulist, V = init_V, prior=prior)
pi_s = get_estimated_pi(m.model, dimension = 'all')
prior.v <- mashr:::set_prior(length(pi_s), prior)
# compute loglikelihood
log_liks <- numeric(max_iter+1)
log_liks[1] <- get_loglik(m.model)+penalty(prior.v, pi_s)
V = init_V
result = list(V = V, logliks = log_liks[1], mash.model = m.model)
for(i in 1:max_iter){
if(track_fit){
tracking[[i]] = result
}
# max_V
V = E_V(mash.data, m.model)
if(cor){
V = cov2cor(V)
}
m.model = fit_mash_V(mash.data, Ulist, V, prior=prior)
pi_s = get_estimated_pi(m.model, dimension = 'all')
log_liks[i+1] <- get_loglik(m.model)+penalty(prior.v, pi_s)
result = list(V = V, logliks = log_liks[1:(i+1)], mash.model = m.model)
# Update delta
delta.ll <- log_liks[i+1] - log_liks[i]
if(delta.ll<=tol) break;
}
if(track_fit){
result$trace = tracking
}
return(result)
}
E_V = function(mash.data, m.model){
n = mashr:::n_effects(mash.data)
Z = mash.data$Bhat/mash.data$Shat
post.m.shat = m.model$result$PosteriorMean / mash.data$Shat
post.sec.shat = plyr::laply(1:n, function(i) (t(m.model$result$PosteriorCov[,,i]/mash.data$Shat[i,])/mash.data$Shat[i,]) + tcrossprod(post.m.shat[i,])) # nx2x2 array
temp1 = crossprod(Z)
temp2 = crossprod(post.m.shat, Z) + crossprod(Z, post.m.shat)
temp3 = unname(plyr::aaply(post.sec.shat, c(2,3), sum))
(temp1 - temp2 + temp3)/n
}
fit_mash_V <- function(mash.data, Ulist, V, prior=c('nullbiased', 'uniform')){
m.data = mashr::mash_set_data(Bhat=mash.data$Bhat, Shat=mash.data$Shat, V = V, alpha = mash.data$alpha)
m.model = mashr::mash(m.data, Ulist, prior=prior, verbose = FALSE, outputlevel = 3)
return(m.model)
}Data
Simple simulation in \(R^2\): \[ \hat{\beta}|\beta \sim N_{2}(\hat{\beta}; \beta, \left(\begin{matrix} 1 & 0.5 \\ 0.5 & 1 \end{matrix}\right)) \]
\[ \beta \sim \frac{1}{4}\delta_{0} + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}\right)) \]
set.seed(1)
n = 4000; p = 2
Sigma = matrix(c(1,0.5,0.5,1),p,p)
U0 = matrix(0,2,2)
U1 = U0; U1[1,1] = 1
U2 = U0; U2[2,2] = 1
U3 = matrix(1,2,2)
Utrue = list(U0=U0, U1=U1, U2=U2, U3=U3)
data = generate_data(n, p, Sigma, Utrue)m.data = mash_set_data(data$Bhat, data$Shat)
U.c = cov_canonical(m.data)Initialize V as I VS V.simple
result.mV.I <- mixture.MV(m.data, U.c)
result.mV.simple <- mixture.MV(m.data, U.c, init_V = estimate_null_correlation_simple(m.data)){
plot(result.mV.I$loglik, xlab = 'iter', ylab = 'loglik')
points(result.mV.simple$loglik, col='green')
legend('bottomright', legend=c('I', 'simple'), col=c('black', 'green'), pch=c(1,1))
}
Different number of iterations
I randomly generate 10 positive definite correlation matrices, V. The sample size is 4000.
