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<h1 class="title toc-ignore">Estimate cor – optim</h1>
<h4 class="author"><em>Yuxin Zou</em></h4>
<h4 class="date"><em>2018-07-24</em></h4>
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<p><strong>Last updated:</strong> 2018-10-09</p>
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</code></pre>
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<p></details></p>
<hr />
<pre class="r"><code>library(mashr)</code></pre>
<pre><code>Loading required package: ashr</code></pre>
<pre class="r"><code>source('../code/generateDataV.R')
source('../code/summary.R')
library(kableExtra)
library(knitr)</code></pre>
<p>We want to estimate <span class="math inline">\(\rho\)</span> <span class="math display">\[
\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) | \left(\begin{matrix} x \\ y \end{matrix} \right) \sim N(\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) ; \left(\begin{matrix} x \\ y \end{matrix} \right), \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right))
\]</span> <span class="math display">\[
\left(\begin{matrix} x \\ y \end{matrix} \right) \sim \sum_{p=0}^{P} \pi_{p} N( \left(\begin{matrix} x \\ y \end{matrix} \right); 0, \Sigma_{p} )
\]</span> <span class="math inline">\(\Rightarrow\)</span> <span class="math display">\[
\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) \sim \sum_{p=0}^{P} \pi_{p} N( \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right); 0, \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + \Sigma_{p} )
\]</span> <span class="math display">\[
\Omega_{p} = \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + \Sigma_{p} = \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + \left( \begin{matrix} \sigma_{p11} & \sigma_{p12} \\ \sigma_{p21} & \sigma_{p22} \end{matrix} \right) = \left( \begin{matrix} 1+\sigma_{p11} & \rho+\sigma_{p12} \\ \rho+\sigma_{p21} & 1+\sigma_{p22} \end{matrix} \right)
\]</span> Let <span class="math inline">\(\omega_{p11} = \sqrt{1+\sigma_{p11}}\)</span>, <span class="math inline">\(\omega_{p22} = \sqrt{1+\sigma_{p22}}\)</span>, <span class="math inline">\(\phi_{p}=\frac{\rho+\sigma_{p12}}{\omega_{k11}\omega_{p22}}\)</span></p>
<div id="mle" class="section level2">
<h2>MLE</h2>
<p>The loglikelihood is (with penalty) <span class="math display">\[
l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{p=0}^{P} \pi_{p}N(x_{i}; 0, \Omega_{p}) + \sum_{p=0}^{P} (\lambda_{p}-1) \log \pi_{p}
\]</span></p>
<p>The penalty on <span class="math inline">\(\pi\)</span> encourages over-estimation of <span class="math inline">\(\pi_{0}\)</span>, <span class="math inline">\(\lambda_{p}\geq 1\)</span>.</p>
<p><span class="math display">\[
l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{p=0}^{P} \pi_{p}\frac{1}{2\pi\omega_{p11}\omega_{p22}\sqrt{1-\phi_{p}^2}} \exp\left( -\frac{1}{2(1-\phi_{p}^2)}\left[ \frac{x_{i}^2}{\omega_{p11}^2} + \frac{y_{i}^2}{\omega_{p22}^2} - \frac{2\phi_{p}x_{i}y_{i}}{\omega_{p11}\omega_{p22}}\right] \right) + \sum_{p=0}^{P} (\lambda_{p}-1) \log \pi_{p}
\]</span></p>
<p><strong>Note:</strong> This probelm is convex with respect to <span class="math inline">\(\pi\)</span>. In terms of <span class="math inline">\(\rho\)</span>, the covenxity depends on the data.</p>
<p>Algorithm:</p>
<pre class="text"><code>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</code></pre>
<pre class="r"><code>#' @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)
}</code></pre>
<pre class="r"><code>#' @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$loglik_matrix),prior=prior,optmethod='mixIP')
log_liks <- c()
ll <- compute.log.lik(lL, pi_s, prior)
log_liks <- c(log_liks, ll)
delta.ll <- 1
niter <- 0
rho_s <- init_rho
