Last updated: 2018-10-09

workflowr checks: (Click a bullet for more information)

  • ✔ R Markdown file: up-to-date

    Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.

  • ✔ Environment: empty

    Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

  • ✔ Seed: set.seed(1)

    The command set.seed(1) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

  • ✔ Session information: recorded

    Great job! Recording the operating system, R version, and package versions is critical for reproducibility.

  • ✔ Repository version: 8a7ee7c

    Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated.

    Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated: 
    
Ignored files:
    Ignored:    .DS_Store
    Ignored:    .Rhistory
    Ignored:    .Rproj.user/
    Ignored:    analysis/.DS_Store
    Ignored:    analysis/.Rhistory
    Ignored:    analysis/include/.DS_Store
    Ignored:    code/.DS_Store
    Ignored:    data/.DS_Store
    Ignored:    docs/.DS_Store
    Ignored:    output/.DS_Store

Untracked files:
    Untracked:  analysis/Classify.Rmd
    Untracked:  analysis/EstimateCorEM3W2.Rmd
    Untracked:  analysis/EstimateCorMaxEMGD.Rmd
    Untracked:  analysis/EstimateCorMaxGD.Rmd
    Untracked:  analysis/EstimateCorOptimEM.Rmd
    Untracked:  analysis/EstimateCorPrior.Rmd
    Untracked:  analysis/EstimateCorSol.Rmd
    Untracked:  analysis/HierarchicalFlashSim.Rmd
    Untracked:  analysis/MashLowSignalGTEx4.Rmd
    Untracked:  analysis/Mash_GTEx.Rmd
    Untracked:  analysis/MeanAsh.Rmd
    Untracked:  analysis/OutlierDetection.Rmd
    Untracked:  analysis/OutlierDetection2.Rmd
    Untracked:  analysis/OutlierDetection3.Rmd
    Untracked:  analysis/OutlierDetection4.Rmd
    Untracked:  analysis/mash_missing_row.Rmd
    Untracked:  code/GTExNullModel.R
    Untracked:  code/MASH.result.1.rds
    Untracked:  code/MashClassify.R
    Untracked:  code/MashCorResult.R
    Untracked:  code/MashNULLCorResult.R
    Untracked:  code/MashSource.R
    Untracked:  code/Weight_plot.R
    Untracked:  code/addemV.R
    Untracked:  code/estimate_cor.R
    Untracked:  code/generateDataV.R
    Untracked:  code/johnprocess.R
    Untracked:  code/sim_mean_sig.R
    Untracked:  code/summary.R
    Untracked:  data/Blischak_et_al_2015/
    Untracked:  data/scale_data.rds
    Untracked:  docs/figure/Classify.Rmd/
    Untracked:  docs/figure/OutlierDetection.Rmd/
    Untracked:  docs/figure/OutlierDetection2.Rmd/
    Untracked:  docs/figure/OutlierDetection3.Rmd/
    Untracked:  docs/figure/Test.Rmd/
    Untracked:  docs/figure/mash_missing_whole_row_5.Rmd/
    Untracked:  docs/include/
    Untracked:  output/AddEMV/
    Untracked:  output/CovED_UKBio_strong.rds
    Untracked:  output/CovED_UKBio_strong_Z.rds
    Untracked:  output/Flash_UKBio_strong.rds
    Untracked:  output/GTExNULLres/
    Untracked:  output/GTEx_2.5_nullData.rds
    Untracked:  output/GTEx_2.5_nullModel.rds
    Untracked:  output/GTEx_2.5_nullPermData.rds
    Untracked:  output/GTEx_2.5_nullPermModel.rds
    Untracked:  output/GTEx_3.5_nullData.rds
    Untracked:  output/GTEx_3.5_nullModel.rds
    Untracked:  output/GTEx_3.5_nullPermData.rds
    Untracked:  output/GTEx_3.5_nullPermModel.rds
    Untracked:  output/GTEx_3_nullData.rds
    Untracked:  output/GTEx_3_nullModel.rds
    Untracked:  output/GTEx_3_nullPermData.rds
    Untracked:  output/GTEx_3_nullPermModel.rds
    Untracked:  output/GTEx_4.5_nullData.rds
    Untracked:  output/GTEx_4.5_nullModel.rds
    Untracked:  output/GTEx_4.5_nullPermData.rds
    Untracked:  output/GTEx_4.5_nullPermModel.rds
    Untracked:  output/GTEx_4_nullData.rds
    Untracked:  output/GTEx_4_nullModel.rds
    Untracked:  output/GTEx_4_nullPermData.rds
    Untracked:  output/GTEx_4_nullPermModel.rds
    Untracked:  output/MASH.10.em2.result.rds
    Untracked:  output/MASH.10.mle.result.rds
    Untracked:  output/MASHNULL.V.result.1.rds
    Untracked:  output/MASHNULL.V.result.10.rds
    Untracked:  output/MASHNULL.V.result.11.rds
    Untracked:  output/MASHNULL.V.result.12.rds
    Untracked:  output/MASHNULL.V.result.13.rds
    Untracked:  output/MASHNULL.V.result.14.rds
    Untracked:  output/MASHNULL.V.result.15.rds
    Untracked:  output/MASHNULL.V.result.16.rds
    Untracked:  output/MASHNULL.V.result.17.rds
    Untracked:  output/MASHNULL.V.result.18.rds
    Untracked:  output/MASHNULL.V.result.19.rds
    Untracked:  output/MASHNULL.V.result.2.rds
    Untracked:  output/MASHNULL.V.result.20.rds
    Untracked:  output/MASHNULL.V.result.3.rds
    Untracked:  output/MASHNULL.V.result.4.rds
    Untracked:  output/MASHNULL.V.result.5.rds
    Untracked:  output/MASHNULL.V.result.6.rds
    Untracked:  output/MASHNULL.V.result.7.rds
    Untracked:  output/MASHNULL.V.result.8.rds
    Untracked:  output/MASHNULL.V.result.9.rds
    Untracked:  output/MashCorSim--midway/
    Untracked:  output/Mash_EE_Cov_0_plusR1.rds
    Untracked:  output/UKBio_mash_model.rds
    Untracked:  output/result.em.rds

