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<h1 class="title toc-ignore">Mash vs. Flash</h1>
<h4 class="author"><em>Jason Willwerscheid</em></h4>
<h4 class="date"><em>4/20/2018</em></h4>
</div>
<p><strong>Last updated:</strong> 2018-06-11</p>
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readd mashvflash analysis
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<div id="code" class="section level2">
<h2>Code</h2>
<p>Setup code.</p>
<pre class="r"><code># FLASH v MASH ------------------------------------------------------
flash_v_mash <- function(Y, true_Y, nfactors) {
data <- flash_set_data(Y, S = 1)
res <- list()
t <- Sys.time()
fl <- fit_flash(data, nfactors)
res$fl_time <- Sys.time() - t
t <- Sys.time()
m <- fit_mash(Y)
res$m_time <- Sys.time() - t
# Sample from FLASH fit
fl_sampler <- flash_lf_sampler(Y, fl, ebnm_fn=ebnm_pn, fixed="factors")
nsamp <- 200
fl_samp <- fl_sampler(nsamp)
res$fl_mse <- flash_pm_mse(fl_samp, true_Y)
res$m_mse <- mash_pm_mse(m, true_Y)
res$fl_ci <- flash_ci_acc(fl_samp, true_Y)
res$m_ci <- mash_ci_acc(m, true_Y)
res$fl_lfsr <- flash_lfsr(fl_samp, true_Y)
res$m_lfsr <- mash_lfsr(m, true_Y)
res
}
plot_res <- function(res) {
old_par <- par("mfrow")
par(mfrow=c(1, 2))
x <- seq(0.025, 0.475, by=0.05)
plot(x, res$fl_lfsr, type='l', ylim=c(0, 0.6), xlab="FLASH", ylab="lfsr")
abline(0, 1)
plot(x, res$m_lfsr, type='l', ylim=c(0, 0.6), xlab="MASH", ylab="lfsr")
abline(0, 1)
par(mfrow=old_par)
}
# Fit using FLASH ---------------------------------------------------
fit_flash <- function(data, nfactors) {
p <- ncol(data$Y)
fl <- flash_add_greedy(data, nfactors, var_type = "zero")
fl <- flash_add_fixed_f(data, diag(rep(1, p)), fl)
flash_backfit(data, fl, nullcheck = F, var_type = "zero")
}
# Fit using MASH ---------------------------------------------------
fit_mash <- function(Y) {
data <- mash_set_data(Y)
U.c = cov_canonical(data)
m.1by1 <- mash_1by1(data)
strong <- get_significant_results(m.1by1, 0.05)
U.pca <- cov_pca(data, 5, strong)
U.ed <- cov_ed(data, U.pca, strong)
mash(data, c(U.c,U.ed))
}
# MSE of posterior means (FLASH) ------------------------------------
flash_pm_mse <- function(fl_samp, true_Y) {
n <- nrow(true_Y)
p <- ncol(true_Y)
nsamp <- length(fl_samp)
post_means <- matrix(0, nrow=n, ncol=p)
for (i in 1:nsamp) {
post_means <- post_means + fl_samp[[i]]
}
post_means <- post_means / nsamp
sum((post_means - true_Y)^2) / (n * p)
}
# Compare with just using FLASH LF:
# sum((flash_get_lf(fl)- true_flash_Y)^2) / (n * p)
# MSE for MASH ------------------------------------------------------
mash_pm_mse <- function(m, true_Y) {
n <- nrow(true_Y)
p <- ncol(true_Y)
sum((get_pm(m) - true_Y)^2) / (n * p)
}
# CI coverage for FLASH ---------------------------------------------
flash_ci_acc <- function(fl_samp, true_Y) {
n <- nrow(true_Y)
p <- ncol(true_Y)
nsamp <- length(fl_samp)
flat_samp <- matrix(0, nrow=n*p, ncol=nsamp)
for (i in 1:nsamp) {
flat_samp[, i] <- as.vector(fl_samp[[i]])
}
CI <- t(apply(flat_samp, 1, function(x) {quantile(x, c(0.025, 0.975))}))
sum((as.vector(true_Y) > CI[, 1])