\[ \hat{b}_{j}|b_{j} \sim N_{5}(z, S_{j}VS_{j}) \] \[ b_{j}\sim\frac{1}{4}\delta_{0} + \frac{1}{4}N_{5}(0,\left(\begin{matrix} 1 & \mathbf{0}_{1\times 4} \\ \mathbf{0}_{4\times 1} & \mathbf{0}_{4\times 4} \end{matrix}\right)) + \frac{1}{4}N_{5}(0,\left(\begin{matrix} \mathbf{1}_{2\times 2} & \mathbf{0}_{1\times 3} \\ \mathbf{0}_{3\times 1} & \mathbf{0}_{3\times 3} \end{matrix}\right)) + \frac{1}{4}N_{5}(0,\mathbf{1}_{5\times 5}) \]
set.seed(20181006)
n=4000; p = 5
U0 = matrix(0,p,p)
U1 = U0; U1[1,1] = 1
U2 = U0; U2[c(1:2), c(1:2)] = 1
U3 = matrix(1, p,p)
Utrue = list(U0 = U0, U1 = U1, U2 = U2, U3 = U3)
for(t in 1:10){
print(paste0('Data: ', t))
Vtrue = clusterGeneration::rcorrmatrix(p)
data = generate_data(n, p, Vtrue, Utrue)
# mash cov
m.data = mash_set_data(Bhat = data$Bhat, Shat = data$Shat)
m.1by1 = mash_1by1(m.data)
strong = get_significant_results(m.1by1)
U.pca = cov_pca(m.data, 3, subset = strong)
U.ed = cov_ed(m.data, U.pca, subset = strong)
U.c = cov_canonical(m.data)
print('Method: mV')
Vhat.mV <- mixture.MV(m.data, c(U.c, U.ed), init_V = estimate_null_correlation_simple(m.data),
tol=1e-4, max_iter = 5, track_fit = TRUE)
saveRDS(list(V.true = Vtrue, V.mV = Vhat.mV, data = data, strong=strong),
paste0('../output/MASH.mV.result.',t,'.rds'))
}files = dir("../output/mVIterations/"); files = files[grep("MASH.mV.result",files)]
times = length(files)
result = vector(mode="list",length = times)
for(i in 1:times) {
result[[i]] = readRDS(paste("../output/mVIterations/", files[[i]], sep=""))
}for(i in 1:times){
m.data = mash_set_data(result[[i]]$data$Bhat, result[[i]]$data$Shat)
m.1by1 = mash_1by1(m.data)
strong = get_significant_results(m.1by1)
U.c = cov_canonical(m.data)
U.pca = cov_pca(m.data, 3, subset = strong)
U.ed = cov_ed(m.data, U.pca, subset = strong)
m.data.true = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = result[[i]]$V.true)
m.model.true = mash(m.data.true, c(U.c,U.ed), verbose = FALSE)
# mV
result[[i]]$V.simple = result[[i]]$V.mV$trace[[1]]$V
result[[i]]$V.mV1 = result[[i]]$V.mV$trace[[2]]$V
result[[i]]$V.mV2 = result[[i]]$V.mV$trace[[3]]$V
result[[i]]$V.mV3 = result[[i]]$V.mV$trace[[4]]$V
result[[i]]$V.mV4 = result[[i]]$V.mV$trace[[5]]$V
result[[i]]$V.mV5 = result[[i]]$V.mV$V
m.model.simple = result[[i]]$V.mV$trace[[1]]$mash.model
m.model.mV1 = result[[i]]$V.mV$trace[[2]]$mash.model
m.model.mV2 = result[[i]]$V.mV$trace[[3]]$mash.model
m.model.mV3 = result[[i]]$V.mV$trace[[4]]$mash.model
m.model.mV4 = result[[i]]$V.mV$trace[[5]]$mash.model
m.model.mV5 = result[[i]]$V.mV$mash.model
result[[i]]$m.model = list(m.model.true = m.model.true, m.model.simple = m.model.simple,
m.model.mV1 = m.model.mV1, m.model.mV2 = m.model.mV2, m.model.mV3 = m.model.mV3,
m.model.mV4 = m.model.mV4, m.model.mV5 = m.model.mV5)
}Error
The Frobenius norm is
