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

We want to estimate \(\rho\) \[ \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) | \left(\begin{matrix} x \\ y \end{matrix} \right) \sim N(\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) ; \left(\begin{matrix} x \\ y \end{matrix} \right), \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right)) \] \[ \left(\begin{matrix} x \\ y \end{matrix} \right) \sim \sum_{p=0}^{P} \pi_{p} N( \left(\begin{matrix} x \\ y \end{matrix} \right); 0, \Sigma_{p} ) \] \(\Rightarrow\) \[ \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) \sim \sum_{p=0}^{P} \pi_{p} N( \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right); 0, \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + \Sigma_{p} ) \] \[ \Omega_{p} = \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + \Sigma_{p} = \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + \left( \begin{matrix} \sigma_{p11} & \sigma_{p12} \\ \sigma_{p21} & \sigma_{p22} \end{matrix} \right) = \left( \begin{matrix} 1+\sigma_{p11} & \rho+\sigma_{p12} \\ \rho+\sigma_{p21} & 1+\sigma_{p22} \end{matrix} \right) \] Let \(\omega_{p11} = \sqrt{1+\sigma_{p11}}\), \(\omega_{p22} = \sqrt{1+\sigma_{p22}}\), \(\phi_{p}=\frac{\rho+\sigma_{p12}}{\omega_{k11}\omega_{p22}}\)

MLE

The loglikelihood is (with penalty) \[ l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{p=0}^{P} \pi_{p}N(x_{i}; 0, \Omega_{p}) + \sum_{p=0}^{P} (\lambda_{p}-1) \log \pi_{p} \]

The penalty on \(\pi\) encourages over-estimation of \(\pi_{0}\), \(\lambda_{p}\geq 1\).

\[ l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{p=0}^{P} \pi_{p}\frac{1}{2\pi\omega_{p11}\omega_{p22}\sqrt{1-\phi_{p}^2}} \exp\left( -\frac{1}{2(1-\phi_{p}^2)}\left[ \frac{x_{i}^2}{\omega_{p11}^2} + \frac{y_{i}^2}{\omega_{p22}^2} - \frac{2\phi_{p}x_{i}y_{i}}{\omega_{p11}\omega_{p22}}\right] \right) + \sum_{p=0}^{P} (\lambda_{p}-1) \log \pi_{p} \]

Note: This probelm is convex with respect to \(\pi\). In terms of \(\rho\), the covenxity depends on the data.

Algorithm:

Input: X, init_rho, Ulist
Given rho, estimate pi by max loglikelihood (convex problem)
Compute loglikelihood
delta = 1
while delta > tol
  Given pi, estimate rho by max loglikelihood (optim function)
  Given rho, estimate pi by max loglikelihood (convex problem)
  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)
}
#' @title Optimize rho with several initial values
#' @param X data, Z scores
#' @param Ulist a list of covariance matrices (expand)
#' @param init_rho initial value for rho. The user could provide several initial values as a vector.
#' @param prior indicates what penalty to use on the likelihood, if any
#' @return list of result
#' \item{result}{result from the rho which gives the highest log likelihood}
#' \item{status}{whether the result is global max or local max}
#' \item{loglik}{the loglikelihood value}
#' \item{rho}{the estimated rho}
#' \item{time}{the running time for each initial rho}
#'
optimize_pi_rho_times <- function(X, Ulist, init_rho=0, prior=c("nullbiased", "uniform"), tol=1e-5){
  times = length(init_rho)
  result = list()
  loglik = c()
  rho = c()
  time.t = c()
  for(i in 1:times){
    out.time = system.time(result[[i]] <- optimize_pi_rho(X, Ulist,
                                                          init_rho=init_rho[i],
                                                          prior=prior,
                                                          tol=tol))
    time.t = c(time.t, out.time['elapsed'])
    loglik = c(loglik, tail(result[[i]]$loglik, n=1))
    rho = c(rho, result[[i]]$rho)
  }
  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, time = time.t, rho=rho))
}

#' @title optimize rho
#' @param X data, Z scores
#' @param Ulist a list of covariance matrices
#' @param init_rho an initial value for rho
#' @param tol tolerance for optimizaiton stop
#' @param prior indicates what penalty to use on the likelihood, if any
#' @return list of result
#' \item{pi}{estimated pi}
#' \item{rho}{estimated rho}
#' \item{loglik}{the loglikelihood value at each iteration}
#' \item{niter}{the number of iteration}
#'
optimize_pi_rho <- function(X, Ulist, init_rho=0, tol=1e-5, prior=c("nullbiased", "uniform")){
  prior <- match.arg(prior)
  if(length(Ulist) <= 1){
    stop('Please provide more U! With only one U, the correlation could be estimated directly using mle.')
  }
  prior <- mashr:::set_prior(length(Ulist), prior)

