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<h1 class="title toc-ignore">Estimate cor—max MASH</h1>
<h4 class="author"><em>Yuxin Zou</em></h4>
<h4 class="date"><em>2018-07-25</em></h4>
</div>
<!-- Update knitr chunk options -->
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<p><strong>Last updated:</strong> 2018-08-15</p>
<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/estimate_cor.R')
source('../code/summary.R')
library(knitr)
library(kableExtra)</code></pre>
<p>Apply the max methods for correlation matrix on mash data.</p>
<p>The estimated V from MLE(<code>optim</code> function) and EM perform better than the truncated correlation (error, mash log likelihood, ROC).</p>
<p>Comparing the estimated V from MLE and EM, EM algorithm tends to compute faster, and the estimated correlation is slightly better than the one from MLE in terms of estimation error, mash log likelihood, ROC.</p>
<div id="one-example" class="section level1">
<h1>One example</h1>
<p><span class="math display">\[
\hat{\beta}|\beta \sim N_{3}(\hat{\beta}; \beta, \left(\begin{matrix} 1 & 0.7 & 0.2 \\
0.7 & 1 & 0.4 \\
0.2 & 0.4 & 1 \end{matrix}\right))
\]</span></p>
<p><span class="math display">\[
\beta \sim \frac{1}{4}\delta_{0} + \frac{1}{4}N_{3}(0, \left(\begin{matrix} 1 & 0 &0\\
0 & 0 & 0 \\
0 & 0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{3}(0, \left(\begin{matrix} 1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{3}(0, \left(\begin{matrix} 1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1 \end{matrix}\right))
\]</span></p>
<pre class="r"><code>set.seed(1)
Sigma = cbind(c(1,0.7,0.2), c(0.7,1,0.4), c(0.2,0.4,1))
U0 = matrix(0,3,3)
U1 = matrix(0,3,3); U1[1,1] = 1
U2 = diag(3); U2[3,3] = 0
U3 = matrix(1,3,3)
data = generate_data(n=4000, p=3, V=Sigma, Utrue = list(U0=U0, U1=U1,U2=U2,U3=U3))</code></pre>
<p>We find the estimate of V with canonical covariances and the PCA covariances.</p>
<pre class="r"><code>m.data = mash_set_data(data$Bhat, data$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)</code></pre>
<p>The PCA correlation matrices are: <img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-4-1.png" width="768" style="display: block; margin: auto;" /></p>
<ol style="list-style-type: decimal">
<li>We run the algorithm in <a href="EstimateCorMax.html">estimate cor mle</a> with 3 different initial points for <span class="math inline">\(\rho\)</span> (-0.5,0,0.5). The <span class="math inline">\(\rho\)</span> in each iteration is estimated using <code>optim</code> function. The estimated correlation is</li>
</ol>
<pre class="r"><code>Vhat.mle = estimateV(m.data, c(U.c, U.ed), init_rho = c(-0.5,0,0.5), tol=1e-4, optmethod = 'mle')
Vhat.mle$V</code></pre>
<pre><code> [,1] [,2] [,3]
[1,] 1.0000000 0.7009268 0.1588733
[2,] 0.7009268 1.0000000 0.4001842
[3,] 0.1588733 0.4001842 1.0000000</code></pre>
<ol start="2" style="list-style-type: decimal">
<li>The result uses algorithm in <a href="EstimateCorMaxEM2.html">estimate cor em</a>. <span class="math inline">\(\rho\)</span> in each iteration is the root of a third degree polynomial.</li>
</ol>
<pre class="r"><code>Vhat.em = estimateV(m.data, c(U.c, U.ed), init_rho = c(-0.5,0,0.5), tol = 1e-4, optmethod = 'em2')
Vhat.em$V</code></pre>
<pre><code> [,1] [,2] [,3]
[1,] 1.0000000 0.7086927 0.1706391
[2,] 0.7086927 1.0000000 0.4187296
[3,] 0.1706391 0.4187296 1.0000000</code></pre>
<p>The running time (in sec.) for each pairwise correlation is</p>
<pre class="r"><code>table = data.frame(rbind(Vhat.mle$ttime, Vhat.em$ttime), row.names = c('mle', 'em'))
colnames(table) = c('12','13','23')
table %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
12
</th>
<th style="text-align:right;">
13
</th>
<th style="text-align:right;">
23
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
mle
</td>
<td style="text-align:right;">
290.737
</td>
<td style="text-align:right;">
207.046
</td>
<td style="text-align:right;">
68.977
</td>
</tr>
<tr>
<td style="text-align:left;">
em
</td>
<td style="text-align:right;">
168.826
</td>
<td style="text-align:right;">
63.495
</td>
<td style="text-align:right;">
51.343
</td>
</tr>
