Estimate V – shorter Ulist

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Last updated: 2020-12-21

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---

library(mashr)

Loading required package: ashr

library(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.full = cov_canonical(m.data)
U.c.redu = cov_canonical(m.data, cov_methods = "simple_het")

Full VS Reduce

The reduced Ulist only contains simple_het.

result.mV.full <- mixture.MV(m.data, U.c.full, init_V = estimate_null_correlation_simple(m.data))
result.mV.redu <- mixture.MV(m.data, U.c.redu, init_V = estimate_null_correlation_simple(m.data))

{
plot(result.mV.full$loglik, xlab = 'iter', ylab = 'loglik')
points(result.mV.redu$loglik, col='green')
legend('bottomright', legend=c('full', 'reduce'), col=c('black', 'green'), pch=c(1,1))
}

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Version	Author	Date
bd5d3af	zouyuxin	2018-10-22
eb57faf	zouyuxin	2018-10-22

The estimated V from full Ulist is

result.mV.full$V

         [,1]     [,2]
[1,] 1.000000 0.509533
[2,] 0.509533 1.000000

The estimated V from reduced Ulist is

result.mV.redu$V

          [,1]      [,2]
[1,] 1.0000000 0.4822162
[2,] 0.4822162 1.0000000

The mash model from the full Ulist has the log likelihood -12302.51.

The mash model from the reduced Ulist has the log likelihood -12304.54.

m.data = mash_set_data(data$Bhat, data$Shat, V = result.mV.redu$V)
m.model = mash(m.data, U.c.full, verbose = FALSE)

Using the V from the reduced Ulist, the mash model using full Ulist has the log likelihood -12302.99.

Different Ulist

I randomly generate 10 positive definite correlation matrices, V. The sample size is 4000. The reduced Ulist only contain identity and singletons.

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

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')
  full.time = system.time(Vhat.mV.full <- mixture.MV(m.data, c(U.c, U.ed), init_V = estimate_null_correlation(m.data, apply_lower_bound = FALSE),
                        tol=1e-4, track_fit = FALSE))['elapsed']
  reduce.time = system.time(Vhat.mV.redu <- mixture.MV(m.data, cov_canonical(m.data, cov_methods = c("identity", "singletons")),
                            init_V = estimate_null_correlation_simple(m.dat),
                            tol=1e-4, track_fit = FALSE))['elapsed']
  saveRDS(list(V.true = Vtrue, V.mV.full = Vhat.mV.full, V.mV.redu = Vhat.mV.redu, data = data, strong=strong,
               full.time = full.time, reduce.time = reduce.time),
          paste0('../output/MASH.mV.result.',t,'.rds'))
}

files = dir("../output/mVUlist/"); 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/mVUlist/", files[[i]], sep=""))
}

for(i in 1:times){
  m.data = mash_set_data(result[[i]]$data$Bhat, result[[i]]$data$Shat)
  result[[i]]$V.simple = estimate_null_correlation_simple(m.data)
  
  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)
  
  m.data.simple = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = result[[i]]$V.simple)
  m.model.simple = mash(m.data.simple, c(U.c,U.ed), verbose = FALSE)
  
  # mV
  # full
  m.model.full = result[[i]]$V.mV.full$mash.model
  
  # reduce
  m.model.redu = result[[i]]$V.mV.redu$mash.model
  m.data.redu = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = result[[i]]$V.mV.redu$V)
  m.model.redu.full = mash(m.data.redu, c(U.c,U.ed), verbose = FALSE)
  
  result[[i]]$m.model = list(m.model.true = m.model.true, m.model.simple = m.model.simple,
                             m.model.full = m.model.full,
                             m.model.redu = m.model.redu, m.model.redu.full = m.model.redu.full)
}

