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<h1 class="title toc-ignore">Estimate cor – max</h1>
<h4 class="author"><em>Yuxin Zou</em></h4>
<h4 class="date"><em>2018-07-24</em></h4>
</div>
<!-- Update knitr chunk options -->
<!-- Insert the date the file was last updated -->
<p><strong>Last updated:</strong> 2018-08-03</p>
<p><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">\[
\left(\begin{matrix} x \\ y \end{matrix} \right) \sim \sum_{k=0}^{K} \pi_{k} N( \left(\begin{matrix} x \\ y \end{matrix} \right); 0, U_{k} )
\]</span> <span class="math inline">\(\Rightarrow\)</span> <span class="math display">\[
\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) \sim \sum_{k=0}^{K} \pi_{k} N( \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right); 0, \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + U_{k} )
\]</span> <span class="math display">\[
\Sigma_{k} = \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + U_{k} = \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + \left( \begin{matrix} u_{k11} & u_{k12} \\ u_{k21} & u_{k22} \end{matrix} \right) = \left( \begin{matrix} 1+u_{k11} & \rho+u_{k12} \\ \rho+u_{k21} & 1+u_{k22} \end{matrix} \right)
\]</span> Let <span class="math inline">\(\sigma_{k11} = \sqrt{1+u_{k11}}\)</span>, <span class="math inline">\(\sigma_{k22} = \sqrt{1+u_{k22}}\)</span>, <span class="math inline">\(\phi_{k}=\frac{\rho+u_{k12}}{\sigma_{k11}\sigma_{k22}}\)</span></p>
<pre class="r"><code>library(mashr)</code></pre>
<pre><code>Loading required package: ashr</code></pre>
<pre class="r"><code>set.seed(1)
n <- 10000; pi <- c(0.8,0.2); rho <- 0.8
V <- matrix(c(1,rho,rho,1),2,2)
U0 <- matrix(0,2,2)
U1 <- diag(2)
X0 <- mvtnorm::rmvnorm(n*pi[1], sigma = V + U0)
X1 <- mvtnorm::rmvnorm(n*pi[2], sigma = V + U1)
X <- rbind(X0,X1)
U2 <- matrix(1,2,2)
U3 <- matrix(0,2,2); U3[1,1] <- 1
U4 <- matrix(0,2,2); U4[2,2] <- 1
Ulist <- list(U0 = U0, U1 = U1, U2=U2, U3 = U3, U4=U4)</code></pre>
<div id="mle" class="section level1">
<h1>MLE</h1>
<p>The loglikelihood is (with penalty) <span class="math display">\[
l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{k=0}^{K} \pi_{k}N(x_{i}; 0, \Sigma_{k}) + \sum_{k=0}^{K} (\lambda_{k}-1) \log \pi_{k}
\]</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">\(\lambda_{k}\geq 1\)</span>.</p>
<p><span class="math display">\[
l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{k=0}^{K} \pi_{k}\frac{1}{2\pi\sigma_{k11}\sigma_{k22}\sqrt{1-\phi_{k}^2}} \exp\left( -\frac{1}{2(1-\phi_{k}^2)}\left[ \frac{x_{i}^2}{\sigma_{k11}^2} + \frac{y_{i}^2}{\sigma_{k22}^2} - \frac{2\phi_{k}x_{i}y_{i}}{\sigma_{k11}\sigma_{k22}}\right] \right) + \sum_{k=0}^{K} (\lambda_{k}-1) \log \pi_{k}
\]</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_pi, init_rho, Ulist
Compute loglikelihood
delta = 1
while delta > tol
Given rho, estimate pi by max loglikelihood
Given pi, estimate rho by max loglikelihood
Compute loglikelihood
Update delta</code></pre>
<pre class="r"><code>source('../code/estimate_cor.R')</code></pre>
<pre class="r"><code>result <- optimize_pi_rho_times(X, Ulist)
plot(result[[1]]$loglik)</code></pre>
<p><img src="figure/EstimateCorMax.Rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /> The estimated <span class="math inline">\(\rho\)</span> is 0.7998163.</p>
</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] mashr_0.2-9 ashr_2.2-10
loaded via a namespace (and not attached):
[1] Rcpp_0.12.18 knitr_1.20 magrittr_1.5
[4] REBayes_1.3 MASS_7.3-50 doParallel_1.0.11
[7] pscl_1.5.2 SQUAREM_2017.10-1 lattice_0.20-35
[10] foreach_1.4.4 plyr_1.8.4 stringr_1.3.1
[13] tools_3.5.1 parallel_3.5.1 grid_3.5.1
[16] rmeta_3.0 htmltools_0.3.6 iterators_1.0.10
[19] assertthat_0.2.0 yaml_2.2.0 rprojroot_1.3-2
[22] digest_0.6.15 Matrix_1.2-14 codetools_0.2-15
[25] evaluate_0.11 rmarkdown_1.10 stringi_1.2.4
[28] compiler_3.5.1 Rmosek_8.0.69 backports_1.1.2
[31] mvtnorm_1.0-8 truncnorm_1.0-8 </code></pre>
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