</tbody>
</table>
<p>The time is the total running time with different initial point.</p>
<ol start="3" style="list-style-type: decimal">
<li>The result uses algorithm in <a href="EstimateCorMaxEMV.html">estimate cor em V</a>.</li>
</ol>
<pre class="r"><code>Vhat.emV = estimateV(m.data, c(U.c, U.ed), init_V = list(diag(ncol(m.data$Bhat)), clusterGeneration::rcorrmatrix(3), clusterGeneration::rcorrmatrix(3)), tol = 1e-4, optmethod = 'emV')
Vhat.emV$V</code></pre>
<pre><code> [,1] [,2] [,3]
[1,] 1.0000000 0.5487183 0.2669639
[2,] 0.5487183 1.0000000 0.4654711
[3,] 0.2669639 0.4654711 1.0000000</code></pre>
<ol start="4" style="list-style-type: decimal">
<li>Using the original truncated correlation:</li>
</ol>
<pre class="r"><code>Vhat.tru = estimate_null_correlation(m.data)
Vhat.tru</code></pre>
<pre><code> [,1] [,2] [,3]
[1,] 1.0000000 0.4296283 0.1222433
[2,] 0.4296283 1.0000000 0.3324459
[3,] 0.1222433 0.3324459 1.0000000</code></pre>
<p>The truncated correlation underestimates the correlations.</p>
<ol start="5" style="list-style-type: decimal">
<li>mash 1by1</li>
</ol>
<pre class="r"><code>V.mash = cor((data$Bhat/data$Shat)[-strong,])
V.mash</code></pre>
<pre><code> [,1] [,2] [,3]
[1,] 1.0000000 0.5313446 0.2445663
[2,] 0.5313446 1.0000000 0.4490049
[3,] 0.2445663 0.4490049 1.0000000</code></pre>
<div id="error" class="section level2">
<h2>Error</h2>
<p>Check the estimation error:</p>
<pre class="r"><code>FError = c(norm(Vhat.mle$V - Sigma, 'F'),
norm(Vhat.em$V - Sigma, 'F'),
norm(Vhat.emV$V - Sigma, 'F'),
norm(Vhat.tru - Sigma, 'F'),
norm(V.mash - Sigma, 'F'))
OpError = c(norm(Vhat.mle$V - Sigma, '2'),
norm(Vhat.em$V - Sigma, '2'),
norm(Vhat.emV$V - Sigma, '2'),
norm(Vhat.tru - Sigma, '2'),
norm(V.mash - Sigma, '2'))
table = data.frame(FrobeniusError = FError, SpectralError = OpError, row.names = c('mle','em','emV','trunc','m.1by1'))
table %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
FrobeniusError
</th>
<th style="text-align:right;">
SpectralError
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
mle
</td>
<td style="text-align:right;">
0.0581773
</td>
<td style="text-align:right;">
0.0411417
</td>
</tr>
<tr>
<td style="text-align:left;">
em
</td>
<td style="text-align:right;">
0.0507627
</td>
<td style="text-align:right;">
0.0391489
</td>
</tr>
<tr>
<td style="text-align:left;">
emV
</td>
<td style="text-align:right;">
0.2516219
</td>
<td style="text-align:right;">
0.1960202
</td>
</tr>
<tr>
<td style="text-align:left;">
trunc
</td>
<td style="text-align:right;">
0.4091712
</td>
<td style="text-align:right;">
0.3049974
</td>
</tr>
<tr>
<td style="text-align:left;">
m.1by1
</td>
<td style="text-align:right;">
0.2562509
</td>
<td style="text-align:right;">
0.1915171
</td>
</tr>
</tbody>
</table>
</div>
<div id="mash-log-likelihood" class="section level2">
<h2>mash log likelihood</h2>
<p>In mash model, the model with correlation from mle has larger loglikelihood.</p>
<pre class="r"><code>m.data.mle = mash_set_data(data$Bhat, data$Shat, V=Vhat.mle$V)
m.model.mle = mash(m.data.mle, c(U.c,U.ed), verbose = FALSE)</code></pre>
<pre class="r"><code>m.data.em = mash_set_data(data$Bhat, data$Shat, V=Vhat.em$V)
m.model.em = mash(m.data.em, c(U.c,U.ed), verbose = FALSE)</code></pre>
<pre class="r"><code>m.data.emV = mash_set_data(data$Bhat, data$Shat, V=Vhat.emV$V)
m.model.emV = mash(m.data.emV, c(U.c,U.ed), verbose = FALSE)</code></pre>
<pre class="r"><code>m.data.trunc = mash_set_data(data$Bhat, data$Shat, V=Vhat.tru)
m.model.trunc = mash(m.data.trunc, c(U.c,U.ed), verbose = FALSE)</code></pre>
<pre class="r"><code>m.data.1by1 = mash_set_data(data$Bhat, data$Shat, V=V.mash)
m.model.1by1 = mash(m.data.1by1, c(U.c,U.ed), verbose = FALSE)</code></pre>
<pre class="r"><code>m.data.correct = mash_set_data(data$Bhat, data$Shat, V=Sigma)
m.model.correct = mash(m.data.correct, c(U.c,U.ed), verbose = FALSE)</code></pre>
<p>The results are summarized in table:</p>
<pre class="r"><code>null.ind = which(apply(data$B,1,sum) == 0)
V.trun = c(get_loglik(m.model.trunc), length(get_significant_results(m.model.trunc)), sum(get_significant_results(m.model.trunc) %in% null.ind))