norm.type='F'
temp = matrix(0,nrow = times, ncol = 6)
for(i in 1:times){
temp[i, ] = c(norm(result[[i]]$V.simple - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV1 - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV2 - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV3 - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV4 - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV5 - result[[i]]$V.true, type = norm.type))
}
colnames(temp) = c('Simple','mViter1','mViter2','mViter3','mViter4','mViter5')
temp = reshape2::melt(temp[])
colnames(temp) = c('Data', 'Method', 'FrobError')
temp$Data = as.factor(temp$Data)
ggplot(temp, aes(x = Data, y=FrobError, fill = Method)) + geom_point()
The spectral norm is
norm.type='2'
temp = matrix(0,nrow = times, ncol = 6)
for(i in 1:times){
temp[i, ] = c(norm(result[[i]]$V.simple - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV1 - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV2 - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV3 - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV4 - result[[i]]$V.true, type = norm.type),
norm(result[[i]]$V.mV5 - result[[i]]$V.true, type = norm.type))
}
colnames(temp) = c('Simple','mViter1','mViter2','mViter3','mViter4','mViter5')
temp = reshape2::melt(temp)
colnames(temp) = c('Data', 'Method', 'SpecError')
temp$Data = as.factor(temp$Data)
ggplot(temp, aes(x = Data, y=SpecError, fill = Method)) + geom_point()
mash log likelihood
temp = matrix(0,nrow = times, ncol = 7)
for(i in 1:times){
temp[i, ] = c(get_loglik(result[[i]]$m.model$m.model.true), get_loglik(result[[i]]$m.model$m.model.simple),
get_loglik(result[[i]]$m.model$m.model.mV1), get_loglik(result[[i]]$m.model$m.model.mV2),
get_loglik(result[[i]]$m.model$m.model.mV3), get_loglik(result[[i]]$m.model$m.model.mV4),
get_loglik(result[[i]]$m.model$m.model.mV5))
}
colnames(temp) = c('True', 'simple','mV1', 'mV2', 'mV3', 'mV4', 'mV5')
temp = reshape2::melt(temp)
colnames(temp) = c('Data', 'Method', 'loglikelihood')
temp$Data = as.factor(temp$Data)
ggplot(temp, aes(x = Data, y=loglikelihood, group = Method, color = Method)) + geom_point()
# ggplot(temp, aes(x = Data, y=loglikelihood, fill = Method)) + geom_bar(position="stack", stat="identity")ROC
ROC.table = function(data, model){
sign.test = data*model$result$PosteriorMean
thresh.seq = seq(0, 1, by=0.005)[-1]
m.seq = matrix(0,length(thresh.seq), 2)
colnames(m.seq) = c('TPR', 'FPR')
for(t in 1:length(thresh.seq)){
m.seq[t,] = c(sum(sign.test>0 & model$result$lfsr <= thresh.seq[t])/sum(data!=0),
sum(data==0 & model$result$lfsr <=thresh.seq[t])/sum(data==0))
}
return(m.seq)
}
plotROC = function(data.true, result.model, title){
m.trun.seq = ROC.table(data.true, result.model$m.model.simple)
m.mV1.seq = ROC.table(data.true, result.model$m.model.mV1)