Unstaged changes:
    Deleted:    analysis/EstimateCorMax.Rmd
    Modified:   analysis/EstimateCorMaxEM2.Rmd
    Deleted:    analysis/EstimateCorMaxEMV.Rmd
    Modified:   analysis/EstimateCorMaxMV.Rmd
    Modified:   analysis/EstimateCorMaxMash.Rmd
    Deleted:    analysis/MashLowSignalGTEx3.5P.Rmd
    Modified:   analysis/Mash_UKBio.Rmd
    Modified:   analysis/mash_missing_samplesize.Rmd
    Modified:   output/Flash_T2_0.rds
    Modified:   output/Flash_T2_0_mclust.rds
    Modified:   output/Mash_model_0_plusR1.rds
    Modified:   output/PresiAddVarCol.rds
    Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.
Expand here to see past versions:
    File Version Author Date Message
    Rmd 8a7ee7c zouyuxin 2018-10-09 wflow_publish(c(“analysis/EstimateCorEM.Rmd”, “analysis/EstimateCorEM2.Rmd”, “analysis/EstimateCorEM3.Rmd”))
    html 4db8ea6 zouyuxin 2018-10-09 Build site.
    Rmd 7c1f5a9 zouyuxin 2018-10-09 wflow_publish(“analysis/EstimateCorEM.Rmd”)
    html f79c6e4 zouyuxin 2018-10-09 Build site.
    Rmd 7c9f970 zouyuxin 2018-10-09 wflow_publish(“analysis/EstimateCorEM.Rmd”)
    html 6c38c31 zouyuxin 2018-10-09 Build site.
    Rmd cd7ebf9 zouyuxin 2018-10-09 wflow_publish(“analysis/EstimateCorEM.Rmd”)
    html 597d99e zouyuxin 2018-09-20 Build site.
    Rmd 2c5aa17 zouyuxin 2018-09-20 wflow_publish(“analysis/EstimateCorEM.Rmd”) 

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

EM

E step

\[ P(X,\mathbf{z}|\rho, \pi) = \prod_{i=1}^{n} \prod_{p=0}^{P}\left[\pi_{p}N(x_{i}; 0, \Omega_{p})\right]^{\mathbb{I}(z_{i}=p)} \prod_{p=0}^{P}\pi_{p}^{\lambda_{p}-1} \]

\[ \mathbb{E}_{\mathbf{z}|X} \log P(X,\mathbf{z}|\rho, \pi) = \sum_{i=1}^{n} \sum_{p=0}^{P} P(z_{i}=p|X)\left[ \log \pi_{p} + \log N(x_{i}; 0, \Omega_{p})\right] + \sum_{p=0}^{P} (\lambda_{p}-1)\log \pi_{p} \]

\[ \gamma_{z_{i}}(p) = P(z_{i}=p|X_{i}) = \frac{\pi_{p}N(x_{i}; 0, \Omega_{p})}{\sum_{p'=0}^{P}\pi_{p'}N(x_{i}; 0, \Omega_{p'})} \]