V.mle = c(get_loglik(m.model.mle), length(get_significant_results(m.model.mle)), sum(get_significant_results(m.model.mle) %in% null.ind))
V.em = c(get_loglik(m.model.em), length(get_significant_results(m.model.em)), sum(get_significant_results(m.model.em) %in% null.ind))
V.emV = c(get_loglik(m.model.emV), length(get_significant_results(m.model.emV)), sum(get_significant_results(m.model.emV) %in% null.ind))
V.1by1 = c(get_loglik(m.model.1by1), length(get_significant_results(m.model.1by1)), sum(get_significant_results(m.model.1by1) %in% null.ind))
V.correct = c(get_loglik(m.model.correct), length(get_significant_results(m.model.correct)), sum(get_significant_results(m.model.correct) %in% null.ind))
temp = cbind(V.mle, V.em, V.emV, V.trun, V.1by1, V.correct)
colnames(temp) = c('MLE','EM','EMV', 'Truncate', 'm.1by1', 'True')
row.names(temp) = c('log likelihood', '# significance', '# False positive')
temp %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
MLE
</th>
<th style="text-align:right;">
EM
</th>
<th style="text-align:right;">
EMV
</th>
<th style="text-align:right;">
Truncate
</th>
<th style="text-align:right;">
m.1by1
</th>
<th style="text-align:right;">
True
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
log likelihood
</td>
<td style="text-align:right;">
-17917.69
</td>
<td style="text-align:right;">
-17919.23
</td>
<td style="text-align:right;">
-17945.16
</td>
<td style="text-align:right;">
-17951.46
</td>
<td style="text-align:right;">
-17943.49
</td>
<td style="text-align:right;">
-17913.58
</td>
</tr>
<tr>
<td style="text-align:left;">
# significance
</td>
<td style="text-align:right;">
146.00
</td>
<td style="text-align:right;">
149.00
</td>
<td style="text-align:right;">
82.00
</td>
<td style="text-align:right;">
85.00
</td>
<td style="text-align:right;">
73.00
</td>
<td style="text-align:right;">
149.00
</td>
</tr>
<tr>
<td style="text-align:left;">
# False positive
</td>
<td style="text-align:right;">
1.00
</td>
<td style="text-align:right;">
1.00
</td>
<td style="text-align:right;">
0.00
</td>
<td style="text-align:right;">
1.00
</td>
<td style="text-align:right;">
0.00
</td>
<td style="text-align:right;">
1.00
</td>
</tr>
</tbody>
</table>
<p>The estimated <code>pi</code> is</p>
<pre class="r"><code>par(mfrow=c(2,3))
barplot(get_estimated_pi(m.model.mle), las=2, cex.names = 0.7, main='MLE', ylim=c(0,0.8))
barplot(get_estimated_pi(m.model.em), las=2, cex.names = 0.7, main='EM', ylim=c(0,0.8))
barplot(get_estimated_pi(m.model.emV), las=2, cex.names = 0.7, main='EMV', ylim=c(0,0.8))
barplot(get_estimated_pi(m.model.trunc), las=2, cex.names = 0.7, main='Truncate', ylim=c(0,0.8))
barplot(get_estimated_pi(m.model.1by1), las=2, cex.names = 0.7, main='m.1by1', ylim=c(0,0.8))
barplot(get_estimated_pi(m.model.correct), las=2, cex.names = 0.7, main='True', ylim=c(0,0.8))</code></pre>
<p><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-19-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="roc" class="section level2">
<h2>ROC</h2>
<pre class="r"><code>m.mle.seq = ROC.table(data$B, m.model.mle)
m.em.seq = ROC.table(data$B, m.model.em)
m.emV.seq = ROC.table(data$B, m.model.emV)
m.trun.seq = ROC.table(data$B, m.model.trunc)
m.1by1.seq = ROC.table(data$B, m.model.1by1)
m.correct.seq = ROC.table(data$B, m.model.correct)</code></pre>
<p><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-21-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="rrmse" class="section level2">
<h2>RRMSE</h2>
<pre class="r"><code>rrmse = rbind(RRMSE(data$B, data$Bhat, list(m.model.mle, m.model.em, m.model.emV, m.model.trunc, m.model.1by1, m.model.correct)))
colnames(rrmse) = c('MLE','EM','EMV', 'Truncate','m.1by1','True')
row.names(rrmse) = 'RRMSE'
rrmse %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
MLE
</th>
<th style="text-align:right;">
EM
</th>
<th style="text-align:right;">
EMV
</th>
<th style="text-align:right;">
Truncate
</th>
<th style="text-align:right;">
m.1by1
</th>
<th style="text-align:right;">
True
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
RRMSE
</td>
<td style="text-align:right;">
0.5277488
</td>
<td style="text-align:right;">