m.mV2.seq = ROC.table(data.true, result.model$m.model.mV2)
m.mV3.seq = ROC.table(data.true, result.model$m.model.mV3)
m.mV4.seq = ROC.table(data.true, result.model$m.model.mV4)
m.mV5.seq = ROC.table(data.true, result.model$m.model.mV5)
m.true.seq = ROC.table(data.true, result.model$m.model.true)
plot(m.true.seq[,'FPR'], m.true.seq[,'TPR'],type='l',xlab = 'FPR', ylab='TPR',
main=paste0(title, 'True Pos vs False Pos'), cex=1.5, lwd = 1.5)
lines(m.trun.seq[,'FPR'], m.trun.seq[,'TPR'], col='red', lwd = 1.5)
lines(m.mV1.seq[,'FPR'], m.mV1.seq[,'TPR'], col='darkorchid', lwd = 1.5)
lines(m.mV2.seq[,'FPR'], m.mV2.seq[,'TPR'], col='darkgoldenrod', lwd = 1.5)
lines(m.mV3.seq[,'FPR'], m.mV3.seq[,'TPR'], col='blue', lwd = 1.5)
lines(m.mV4.seq[,'FPR'], m.mV4.seq[,'TPR'], col='cyan', lwd = 1.5)
lines(m.mV5.seq[,'FPR'], m.mV5.seq[,'TPR'], col='green', lwd = 1.5)
legend('bottomright', c('True','simple', 'mViter1', 'mViter2', 'mViter3', 'mViter4', 'mViter5'),col=c('black','red','darkorchid','darkgoldenrod', 'blue','cyan', 'green'),
lty=c(1,1,1,1,1,1,1), lwd=c(1.5,1.5,1.5,1.5,1.5,1.5,1.5))
}par(mfrow=c(1,2))
for(i in 1:times){
plotROC(result[[i]]$data$B, result[[i]]$m.model, title=paste0('Data', i, ' '))
}
RRMSE
RRMSE = function(datatrue, dataobs, model){
model = Filter(length, model)
rrmse = numeric(length(model))
for(k in 1:length(model)){
rrmse[k] = sqrt(mean((datatrue - model[[k]]$result$PosteriorMean)^2)/mean((datatrue - dataobs)^2))
}
rrmse = as.matrix(t(rrmse))
colnames(rrmse) = names(model)
return(rrmse)
}par(mfrow=c(1,2))
for(i in 1:times){
rrmse = rbind(RRMSE(result[[i]]$data$B, result[[i]]$data$Bhat, result[[i]]$m.model))
barplot(rrmse, ylim=c(0,(1+max(rrmse))/2), las=2, cex.names = 0.7, main='RRMSE')
}
sessionInfo()R version 4.0.3 (2020-10-10)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/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] ggplot2_3.3.2 mashr_0.2.40 ashr_2.2-51 workflowr_1.6.2
loaded via a namespace (and not attached):
[1] tidyselect_1.1.0 xfun_0.19 reshape2_1.4.4 purrr_0.3.4
[5] lattice_0.20-41 colorspace_2.0-0 vctrs_0.3.5 generics_0.1.0
[9] htmltools_0.5.0 yaml_2.2.1 rlang_0.4.9 mixsqp_0.3-46
[13] later_1.1.0.1 pillar_1.4.7 glue_1.4.2 withr_2.3.0
[17] lifecycle_0.2.0 plyr_1.8.6 stringr_1.4.0 munsell_0.5.0
[21] gtable_0.3.0 mvtnorm_1.1-1 evaluate_0.14 labeling_0.4.2
[25] knitr_1.30 httpuv_1.5.4 invgamma_1.1 irlba_2.3.3
[29] Rcpp_1.0.5 promises_1.1.1 scales_1.1.1 rmeta_3.0
[33] truncnorm_1.0-8 abind_1.4-5 farver_2.0.3 fs_1.5.0
[37] digest_0.6.27 stringi_1.5.3 dplyr_1.0.2 grid_4.0.3
[41] rprojroot_2.0.2 tools_4.0.3 magrittr_2.0.1 tibble_3.0.4
[45] crayon_1.3.4 whisker_0.4 pkgconfig_2.0.3 MASS_7.3-53
[49] ellipsis_0.3.1 Matrix_1.2-18 SQUAREM_2020.5 assertthat_0.2.1
[53] rmarkdown_2.5 rstudioapi_0.13 R6_2.5.0 git2r_0.27.1
[57] compiler_4.0.3