Estimate V – fewer samples

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)

samples = sample(1:n, 1000)
m.data.subset = mash_set_data(Bhat = data$Bhat[samples,], Shat = data$Shat)

Full data vs Subset

result.mV.full <- estimate_null_correlation(m.data, U.c, tol=1e-3)
result.mV.subset <- estimate_null_correlation(m.data.subset, U.c, tol=1e-3)

Fit full mash model using estimated V

m.data.subset.V = mash_update_data(m.data, V=result.mV.subset$V)
model.subset = mash(m.data.subset.V, U.c)

 - Computing 4000 x 106 likelihood matrix.
 - Likelihood calculations took 0.09 seconds.
 - Fitting model with 106 mixture components.
 - Model fitting took 0.64 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.05 seconds.

          | model with V.Full | model with V.subset

————–|——————-|——————– loglikelihood | get_loglik(result.mV.full$loglik) | get_loglik(model.subset)

More simulations

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(20181124)
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)
  
  samples = sample(1:n, 1000)
  m.data.subset = mash_set_data(Bhat = data$Bhat[samples,], Shat = data$Shat)
  print('Method: mV')
  print('Full')
  Vhat.full <- estimate_null_correlation(m.data, c(U.c, U.ed),
                        tol=1e-3, max_iter = 5)
  print('Subset')
  Vhat.subset <- estimate_null_correlation(m.data.subset, c(U.c, U.ed),
                        tol=1e-3, max_iter = 5)
  m.data.subset.V = mash_update_data(m.data, V = Vhat.subset$V)
  model.subset = mash(m.data.subset.V, c(U.c,U.ed))
  saveRDS(list(V.true = Vtrue, V.mV.Full = Vhat.full, V.mV.subset = Vhat.subset, 
               model.subset = model.subset, data = data, strong=strong),
          paste0('../output/mVsubset/MASH.mV.subset.result.',t,'.rds'))
}

files = dir("../output/mVsubset/"); files = files[grep("MASH.mV.subset.result",files)]
times = length(files)
result = vector(mode="list",length = times)
for(i in 1:times) {
  result[[i]] = readRDS(paste("../output/mVsubset/", files[[i]], sep=""))
}

result_wrap = vector("list", times) 
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_wrap[[i]]$V.true = result[[i]]$V.true
  result_wrap[[i]]$V.full = result[[i]]$V.mV.Full$V
  result_wrap[[i]]$V.subset = result[[i]]$V.mV.subset$V
  
  result_wrap[[i]]$m.model = list(m.model.true = m.model.true, m.model.full = result[[i]]$V.mV.Full$mash.model,
                             m.model.subset = result[[i]]$model.subset)
}

norm.type='F'
temp = matrix(0,nrow = times, ncol = 2)
for(i in 1:times){
  temp[i, ] = c(norm(result_wrap[[i]]$V.full - result_wrap[[i]]$V.true, type = norm.type), 
                norm(result_wrap[[i]]$V.subset - result_wrap[[i]]$V.true, type = norm.type))
}
colnames(temp) = c('Full','Subset')
temp = reshape2::melt(temp[])
colnames(temp) = c('Data', 'Method', 'FrobError')
ggplot(temp, aes(x = Data, y=FrobError, group = Method, color = Method)) + geom_line()

norm.type='2'
temp = matrix(0,nrow = times, ncol = 2)
for(i in 1:times){
  temp[i, ] = c(norm(result_wrap[[i]]$V.full - result_wrap[[i]]$V.true, type = norm.type), 
                norm(result_wrap[[i]]$V.subset - result_wrap[[i]]$V.true, type = norm.type))
}
colnames(temp) = c('Full','Subset')
temp = reshape2::melt(temp[])
colnames(temp) = c('Data', 'Method', 'SpecError')
ggplot(temp, aes(x = Data, y=SpecError, group = Method, color = Method)) + geom_line()

temp = matrix(0,nrow = times, ncol = 3)
for(i in 1:times){
  temp[i, ] = c(get_loglik(result_wrap[[i]]$m.model$m.model.true), get_loglik(result_wrap[[i]]$m.model$m.model.full),
                get_loglik(result_wrap[[i]]$m.model$m.model.subset))
}
colnames(temp) = c('True', 'Full','Subset')
temp = reshape2::melt(temp)
colnames(temp) = c('Data', 'Method', 'loglikelihood')
ggplot(temp, aes(x = Data, y=loglikelihood, group = Method, color = Method)) + geom_line()

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.full.seq = ROC.table(data.true, result.model$m.model.full)
  m.subset.seq = ROC.table(data.true, result.model$m.model.subset)
  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.full.seq[,'FPR'], m.full.seq[,'TPR'], col='red', lwd = 1.5)
  lines(m.subset.seq[,'FPR'], m.subset.seq[,'TPR'], col='darkorchid', lwd = 1.5)
  legend('bottomright', c('True','Full', 'Subset'),col=c('black','red','darkorchid'),
           lty=c(1,1,1), lwd=c(1.5,1.5,1.5))
}

par(mfrow=c(1,2))
for(i in 1:times){
  plotROC(result[[i]]$data$B, result_wrap[[i]]$m.model, title=paste0('Data', i, ' '))
}

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_wrap[[i]]$m.model))
  barplot(rrmse, ylim=c(0,(1+max(rrmse))/2), las=2, cex.names = 0.7, main='RRMSE')
}

Session information

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.14.1

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/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_3.1.0     mashr_0.2.18.0536 ashr_2.2-23      

loaded via a namespace (and not attached):
 [1] tidyselect_0.2.5  purrr_0.2.5       reshape2_1.4.3   
 [4] lattice_0.20-35   Rmosek_8.0.69     colorspace_1.3-2 
 [7] htmltools_0.3.6   yaml_2.2.0        rlang_0.3.0.1    
[10] R.oo_1.22.0       pillar_1.3.0      glue_1.3.0       
[13] withr_2.1.2       R.utils_2.7.0     REBayes_1.3      
[16] bindrcpp_0.2.2    foreach_1.4.4     plyr_1.8.4       
[19] bindr_0.1.1       stringr_1.3.1     munsell_0.5.0    
[22] gtable_0.2.0      workflowr_1.1.1   R.methodsS3_1.7.1
[25] mvtnorm_1.0-8     codetools_0.2-15  evaluate_0.12    
[28] labeling_0.3      knitr_1.20        pscl_1.5.2       
[31] doParallel_1.0.14 parallel_3.5.1    Rcpp_1.0.0       
[34] scales_1.0.0      backports_1.1.2   rmeta_3.0        
[37] truncnorm_1.0-8   abind_1.4-5       digest_0.6.18    
[40] stringi_1.2.4     dplyr_0.7.6       grid_3.5.1       
[43] rprojroot_1.3-2   tools_3.5.1       magrittr_1.5     
[46] lazyeval_0.2.1    tibble_1.4.2      crayon_1.3.4     
[49] whisker_0.3-2     pkgconfig_2.0.2   MASS_7.3-50      
[52] Matrix_1.2-14     SQUAREM_2017.10-1 assertthat_0.2.0 
[55] rmarkdown_1.10    iterators_1.0.10  R6_2.3.0         
[58] git2r_0.23.0      compiler_3.5.1