0.5285126
</td>
<td style="text-align:right;">
0.5437355
</td>
<td style="text-align:right;">
0.5592648
</td>
<td style="text-align:right;">
0.5442074
</td>
<td style="text-align:right;">
0.5283068
</td>
</tr>
</tbody>
</table>
<pre class="r"><code>barplot(rrmse, ylim=c(0,(1+max(rrmse))/2), names.arg = c('MLE','EM', 'EMV','Truncate','m.1by1','True'), las=2, cex.names = 0.7, main='RRMSE')</code></pre>
<p><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-23-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="more-simulations" class="section level1">
<h1>More simulations</h1>
<p>I randomly generate 10 positive definite correlation matrices, V. The sample size is 4000.</p>
<p><span class="math display">\[
\hat{z}|z \sim N_{5}(z, V)
\]</span> <span class="math display">\[
z\sim\frac{1}{4}\delta_{0} + \frac{1}{4}N_{5}(0,\left(\begin{matrix} 1 & \mathbf{0}_{1\times 4} \\ \mathbf{0}_{4\times 1} & \mathbf{0}_{4\times 4} \end{matrix}\right)) + \frac{1}{4}N_{5}(0,\left(\begin{matrix} \mathbf{1}_{2\times 2} & \mathbf{0}_{1\times 3} \\ \mathbf{0}_{3\times 1} & \mathbf{0}_{3\times 3} \end{matrix}\right)) + \frac{1}{4}N_{5}(0,\mathbf{1}_{5\times 5})
\]</span></p>
<pre class="r"><code>set.seed(100)
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){
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)
Vhat.mle <- estimateV(m.data, c(U.c, U.ed), init_rho = c(-0.5,0,0.5), tol=1e-4, optmethod = 'mle')
Vhat.em <- estimateV(m.data, c(U.c, U.ed), init_rho = c(-0.5,0,0.5), tol=1e-4, optmethod = 'em2')
Vhat.emV <- estimateV(m.data, c(U.c, U.ed), init_V = list(diag(ncol(m.data$Bhat)), clusterGeneration::rcorrmatrix(p), clusterGeneration::rcorrmatrix(p)),tol=1e-4, optmethod = 'emV')
saveRDS(list(V.true = Vtrue, V.mle = Vhat.mle, V.em = Vhat.em, V.emV = Vhat.emV, data = data, strong=strong),
paste0('../output/MASH.result.',t,'.rds'))
}</code></pre>
<pre class="r"><code>files = dir("../output/AddEMV/"); files = files[grep("MASH.result",files)]
times = length(files)
result = vector(mode="list",length = times)
for(i in 1:times) {
result[[i]] = readRDS(paste("../output/AddEMV/", files[[i]], sep=""))
}</code></pre>
<pre class="r"><code>mle.pd = numeric(times)
em.pd = numeric(times)
for(i in 1:times){
m.data = mash_set_data(result[[i]]$data$Bhat, result[[i]]$data$Shat)
result[[i]]$V.trun = estimate_null_correlation(m.data, apply_lower_bound = FALSE)
m.1by1 = mash_1by1(m.data)
strong = get_significant_results(m.1by1)
result[[i]]$V.1by1 = cor(m.data$Bhat[-strong,])
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]]$V.true)
m.model.true = mash(m.data.true, c(U.c,U.ed), verbose = FALSE)
m.data.trunc = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = result[[i]]$V.trun)
m.model.trunc = mash(m.data.trunc, c(U.c,U.ed), verbose = FALSE)
m.data.1by1 = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = result[[i]]$V.1by1)
m.model.1by1 = mash(m.data.1by1, c(U.c,U.ed), verbose = FALSE)
m.data.emV = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = result[[i]]$V.emV$V)
m.model.emV = mash(m.data.emV, c(U.c,U.ed), verbose = FALSE)
# MLE
m.model.mle = m.model.mle.F = m.model.mle.2 = list()
R <- tryCatch(chol(result[[i]]$V.mle$V),error = function (e) FALSE)
if(is.matrix(R)){
mle.pd[i] = 1
m.data.mle = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = result[[i]]$V.mle$V)
m.model.mle = mash(m.data.mle, c(U.c,U.ed), verbose = FALSE)
}else{
V.mle.near.F = as.matrix(Matrix::nearPD(result[[i]]$V.mle$V, conv.norm.type = 'F', keepDiag = TRUE)$mat)
V.mle.near.2 = as.matrix(Matrix::nearPD(result[[i]]$V.mle$V, conv.norm.type = '2', keepDiag = TRUE)$mat)
result[[i]]$V.mle.F = V.mle.near.F
result[[i]]$V.mle.2 = V.mle.near.2
# mashmodel
m.data.mle.F = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = V.mle.near.F)
m.model.mle.F = mash(m.data.mle.F, c(U.c,U.ed), verbose = FALSE)
m.data.mle.2 = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = V.mle.near.2)
m.model.mle.2 = mash(m.data.mle.2, c(U.c,U.ed), verbose = FALSE)
}
# EM
m.model.em = m.model.em.F = m.model.em.2 = list()
R <- tryCatch(chol(result[[i]]$V.em$V),error = function (e) FALSE)
if(is.matrix(R)){
em.pd[i] = 1