M step

\(\pi\): \[ \sum_{i=1}^{n} \gamma_{z_{i}}(p) \frac{1}{\pi_{p}} + \frac{\lambda_{p}-1}{\pi_{p}} - \lambda = 0 \quad \rightarrow \pi_{p} = \frac{1}{\lambda} \left(\sum_{i=1}^{n} \gamma_{z_{i}}(p) + \lambda_{p}-1\right) \quad \lambda = n + \sum_{p=1}^{P}\lambda_{p} - P \]

\[ \hat{\pi}_{p} = \frac{\sum_{i=1}^{n} \gamma_{z_{i}}(p) + \lambda_{p} - 1 }{n + \sum_{p=1}^{P}\lambda_{p} - P } \]

\(\rho\): \[ \begin{align*} f(\rho) &= \sum_{i=1}^{n} \sum_{p=1}^{P} \gamma_{z_{i}}(p)\left[ -\frac{1}{2}\log (1-\phi_{p}^2)-\frac{1}{2(1-\phi_{p}^2)}\left[ \frac{x_{i}^2}{\omega_{p11}^2} + \frac{y_{i}^2}{\omega_{p22}^2} - \frac{2\phi_{p}x_{i}y_{i}}{\omega_{p11}\omega_{p22}}\right] \right]\\ f(\rho)' &= \sum_{i=1}^{n} \sum_{p=1}^{P} \gamma_{z_{i}}(p)\left[ \frac{\phi_{p}}{1-\phi_{p}^2}-\frac{\phi_{p}}{(1-\phi_{p}^2)^2}\left[ \frac{x_{i}^2}{\omega_{p11}^2} + \frac{y_{i}^2}{\omega_{p22}^2}\right] - \frac{\phi_{p}+1}{(1-\phi_{p}^2)^2}\frac{x_{i}y_{i}}{\omega_{p11}\omega_{p22}}\right]\frac{1}{\omega_{p11}\omega_{p22}} = 0 \end{align*} \] \(\phi_p = \frac{\rho + \sigma_{p12}}{\omega_{p11}\omega_{p22}}\), \(\phi_{p}\) is a function of \(\rho\).

Algorithm:

Input: X, Ulist, init_rho, init_pi
Compute loglikelihood
delta = 1
while delta > tol
  E step: update z
  M step: update pi, update rho
  Compute loglikelihood
  Update delta
#' @param rho the off diagonal element of V, 2 by 2 correlation matrix
#' @param Ulist a list of covariance matrices, U_{k}
get_sigma <- function(rho, Ulist){
  V <- matrix(c(1,rho,rho,1), 2,2)
  lapply(Ulist, function(U) U + V)
}

penalty <- function(prior, pi_s){
  subset <- (prior != 1.0)
  sum((prior-1)[subset]*log(pi_s[subset]))
}

#' @title compute log likelihood
#' @param L log likelihoods,
#' where the (i,k)th entry is the log probability of observation i
#' given it came from component k of g
#' @param p the vector of mixture proportions
#' @param prior the weight for the penalty
compute.log.lik <- function(lL, p, prior){
  p = normalize(pmax(0,p))
  temp = log(exp(lL$loglik_matrix) %*% p)+lL$lfactors
  return(sum(temp) + penalty(prior, p))
  # return(sum(temp))
}

normalize <- function(x){
  x/sum(x)
}
mixture.EM.times <- function(X, Ulist, init_rho=0, init_pi=NULL, prior = c('nullbiased', 'uniform'), control = list()){
  times = length(init_rho)
  result = list()
  loglik = c()
  rho = c()
  time.t = c()
  converge.status = c()
  for(i in 1:times){
    out.time = system.time(result[[i]] <- mixture.EM(X, Ulist,
                                                     init_pi=init_pi,
                                                     init_rho=init_rho[i],
                                                     prior=prior,
                                                     control = control))
    time.t = c(time.t, out.time['elapsed'])
    loglik = c(loglik, result[[i]]$loglik)
    rho = c(rho, result[[i]]$rhohat)
    converge.status = c(converge.status, result[[i]]$converged)
  }
  if(abs(max(loglik) - min(loglik)) < 1e-4){
    status = 'global'
  }else{
    status = 'local'
  }
  ind = which.max(loglik)
  return(list(result = result[[ind]], status = status, loglik = loglik, rho=rho, time = time.t, converge.status = converge.status))
}

mixture.EM <- function(X, Ulist, init_rho=0, init_pi = NULL, prior = c('nullbiased', 'uniform'), control = list()) {
  prior = match.arg(prior)
  prior <- mashr:::set_prior(length(Ulist), prior)
  k = length(Ulist)
  if (is.null(init_pi)){
    init_pi <- rep(1/k,k)
  }
  control = ashr:::set_control_squarem(control,nrow(X))
  res = SQUAREM::squarem(par=c(init_pi, init_rho),fixptfn=fixpoint_EM, objfn=negpenloglik,X=X, Ulist=Ulist, prior=prior, control=control)

