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library(mashr)
Loading required package: ashr
library(ggplot2)
library(knitr)
library(kableExtra)
source('../code/generateDataV.R')

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 = c(2000:2100)
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, details = TRUE)
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)
model.subset = mash(m.data.subset.V, U.c)
 - Computing 4000 x 106 likelihood matrix.
 - Likelihood calculations took 0.05 seconds.
 - Fitting model with 106 mixture components.
 - Model fitting took 0.52 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.02 seconds.
logliks = rbind(c(get_loglik(result.mV.full$mash.model), get_loglik(model.subset)))
colnames(logliks) = c('Full','Subset')
logliks %>% kable() %>% kable_styling()
Full Subset
-12302.45 -12302.74

More simulations

I randomly generate 10 positive definite correlation matrices, V. The sample size is 4000.

\[ \hat{b}_{j}|b_{j} \sim N_{5}(z, S_{j}VS_{j}) \] \[ b_{j}\sim\frac{1}{4}\delta_{0} + \frac{1}{4}N_{5}(0,\left(\begin{matrix} 1 & \mathbf{0}_{1\times 4} \\ \mathbf{0}_{4\times 1} & \mathbf{0}_{4\times 4} \end{matrix}\right)) + \frac{1}{4}N_{5}(0,\left(\begin{matrix} \mathbf{1}_{2\times 2} & \mathbf{0}_{1\times 3} \\ \mathbf{0}_{3\times 1} & \mathbf{0}_{3\times 3} \end{matrix}\right)) + \frac{1}{4}N_{5}(0,\mathbf{1}_{5\times 5}) \]

generate_data = function(n, p, V, Utrue){
  if (is.null(pi)) {
    pi = rep(1, length(Utrue)) # default to uniform distribution
  }
  assertthat::are_equal(length(pi), length(Utrue))

  for (j in 1:length(Utrue)) {
    assertthat::are_equal(dim(Utrue[j]), c(p, p))
  }

  pi <- pi / sum(pi) # normalize pi to sum to one
  which_U <- sample(1:length(pi), n, replace=TRUE, prob=pi)

  Beta = matrix(0, nrow=n, ncol=p)
  Beta[1:(n/4),] = MASS::mvrnorm(n/4, rep(0, p), Utrue[[1]])
  Beta[((n/4)+1):(2* n/4),] = MASS::mvrnorm(n/4, rep(0, p), Utrue[[2]])
  Beta[(2*(n/4)+1):(3* n/4),] = MASS::mvrnorm(n/4, rep(0, p), Utrue[[3]])
  Beta[(3*(n/4)+1):(n),] = MASS::mvrnorm(n/4, rep(0, p), Utrue[[4]])

  E = MASS::mvrnorm(n, rep(0, p), V)
  Bhat = Beta + E
  Shat = 1
  return(list(B = Beta, Bhat=Bhat, Shat = Shat))
}

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 = c(sample(1:1000, 200))
  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, details=TRUE)
  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)
  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
  
  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)
}

Error

The Frobenius norm is

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_point()

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The spectral norm is

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_point()

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mash log likelihood

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_point()

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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.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, ' '))
}

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

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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] kableExtra_1.3.1 knitr_1.30       ggplot2_3.3.2    mashr_0.2.40    
[5] 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_0.3.6       generics_0.1.0   
 [9] viridisLite_0.3.0 htmltools_0.5.0   yaml_2.2.1        rlang_0.4.9      
[13] mixsqp_0.3-46     later_1.1.0.1     pillar_1.4.7      glue_1.4.2       
[17] withr_2.3.0       lifecycle_0.2.0   plyr_1.8.6        stringr_1.4.0    
[21] munsell_0.5.0     gtable_0.3.0      rvest_0.3.6       mvtnorm_1.1-1    
[25] evaluate_0.14     labeling_0.4.2    httpuv_1.5.4      invgamma_1.1     
[29] irlba_2.3.3       highr_0.8         Rcpp_1.0.5        promises_1.1.1   
[33] scales_1.1.1      rmeta_3.0         webshot_0.5.2     truncnorm_1.0-8  
[37] abind_1.4-5       farver_2.0.3      fs_1.5.0          digest_0.6.27    
[41] stringi_1.5.3     dplyr_1.0.2       grid_4.0.3        rprojroot_2.0.2  
[45] tools_4.0.3       magrittr_2.0.1    tibble_3.0.4      crayon_1.3.4     
[49] whisker_0.4       pkgconfig_2.0.3   MASS_7.3-53       ellipsis_0.3.1   
[53] Matrix_1.2-18     xml2_1.3.2        SQUAREM_2020.5    httr_1.4.2       
[57] assertthat_0.2.1  rmarkdown_2.5     rstudioapi_0.13   R6_2.5.0         
[61] git2r_0.27.1      compiler_4.0.3