m.data.em = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = result[[i]]$V.em$V)
m.model.em = mash(m.data.em, c(U.c,U.ed), verbose = FALSE)
}else{
V.em.near.F = as.matrix(Matrix::nearPD(result[[i]]$V.em$V, conv.norm.type = 'F', keepDiag = TRUE)$mat)
V.em.near.2 = as.matrix(Matrix::nearPD(result[[i]]$V.em$V, conv.norm.type = '2', keepDiag = TRUE)$mat)
result[[i]]$V.em.F = V.em.near.F
result[[i]]$V.em.2 = V.em.near.2
# mashmodel
m.data.em.F = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = V.em.near.F)
m.model.em.F = mash(m.data.em.F, c(U.c,U.ed), verbose = FALSE)
m.data.em.2 = mash_set_data(Bhat = m.data$Bhat, Shat = m.data$Shat, V = V.em.near.2)
m.model.em.2 = mash(m.data.em.2, c(U.c,U.ed), verbose = FALSE)
}
result[[i]]$m.model = list(m.model.true = m.model.true, m.model.trunc = m.model.trunc,
m.model.1by1 = m.model.1by1, m.model.emV = m.model.emV,
m.model.mle = m.model.mle,
m.model.mle.F = m.model.mle.F, m.model.mle.2 = m.model.mle.2,
m.model.em = m.model.em,
m.model.em.F = m.model.em.F, m.model.em.2 = m.model.em.2)
}</code></pre>
<div id="error-1" class="section level2">
<h2>Error</h2>
<p>Some estimated correlation matrices are not positive definite. So I estimate the nearest PD cor matrix with <code>nearPD</code> function.</p>
<p>The column with <code>.F</code>, <code>.2</code> are from the nearest positive definite matrix with respect to Frobenius norm and spectral norm.</p>
<p>The Frobenius norm is</p>
<pre class="r"><code>temp = matrix(0,nrow = times, ncol = 7)
for(i in 1:times){
temp[i, ] = error.cor(result[[i]], norm.type='F', mle.pd = mle.pd[i], em.pd = em.pd[i])
}
colnames(temp) = c('Trunc','m.1by1', 'MLE','MLE.F', 'EM', 'EM.F', 'EMV')
temp[temp==0] = NA
temp %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:right;">
Trunc
</th>
<th style="text-align:right;">
m.1by1
</th>
<th style="text-align:right;">
MLE
</th>
<th style="text-align:right;">
MLE.F
</th>
<th style="text-align:right;">
EM
</th>
<th style="text-align:right;">
EM.F
</th>
<th style="text-align:right;">
EMV
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:right;">
0.5847549
</td>
<td style="text-align:right;">
0.9286016
</td>
<td style="text-align:right;">
0.3185293
</td>
<td style="text-align:right;">
0.3053278
</td>
<td style="text-align:right;">
0.0941757
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
1.0198618
</td>
</tr>
<tr>
<td style="text-align:right;">
0.6345196
</td>
<td style="text-align:right;">
0.8140108
</td>
<td style="text-align:right;">
0.1169154
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.0981075
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.8913631
</td>
</tr>
<tr>
<td style="text-align:right;">
0.7201453
</td>
<td style="text-align:right;">
0.9531300
</td>
<td style="text-align:right;">
0.2344733
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.1734402
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
1.0216638
</td>
</tr>
<tr>
<td style="text-align:right;">
0.8370832
</td>
<td style="text-align:right;">
1.0534335
</td>
<td style="text-align:right;">
0.1968091
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.2411372
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
1.1115619
</td>
</tr>
<tr>
<td style="text-align:right;">
0.8206008
</td>
<td style="text-align:right;">
0.8466408
</td>
<td style="text-align:right;">
0.1194142
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.1189734
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.9176794
</td>
</tr>
<tr>
<td style="text-align:right;">
0.8455747
</td>
<td style="text-align:right;">
1.1393764
</td>
<td style="text-align:right;">
0.1650726
</td>
<td style="text-align:right;">
0.1479670
</td>
<td style="text-align:right;">
0.1653665
</td>
<td style="text-align:right;">
0.1516188
</td>
<td style="text-align:right;">
1.2178752
</td>
</tr>
<tr>
<td style="text-align:right;">
0.5173056
</td>
<td style="text-align:right;">
0.8194980
</td>
<td style="text-align:right;">
0.1211599
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.0861518
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.8914736
</td>
</tr>
<tr>
<td style="text-align:right;">
0.8840057