& (as.vector(true_Y < CI[, 2]))) / (n * p)
}
# CI coverage for MASH ----------------------------------------------
mash_ci_acc <- function(m, true_Y) {
sum((true_Y > get_pm(m) - 1.96 * get_psd(m))
& (true_Y < get_pm(m) + 1.96 * get_psd(m))) / (n * p)
}
# LFSR for FLASH ----------------------------------------------------
flash_lfsr <- function(fl_samp, true_Y, step=0.05) {
n <- nrow(true_Y)
p <- ncol(true_Y)
nsamp <- length(fl_samp)
lfsr <- matrix(0, nrow=n, ncol=p)
for (i in 1:nsamp) {
lfsr <- lfsr + (fl_samp[[i]] > 0) + 0.5*(fl_samp[[i]] == 0)
}
signs <- lfsr >= nsamp / 2
correct_signs <- true_Y > 0
gotitright <- signs == correct_signs
lfsr <- pmin(lfsr, 100 - lfsr) / 100
nsteps <- floor(.5 / step)
fsr_by_lfsr <- rep(0, nsteps)
for (k in 1:nsteps) {
idx <- (lfsr >= (step * (k - 1)) & lfsr < (step * k))
fsr_by_lfsr[k] <- ifelse(sum(idx) == 0, 0,
1 - sum(gotitright[idx]) / sum(idx))
}
fsr_by_lfsr
}
# LFSR for MASH -----------------------------------------------------
mash_lfsr <- function(m, true_Y, step=0.05) {
lfsr <- get_lfsr(m)
signs <- get_pm(m) > 0
correct_signs <- true_Y > 0
gotitright <- signs == correct_signs
nsteps <- floor(.5 / step)
fsr_by_lfsr <- rep(0, nsteps)
for (k in 1:nsteps) {
idx <- (lfsr >= (step * (k - 1)) & lfsr < (step * k))
fsr_by_lfsr[k] <- ifelse(sum(idx) == 0, 0,
1 - sum(gotitright[idx]) / sum(idx))
}
fsr_by_lfsr
}</code></pre>
<p>Augmented FLASH simulation.</p>
<pre class="r"><code># Simulate from FLASH model -----------------------------------------
n <- 1000
p <- 10
flash_factors <- 5
# Use one factor of all ones and one more interesting factor
nfactors <- 2
k <- p + nfactors
ff <- matrix(0, nrow=k, ncol=p)
ff[1, ] <- rep(10, p)
ff[2, ] <- c(seq(10, 2, by=-2), rep(0, p - 5))
diag(ff[3:k, ]) <- 3
ll <- matrix(rnorm(n * k), nrow=n, ncol=k)
true_flash_Y <- ll %*% ff
flash_Y <- true_flash_Y + rnorm(n*p)
# RESULTS
flash_res <- flash_v_mash(flash_Y, true_flash_Y, flash_factors)</code></pre>
<pre><code>fitting factor/loading 1</code></pre>
<pre><code>fitting factor/loading 2</code></pre>
<pre><code>fitting factor/loading 3</code></pre>
<pre><code>fitting factor/loading 4</code></pre>
<pre><code>fitting factor/loading 5</code></pre>
<pre><code> - Computing 1000 x 463 likelihood matrix.
- Likelihood calculations took 0.12 seconds.
- Fitting model with 463 mixture components.
- Model fitting took 0.26 seconds.
- Computing posterior matrices.
- Computation allocated took 0.03 seconds.</code></pre>
<p>FLASH simulation.</p>
<pre class="r"><code># Simulate from basic FLASH model -----------------------------------
ff <- ff[1:nfactors, ]
ll <- matrix(rnorm(n * nfactors), nrow=n, ncol=nfactors)
true_basic_Y <- ll %*% ff
basic_Y <- true_basic_Y + rnorm(n*p)
# RESULTS
basic_res <- flash_v_mash(basic_Y, true_basic_Y, flash_factors)</code></pre>
<pre><code>fitting factor/loading 1</code></pre>
<pre><code>fitting factor/loading 2</code></pre>
<pre><code>fitting factor/loading 3</code></pre>
<pre><code> - Computing 1000 x 463 likelihood matrix.
- Likelihood calculations took 0.12 seconds.