  return(list(pihat = normalize(pmax(0,head(res$par, -1))), rhohat = tail(res$par, 1), loglik=-res$value.objfn, niter = res$iter, converged=res$convergence, control=control))
}

fixpoint_EM = function(par, X, Ulist, prior){
  rho = tail(par,1)
  pi_s = head(par, -1)
  pi_s = normalize(pmax(0,pi_s)) #avoid occasional problems with negative pis due to rounding

  # compute L
  Sigma <- get_sigma(rho, Ulist)
  L <- t(plyr::laply(Sigma,function(U){mvtnorm::dmvnorm(x=X,sigma=U)}))

  # E
  m  = t(pi_s * t(L)) # matrix_lik is n by k; so this is also n by k
  m.rowsum = rowSums(m)
  classprob = m/m.rowsum #an n by k matrix

  # M
  pinew = normalize(colSums(classprob) + prior - 1)

  rhonew = optimize(EMloglikelihood, interval = c(-1,1), maximum = TRUE, X = X, Ulist = Ulist, z = classprob)$maximum

  return(c(pinew,rhonew))
}

EMloglikelihood = function(rho, X, Ulist, z){
  Sigma = get_sigma(rho, Ulist)
  L = t(plyr::laply(Sigma,function(U){mvtnorm::dmvnorm(x=X,sigma=U, log=TRUE)}))
  sum(L * z)
}

negpenloglik = function(par, X, Ulist, prior){
  Sigma <- get_sigma(tail(par,1), Ulist)
  lL <- t(plyr::laply(Sigma,function(U){mvtnorm::dmvnorm(x=X,sigma=U, log=TRUE)}))
  lfactors <- apply(lL,1,max)
  matrix_llik <- lL - lfactors
  lL = list(loglik_matrix = matrix_llik,
            lfactors   = lfactors)
  ll <- compute.log.lik(lL, head(par, -1), prior)
  return(-ll)
}

Data

\[ \hat{\beta}|\beta \sim N_{2}(\hat{\beta}; \beta, \left(\begin{matrix} 1 & 0.5 \\ 0.5 & 1 \end{matrix}\right)) \]

\[ \beta \sim \frac{1}{4}\delta_{0} + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}\right)) \]

n = 4000

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)
grid = mashr:::autoselect_grid(m.data, sqrt(2))
Ulist = mashr:::normalize_Ulist(U.c)
xUlist = mashr:::expand_cov(Ulist,grid,usepointmass =  TRUE)

result.em <- mixture.EM.times(m.data$Bhat, xUlist)
saveRDS(result.em, '../output/result.em.rds')
result.em = readRDS('../output/result.em.rds')

The estimated \(\rho\) is 0.5066755. The running time is 739.486 seconds.

m.data.em = mash_set_data(data$Bhat, data$Shat, V = matrix(c(1,result.em$rho,result.em$rho,1),2,2))
U.c = cov_canonical(m.data.em)
m.em = mash(m.data.em, U.c, verbose= FALSE)
null.ind = which(apply(data$B,1,sum) == 0)

The log likelihood is -12302.54. There are 26 significant samples, 0 false positives. The RRMSE is 0.582108.

Session information

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

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] mashr_0.2-15 ashr_2.2-14 

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.19      knitr_1.20        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.1.1   REBayes_1.3      
 [7] MASS_7.3-50       pscl_1.5.2        doParallel_1.0.14
[10] SQUAREM_2017.10-1 lattice_0.20-35   foreach_1.4.4    
[13] plyr_1.8.4        stringr_1.3.1     tools_3.5.1      
[16] parallel_3.5.1    grid_3.5.1        R.oo_1.22.0      
[19] rmeta_3.0         git2r_0.23.0      htmltools_0.3.6  
[22] iterators_1.0.10  assertthat_0.2.0  abind_1.4-5      
[25] yaml_2.2.0        rprojroot_1.3-2   digest_0.6.15    
[28] Matrix_1.2-14     codetools_0.2-15  R.utils_2.6.0    
[31] evaluate_0.11     rmarkdown_1.10    stringi_1.2.4    
[34] compiler_3.5.1    Rmosek_8.0.69     backports_1.1.2  
[37] R.methodsS3_1.7.1 mvtnorm_1.0-8     truncnorm_1.0-8

This reproducible R Markdown analysis was created with workflowr 1.1.1