</td>
<td style="text-align:right;">
0.9546138
</td>
<td style="text-align:right;">
0.1642744
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.1563607
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
1.0169747
</td>
</tr>
<tr>
<td style="text-align:right;">
0.6535878
</td>
<td style="text-align:right;">
1.0554833
</td>
<td style="text-align:right;">
0.1240352
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.1323659
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
1.1491978
</td>
</tr>
<tr>
<td style="text-align:right;">
0.6425639
</td>
<td style="text-align:right;">
0.9346341
</td>
<td style="text-align:right;">
0.0794776
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.0700813
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
1.0196620
</td>
</tr>
</tbody>
</table>
<p>The spectral norm is</p>
<pre class="r"><code>temp = matrix(0,nrow = times, ncol = 7)
for(i in 1:times){
temp[i, ] = error.cor(result[[i]], norm.type='2', mle.pd = mle.pd[i], em.pd = em.pd[i])
}
colnames(temp) = c('Trunc','m.1by1', 'MLE','MLE.2', 'EM', 'EM.2', 'EMV')
temp[temp==0] = NA
temp %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:right;">
Trunc
</th>
<th style="text-align:right;">
m.1by1
</th>
<th style="text-align:right;">
MLE
</th>
<th style="text-align:right;">
MLE.2
</th>
<th style="text-align:right;">
EM
</th>
<th style="text-align:right;">
EM.2
</th>
<th style="text-align:right;">
EMV
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:right;">
0.4230698
</td>
<td style="text-align:right;">
0.7526315
</td>
<td style="text-align:right;">
0.2258153
</td>
<td style="text-align:right;">
0.2205340
</td>
<td style="text-align:right;">
0.0662442
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.8335745
</td>
</tr>
<tr>
<td style="text-align:right;">
0.5228925
</td>
<td style="text-align:right;">
0.6281308
</td>
<td style="text-align:right;">
0.0783098
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.0677285
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.7069909
</td>
</tr>
<tr>
<td style="text-align:right;">
0.5156642
</td>
<td style="text-align:right;">
0.7945237
</td>
<td style="text-align:right;">
0.1750836
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.1239438
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.8577274
</td>
</tr>
<tr>
<td style="text-align:right;">
0.6529121
</td>
<td style="text-align:right;">
0.8335401
</td>
<td style="text-align:right;">
0.1484773
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.1768351
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.8944724
</td>
</tr>
<tr>
<td style="text-align:right;">
0.6500336
</td>
<td style="text-align:right;">
0.6377762
</td>
<td style="text-align:right;">
0.0778356
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.0849870
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.7052165
</td>
</tr>
<tr>
<td style="text-align:right;">
0.5607948
</td>
<td style="text-align:right;">
0.8613851
</td>
<td style="text-align:right;">
0.1001500
</td>
<td style="text-align:right;">
0.0937816
</td>
<td style="text-align:right;">
0.1085885
</td>
<td style="text-align:right;">
0.1015186
</td>
<td style="text-align:right;">
0.9343139
</td>
</tr>
<tr>
<td style="text-align:right;">
0.3157614
</td>
<td style="text-align:right;">
0.6406310
</td>
<td style="text-align:right;">
0.0795125
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.0659174
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.6982291
</td>
</tr>
<tr>
<td style="text-align:right;">
0.7134025
</td>
<td style="text-align:right;">
0.7290570
</td>
<td style="text-align:right;">
0.1090477
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.1103719
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.7987348
</td>
</tr>
<tr>
<td style="text-align:right;">
0.4767534
</td>
<td style="text-align:right;">
0.8894951
</td>
<td style="text-align:right;">
0.0964157
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.0957622
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.9760343
</td>
</tr>
<tr>
<td style="text-align:right;">
0.4591215
</td>
<td style="text-align:right;">
0.7926789
</td>