- Fitting model with 463 mixture components.</code></pre>
<pre><code>Warning in REBayes::KWDual(A, rep(1, k), normalize(w), control = control): estimated mixing distribution has some negative values:
consider reducing rtol</code></pre>
<pre><code>Warning in mixIP(matrix_lik = structure(c(4.36775456036503e-15, 0, 0,
0, : Optimization step yields mixture weights that are either too small,
or negative; weights have been corrected and renormalized after the
optimization.</code></pre>
<pre><code> - Model fitting took 0.24 seconds.
- Computing posterior matrices.
- Computation allocated took 0.29 seconds.</code></pre>
<p>MASH simulation.</p>
<pre class="r"><code># Simulate from MASH model ------------------------------------------
Sigma <- list()
Sigma[[1]] <- matrix(1, nrow=p, ncol=p)
Sigma[[2]] <- matrix(0, nrow=p, ncol=p)
for (i in 1:p) {
for (j in 1:p) {
Sigma[[2]][i, j] <- max(1 - abs(i - j) / 4, 0)
}
}
for (k in 1:p) {
Sigma[[k + 2]] <- matrix(0, nrow=p, ncol=p)
Sigma[[k + 2]][k, k] <- 1
}
which_sigma <- sample(1:12, 1000, T, prob=c(.3, .3, rep(.4/p, p)))
true_mash_Y <- matrix(0, nrow=n, ncol=p)
for (i in 1:n) {
true_mash_Y[i, ] <- 5*mvrnorm(1, rep(0, p), Sigma[[which_sigma[i]]])
}
mash_Y <- true_mash_Y + rnorm(n * p)
# RESULTS
mash_res <- flash_v_mash(mash_Y, true_mash_Y, flash_factors)</code></pre>
<pre><code>fitting factor/loading 1</code></pre>
<pre><code>fitting factor/loading 2</code></pre>
<pre><code>fitting factor/loading 3</code></pre>
<pre><code>fitting factor/loading 4</code></pre>
<pre><code>fitting factor/loading 5</code></pre>
<pre><code> - Computing 1000 x 400 likelihood matrix.
- Likelihood calculations took 0.11 seconds.
- Fitting model with 400 mixture components.
- Model fitting took 0.58 seconds.
- Computing posterior matrices.
- Computation allocated took 0.03 seconds.</code></pre>
</div>
<div id="summary" class="section level2">
<h2>Summary</h2>
<p>In each case below, I follow the vignettes to produce a MASH fit (I use both canonical and data-driven covariance matrices). I fit a FLASH object (fixing the standard errors) by adding up to 10 factors greedily, then adding <span class="math inline">\(p\)</span> fixed one-hot vectors, and finally backfitting.</p>
<p>The two fits perform similarly. The MASH fit does somewhat better on data generated from the MASH model; more surprisingly, it performs comparably to FLASH on data generated from both the standard two-factor FLASH model. Both do poorly on the “augmented FLASH model” (described below), with MSEs near 1 (which would be obtained by simply using <span class="math inline">\(Y\)</span> as an estimate).</p>
</div>
<div id="flash-model" class="section level2">
<h2>Flash Model</h2>
<p>First I simulate from the basic FLASH model <span class="math inline">\(Y = LF + E\)</span> with <span class="math inline">\(E_{ij} \sim N(0, 1)\)</span>. Here, <span class="math inline">\(Y \in \mathbb{R}^{1000 \times 10}\)</span>, <span class="math inline">\(L \in \mathbb{R}^{1000 \times 2}\)</span> has i.i.d. <span class="math inline">\(N(0, 1)\)</span> entries, and <span class="math inline">\(F\)</span> is as follows:</p>
<pre><code> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 10 10 10 10 10 10 10 10 10 10
[2,] 10 8 6 4 2 0 0 0 0 0</code></pre>
<p><strong>The MSE of the FLASH fit is 0.2, vs. 0.21 for the MASH fit. The proportion of 95% confidence intervals that contain the true value <span class="math inline">\(LF_{ij}\)</span> is 0.94 for FLASH and 0.96 for MASH.</strong> The true false sign rate vs lfsr appears as follows:</p>
<p><img src="figure/mashvflash.Rmd/lfsr1-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of lfsr1-1.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/willwerscheid/MASHvFLASH/blob/624de1dfb3db11bdc40a4a3d946a8fbd7abf40ce/docs/figure/mashvflash.Rmd/lfsr1-1.png" target="_blank">624de1d</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/9f81f0649020df46c440e91d61356176321aa930/docs/figure/mashvflash.Rmd/lfsr1-1.png" target="_blank">9f81f06</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/17f4dfb20b57e7e3b41a94a7b2dce787af325b3a/docs/figure/mashvflash.Rmd/lfsr1-1.png" target="_blank">17f4dfb</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/6508f2be3666427c3f8bd6c348e59ab7b9355565/docs/figure/mashvflash.Rmd/lfsr1-1.png" target="_blank">6508f2b</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-09