  Sigma <- get_sigma(init_rho, 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)
  pi_s <- mashr:::optimize_pi(exp(lL$loglik_matrix),prior=prior,optmethod='mixSQP')

  log_liks <- c()
  ll       <- compute.log.lik(lL, pi_s, prior)
  log_liks <- c(log_liks, ll)
  delta.ll <- 1
  niter <- 0
  rho_s <- init_rho
  while( delta.ll > tol){
    # max_rho
    rho_s <- optim(rho_s, optimize_rho, lower = -1, upper = 1, X = X, Ulist=Ulist, pi_s = pi_s, prior = prior, method = 'Brent')$par

    Sigma <- get_sigma(rho_s, 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)

    # max pi
    pi_s <- mashr:::optimize_pi(exp(lL$loglik_matrix),prior=prior,optmethod='mixSQP')

    # compute loglike
    ll <- compute.log.lik(lL, pi_s, prior)
    log_liks <- c(log_liks, ll)
    # Update delta
    delta.ll <- log_liks[length(log_liks)] - log_liks[length(log_liks)-1]
    niter <- niter + 1
  }
  return(list(pi = pi_s, rho=rho_s, loglik = log_liks, niter = niter))
}

optimize_rho <- function(rho, X, Ulist, pi_s, prior){
  Sigma <- get_sigma(rho, 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)

  return(-compute.log.lik(lL, pi_s, prior))
}

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.optim <- optimize_pi_rho_times(data$Bhat, xUlist, init_rho = 0)

The log likelihood at each iteration:

plot(result.optim$result$loglik, ylab = 'log likelihood', xlab = 'iteration')

Version Author Date
e0808e1 zouyuxin 2018-10-09

The estimated \(\rho\) is 0.5062745. The running time is 109.119 seconds.

m.data.optim = mash_set_data(data$Bhat, data$Shat, V = matrix(c(1,result.optim$rho,result.optim$rho,1),2,2))
U.c = cov_canonical(m.data.optim)
m.optim = mash(m.data.optim, 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 0.5820856.

The ROC curve:

m.data.correct = mash_set_data(data$Bhat, data$Shat, V=Sigma)
m.correct = mash(m.data.correct, U.c, verbose = FALSE)
m.correct.seq = ROC.table(data$B, m.correct)
m.optim.seq = ROC.table(data$B, m.optim)

Version Author Date
e0808e1 zouyuxin 2018-10-09
fcfbe32 zouyuxin 2018-10-09
a10e3f3 zouyuxin 2018-09-20 

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] knitr_1.30       kableExtra_1.3.1 mashr_0.2.40     ashr_2.2-51     
[5] workflowr_1.6.2 

loaded via a namespace (and not attached):
 [1] xfun_0.19         lattice_0.20-41   colorspace_2.0-0  vctrs_0.3.5      
 [5] htmltools_0.5.0   viridisLite_0.3.0 yaml_2.2.1        rlang_0.4.9      
 [9] mixsqp_0.3-46     later_1.1.0.1     pillar_1.4.7      glue_1.4.2       
[13] lifecycle_0.2.0   plyr_1.8.6        stringr_1.4.0     munsell_0.5.0    
[17] rvest_0.3.6       mvtnorm_1.1-1     evaluate_0.14     httpuv_1.5.4     
[21] invgamma_1.1      irlba_2.3.3       Rcpp_1.0.5        promises_1.1.1   
[25] scales_1.1.1      rmeta_3.0         webshot_0.5.2     truncnorm_1.0-8  
[29] abind_1.4-5       fs_1.5.0          digest_0.6.27     stringi_1.5.3    
[33] grid_4.0.3        rprojroot_2.0.2   tools_4.0.3       magrittr_2.0.1   
[37] tibble_3.0.4      crayon_1.3.4      whisker_0.4       pkgconfig_2.0.3  
[41] ellipsis_0.3.1    MASS_7.3-53       Matrix_1.2-18     SQUAREM_2020.5   
[45] xml2_1.3.2        assertthat_0.2.1  rmarkdown_2.5     httr_1.4.2       
[49] rstudioapi_0.13   R6_2.5.0          git2r_0.27.1      compiler_4.0.3