<td style="text-align:right;">
0.0568252
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.0558999
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
0.8740268
</td>
</tr>
</tbody>
</table>
</div>
<div id="time" class="section level2">
<h2>Time</h2>
<p>The total running time for each matrix is</p>
<pre class="r"><code>mle.time = em.time = numeric(times)
for(i in 1:times){
mle.time[i] = sum(result[[i]]$V.mle$ttime)
em.time[i] = sum(result[[i]]$V.em$ttime)
}
temp = cbind(mle.time, em.time)
colnames(temp) = c('MLE', 'EM')
row.names(temp) = 1:10
temp %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:right;">
MLE
</th>
<th style="text-align:right;">
EM
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:right;">
3171.983
</td>
<td style="text-align:right;">
848.351
</td>
</tr>
<tr>
<td style="text-align:right;">
2341.424
</td>
<td style="text-align:right;">
704.601
</td>
</tr>
<tr>
<td style="text-align:right;">
1990.764
</td>
<td style="text-align:right;">
740.046
</td>
</tr>
<tr>
<td style="text-align:right;">
3249.162
</td>
<td style="text-align:right;">
1072.233
</td>
</tr>
<tr>
<td style="text-align:right;">
1988.220
</td>
<td style="text-align:right;">
717.808
</td>
</tr>
<tr>
<td style="text-align:right;">
2580.794
</td>
<td style="text-align:right;">
958.710
</td>
</tr>
<tr>
<td style="text-align:right;">
1928.634
</td>
<td style="text-align:right;">
597.511
</td>
</tr>
<tr>
<td style="text-align:right;">
2992.277
</td>
<td style="text-align:right;">
1114.932
</td>
</tr>
<tr>
<td style="text-align:right;">
2339.513
</td>
<td style="text-align:right;">
708.779
</td>
</tr>
<tr>
<td style="text-align:right;">
2727.560
</td>
<td style="text-align:right;">
772.780
</td>
</tr>
</tbody>
</table>
</div>
<div id="mash-log-likelihood-1" class="section level2">
<h2>mash log likelihood</h2>
<p>The <code>NA</code> means the estimated correlation matrix is not positive definite.</p>
<pre class="r"><code>temp = matrix(0,nrow = times, ncol = 10)
for(i in 1:times){
temp[i, ] = loglik.cor(result[[i]]$m.model, mle.pd = mle.pd[i], em.pd = em.pd[i])
}
colnames(temp) = c('True', 'Trunc','m.1by1', 'MLE','MLE.F', 'MLE.2', 'EM', 'EM.F', 'EM.2','EMV')
temp[temp == 0] = NA
temp[,-c(6,9)] %>% kable() %>% kable_styling()</code></pre>
<table class="table" style="margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:right;">
True
</th>
<th style="text-align:right;">
Trunc
</th>
<th style="text-align:right;">
m.1by1
</th>
<th style="text-align:right;">
MLE
</th>
<th style="text-align:right;">
MLE.F
</th>
<th style="text-align:right;">
EM
</th>
<th style="text-align:right;">
EM.F
</th>
<th style="text-align:right;">
EMV
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:right;">
-26039.92
</td>
<td style="text-align:right;">
-26130.50
</td>
<td style="text-align:right;">
-26112.94
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-26265.49
</td>
<td style="text-align:right;">
-26120.81
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-26136.19
</td>
</tr>
<tr>
<td style="text-align:right;">
-25669.59
</td>
<td style="text-align:right;">
-26997.92
</td>
<td style="text-align:right;">
-26950.67
</td>
<td style="text-align:right;">
-26028.96
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-25859.86
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-26967.47
</td>
</tr>
<tr>
<td style="text-align:right;">
-27473.71
</td>
<td style="text-align:right;">
-27547.11
</td>
<td style="text-align:right;">
-27535.67
</td>
<td style="text-align:right;">
-27465.76
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-27463.43
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-27551.75
</td>
</tr>
<tr>
<td style="text-align:right;">
-28215.48
</td>
<td style="text-align:right;">
-28604.98
</td>
<td style="text-align:right;">
-28646.72
</td>
<td style="text-align:right;">
-28301.41
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-28338.57
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-28669.24
</td>
</tr>
<tr>
<td style="text-align:right;">
-24988.68
</td>
<td style="text-align:right;">
-25236.18
</td>
<td style="text-align:right;">
-25110.41
</td>
<td style="text-align:right;">