Error

The Frobenius norm is

norm.type='F'
temp = matrix(0,nrow = times, ncol = 3)
for(i in 1:times){
  temp[i, ] = c(norm(result[[i]]$V.simple - result[[i]]$V.true, type = norm.type),
                norm(result[[i]]$V.mV.full$V - result[[i]]$V.true, type = norm.type), 
                norm(result[[i]]$V.mV.redu$V - result[[i]]$V.true, type = norm.type))
}
colnames(temp) = c('simple', 'Full','Reduce')
temp = reshape2::melt(temp[])
colnames(temp) = c('Data', 'Method', 'FrobError')
temp$Data = as.factor(temp$Data)
ggplot(temp, aes(x = Data, y=FrobError, color = Method)) + geom_point()

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Version	Author	Date
eb57faf	zouyuxin	2018-10-22

The spectral norm is

norm.type='2'
temp = matrix(0,nrow = times, ncol = 3)
for(i in 1:times){
  temp[i, ] = c(norm(result[[i]]$V.simple - result[[i]]$V.true, type = norm.type),
                norm(result[[i]]$V.mV.full$V - result[[i]]$V.true, type = norm.type), 
                norm(result[[i]]$V.mV.redu$V - result[[i]]$V.true, type = norm.type))
}
colnames(temp) = c('simple', 'Full','Reduce')
temp = reshape2::melt(temp)
colnames(temp) = c('Data', 'Method', 'SpecError')
temp$Data = as.factor(temp$Data)
ggplot(temp, aes(x = Data, y=SpecError, group = Method, color = Method)) + geom_point()

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Version	Author	Date
eb57faf	zouyuxin	2018-10-22

Time

The total running time for each matrix is

mV.full.time = mV.redu.time = numeric(times)
for(i in 1:times){
  mV.full.time[i] = result[[i]]$full.time
  mV.redu.time[i] = result[[i]]$reduce.time
}
temp = cbind(mV.full.time, mV.redu.time)
colnames(temp) = c('Full', 'Reduce')
row.names(temp) = 1:10

temp = reshape2::melt(temp)
colnames(temp) = c('Data', 'Method', 'Time')
temp$Data = as.factor(temp$Data)
ggplot(temp, aes(x = Data, y=Time, group = Method, color = Method)) + geom_point()

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Version	Author	Date
eb57faf	zouyuxin	2018-10-22

mash log likelihood

temp = matrix(0,nrow = times, ncol = 5)
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.full),
                get_loglik(result[[i]]$m.model$m.model.redu), 
                get_loglik(result[[i]]$m.model$m.model.redu.full))
}
colnames(temp) = c('True','simple', 'Full','Reduce', 'Reduce+Full')
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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Version	Author	Date
eb57faf	zouyuxin	2018-10-22

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.true.seq = ROC.table(data.true, result.model$m.model.true)
  m.simple.seq = ROC.table(data.true, result.model$m.model.simple)
  m.mV.full.seq = ROC.table(data.true, result.model$m.model.full)
  m.mV.reduce.seq = ROC.table(data.true, result.model$m.model.redu)
  m.mV.reduce.full.seq = ROC.table(data.true, result.model$m.model.redu.full)

  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.mV.full.seq[,'FPR'], m.mV.full.seq[,'TPR'], col='red', lwd = 1.5)
  lines(m.mV.reduce.seq[,'FPR'], m.mV.reduce.seq[,'TPR'], col='darkorchid', lwd = 1.5)
  lines(m.mV.reduce.full.seq[,'FPR'], m.mV.reduce.full.seq[,'TPR'], col='darkgoldenrod', lwd = 1.5)
  lines(m.simple.seq[,'FPR'], m.simple.seq[,'TPR'], col='blue', lwd = 1.5)
  legend('bottomright', c('True','simple', 'Full', 'Reduce', 'Reduce+Full'),col=c('black','blue', 'red','darkorchid','darkgoldenrod'),
           lty=c(1,1,1,1,1), lwd=c(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, ' '))
}

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eb57faf	zouyuxin	2018-10-22

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eb57faf	zouyuxin	2018-10-22

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')
}

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eb57faf	zouyuxin	2018-10-22

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Version	Author	Date
eb57faf	zouyuxin	2018-10-22

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Version	Author	Date
eb57faf	zouyuxin	2018-10-22

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Version	Author	Date
eb57faf	zouyuxin	2018-10-22

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Version	Author	Date
eb57faf	zouyuxin	2018-10-22

Session information

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