while( delta.ll > tol){
# max_rho
rho_s <- optim(rho_s, optimize_rho, lower = -1, upper = 1, X = X, Ulist=Ulist, pi_s = pi_s, prior = prior, method = 'Brent')$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='mixIP')
# 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))
}</code></pre>
</div>
<div id="data" class="section level2">
<h2>Data</h2>
<p><span class="math display">\[
\hat{\beta}|\beta \sim N_{2}(\hat{\beta}; \beta, \left(\begin{matrix} 1 & 0.5 \\
0.5 & 1 \end{matrix}\right))
\]</span></p>
<p><span class="math display">\[
\beta \sim \frac{1}{4}\delta_{0} + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 0 \\
0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 0 & 0 \\
0 & 1 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 1 \\
1 & 1 \end{matrix}\right))
\]</span></p>
<p>n = 4000</p>
<pre class="r"><code>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)</code></pre>
<pre class="r"><code>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$Bhat, xUlist, init_rho = 0)</code></pre>
<p>The log likelihood at each iteration:</p>
<pre class="r"><code>plot(result.optim$result$loglik, ylab = 'log likelihood', xlab = 'iteration')</code></pre>
<p><img src="figure/EstimateCorOptim.Rmd/unnamed-chunk-7-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of unnamed-chunk-7-1.png:</em></summary>
<table style="border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/e0808e1af2c300180c9e69d73f4820422b83f3b3/docs/figure/EstimateCorOptim.Rmd/unnamed-chunk-7-1.png" target="_blank">e0808e1</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-10-08
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>The estimated <span class="math inline">\(\rho\)</span> is 0.5062776. The running time is 64.397 seconds.</p>
<pre class="r"><code>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$B,1,sum) == 0)</code></pre>
<p>The log likelihood is -12302.54. There are 26 significant samples, 0 false positives. The RRMSE is 0.582086.</p>
<p>The ROC curve:</p>
<pre class="r"><code>m.data.correct = mash_set_data(data$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)</code></pre>
<p><img src="figure/EstimateCorOptim.Rmd/unnamed-chunk-10-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of unnamed-chunk-10-1.png:</em></summary>
<table style="border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/zouyuxin/mash_application/blob/a10e3f331e17ad54f38b28b5ad4a2336c8e7598f/docs/figure/EstimateCorOptim.Rmd/unnamed-chunk-10-1.png" target="_blank">a10e3f3</a>
</td>
<td style="text-align:left;">
zouyuxin
</td>
<td style="text-align:left;">
2018-09-20
</td>
</tr>
</tbody>
</table>
<p></details></p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>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] knitr_1.20 kableExtra_0.9.0 mashr_0.2-15 ashr_2.2-14
loaded via a namespace (and not attached):
[1] Rcpp_0.12.19 pillar_1.3.0 compiler_3.5.1
[4] git2r_0.23.0 plyr_1.8.4 workflowr_1.1.1
[7] R.methodsS3_1.7.1 R.utils_2.6.0 iterators_1.0.10
[10] tools_3.5.1 digest_0.6.15 viridisLite_0.3.0
[13] tibble_1.4.2 evaluate_0.11 lattice_0.20-35
[16] pkgconfig_2.0.2 rlang_0.2.2 Matrix_1.2-14
[19] foreach_1.4.4 rstudioapi_0.7 yaml_2.2.0
[22] parallel_3.5.1 mvtnorm_1.0-8 xml2_1.2.0
[25] httr_1.3.1 stringr_1.3.1 REBayes_1.3
[28] hms_0.4.2 rprojroot_1.3-2 grid_3.5.1
[31] R6_2.2.2 rmarkdown_1.10 rmeta_3.0
[34] readr_1.1.1 magrittr_1.5 whisker_0.3-2
[37] scales_1.0.0 backports_1.1.2 codetools_0.2-15
[40] htmltools_0.3.6 MASS_7.3-50 rvest_0.3.2
[43] abind_1.4-5 assertthat_0.2.0 colorspace_1.3-2
[46] stringi_1.2.4 Rmosek_8.0.69 munsell_0.5.0
[49] doParallel_1.0.14 pscl_1.5.2 truncnorm_1.0-8
[52] SQUAREM_2017.10-1 crayon_1.3.4 R.oo_1.22.0 </code></pre>
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