-25056.59
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-25048.86
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-25123.96
</td>
</tr>
<tr>
<td style="text-align:right;">
-24299.89
</td>
<td style="text-align:right;">
-24978.79
</td>
<td style="text-align:right;">
-24972.25
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-24492.32
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-24478.09
</td>
<td style="text-align:right;">
-25020.38
</td>
</tr>
<tr>
<td style="text-align:right;">
-27574.71
</td>
<td style="text-align:right;">
-27698.40
</td>
<td style="text-align:right;">
-27662.48
</td>
<td style="text-align:right;">
-27517.65
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-27540.65
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-27684.83
</td>
</tr>
<tr>
<td style="text-align:right;">
-27941.86
</td>
<td style="text-align:right;">
-28159.65
</td>
<td style="text-align:right;">
-28182.04
</td>
<td style="text-align:right;">
-27979.99
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-27953.02
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-28222.45
</td>
</tr>
<tr>
<td style="text-align:right;">
-29788.75
</td>
<td style="text-align:right;">
-29824.45
</td>
<td style="text-align:right;">
-29921.87
</td>
<td style="text-align:right;">
-29760.81
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-29759.50
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-29954.96
</td>
</tr>
<tr>
<td style="text-align:right;">
-28542.34
</td>
<td style="text-align:right;">
-28922.58
</td>
<td style="text-align:right;">
-29163.51
</td>
<td style="text-align:right;">
-28537.96
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-28550.73
</td>
<td style="text-align:right;">
NA
</td>
<td style="text-align:right;">
-29221.61
</td>
</tr>
</tbody>
</table>
</div>
<div id="roc-1" class="section level2">
<h2>ROC</h2>
<pre class="r"><code>par(mfrow=c(1,2))
for(i in 1:times){
plotROC(result[[i]]$data$B, result[[i]]$m.model, mle.pd = mle.pd[i], em.pd = em.pd[i], title=paste0('Data', i, ' '))
}</code></pre>
<p><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-31-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-31-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-31-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-31-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-31-5.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="rrmse-1" class="section level2">
<h2>RRMSE</h2>
<pre class="r"><code>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')
}</code></pre>
<p><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-32-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-32-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-32-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-32-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/EstimateCorMaxMash.Rmd/unnamed-chunk-32-5.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="session-information" class="section level1">
<h1>Session information</h1>
<!-- Insert the session information into the document -->
<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] kableExtra_0.9.0 knitr_1.20 mashr_0.2-11 ashr_2.2-10
loaded via a namespace (and not attached):
[1] Rcpp_0.12.18 highr_0.7
[3] compiler_3.5.1 pillar_1.3.0
[5] plyr_1.8.4 iterators_1.0.10
[7] tools_3.5.1 corrplot_0.84
[9] digest_0.6.15 viridisLite_0.3.0
[11] evaluate_0.11 tibble_1.4.2
[13] lattice_0.20-35 pkgconfig_2.0.1
[15] rlang_0.2.1 Matrix_1.2-14
[17] foreach_1.4.4 rstudioapi_0.7
[19] yaml_2.2.0 parallel_3.5.1
[21] mvtnorm_1.0-8 xml2_1.2.0
[23] httr_1.3.1 stringr_1.3.1
[25] REBayes_1.3 hms_0.4.2
[27] rprojroot_1.3-2 grid_3.5.1
[29] R6_2.2.2 rmarkdown_1.10
[31] rmeta_3.0 readr_1.1.1
[33] magrittr_1.5 scales_0.5.0
[35] backports_1.1.2 codetools_0.2-15
[37] htmltools_0.3.6 MASS_7.3-50
[39] rvest_0.3.2 assertthat_0.2.0
[41] colorspace_1.3-2 stringi_1.2.4
[43] Rmosek_8.0.69 munsell_0.5.0
[45] pscl_1.5.2 doParallel_1.0.11
[47] truncnorm_1.0-8 SQUAREM_2017.10-1
[49] clusterGeneration_1.3.4 ExtremeDeconvolution_1.3
[51] crayon_1.3.4 </code></pre>
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