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/861d4b3595125a0b8a304aa13375835acf4c8505/docs/figure/mashvflash.Rmd/lfsr1-1.png" target="_blank">861d4b3</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-09
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/a76f3cf9f3dad726abb36bc409df06e9c31f57d2/docs/figure/mashvflash.Rmd/lfsr1-1.png" target="_blank">a76f3cf</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-09
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>The FLASH fit took 0.61 s. The MASH fit took 8.86 s.</p>
</div>
<div id="augmented-flash-model" class="section level2">
<h2>Augmented Flash Model</h2>
<p>Next I simulate from the “augmented” FLASH model <span class="math display">\[ Y = L \begin{pmatrix} F \\ 3I_{10} \end{pmatrix} + E \]</span> with <span class="math inline">\(F\)</span> as above.</p>
<p><strong>The MSE of the FLASH fit is 0.93, vs. 1.05 for the MASH fit. The proportion of 95% confidence intervals that contain the true value is 0.94 for FLASH and 0.93 for MASH.</strong> The true false sign rate vs lfsr appears as follows:</p>
<p><img src="figure/mashvflash.Rmd/lfsr2-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of lfsr2-1.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/willwerscheid/MASHvFLASH/blob/a1180b51f9cae5996e43a7bb368317bad76cf981/docs/figure/mashvflash.Rmd/lfsr2-1.png" target="_blank">a1180b5</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/624de1dfb3db11bdc40a4a3d946a8fbd7abf40ce/docs/figure/mashvflash.Rmd/lfsr2-1.png" target="_blank">624de1d</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/9f81f0649020df46c440e91d61356176321aa930/docs/figure/mashvflash.Rmd/lfsr2-1.png" target="_blank">9f81f06</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/17f4dfb20b57e7e3b41a94a7b2dce787af325b3a/docs/figure/mashvflash.Rmd/lfsr2-1.png" target="_blank">17f4dfb</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/6508f2be3666427c3f8bd6c348e59ab7b9355565/docs/figure/mashvflash.Rmd/lfsr2-1.png" target="_blank">6508f2b</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-09
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/861d4b3595125a0b8a304aa13375835acf4c8505/docs/figure/mashvflash.Rmd/lfsr2-1.png" target="_blank">861d4b3</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-09
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/a76f3cf9f3dad726abb36bc409df06e9c31f57d2/docs/figure/mashvflash.Rmd/lfsr2-1.png" target="_blank">a76f3cf</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-09
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>The FLASH fit took 18.36 s. The MASH fit took 3.57 s.</p>
</div>
<div id="mash-model" class="section level2">
<h2>MASH Model</h2>
<p>Finally I simulate from the MASH model <span class="math display">\[ X \sim \sum \pi_i N(0, \Sigma_i),\ Y = X + E \]</span> with <span class="math inline">\(E_{ij} \sim N(0, 1)\)</span>. I set <span class="math inline">\(\Sigma_1\)</span> to be the all ones matrix, <span class="math inline">\(\Sigma_2\)</span> to be a banded covariance matrix with non-zero entries on the first three off-diagonals, and <span class="math inline">\(\Sigma_3\)</span> through <span class="math inline">\(\Sigma_{12}\)</span> to have a single non-zero entry (corresponding to tissue-specific effects). <span class="math inline">\(\pi\)</span> is set to <span class="math inline">\((0.3, 0.3, 0.04, 0.04, \ldots, 0.04)\)</span>.</p>
<p><strong>The MSE of the FLASH fit is 0.56, vs. 0.43 for the MASH fit. The proportion of 95% confidence intervals that contain the true value is 0.9 for FLASH and 0.94 for MASH.</strong> The true false sign rate vs lfsr appears as follows:</p>
<p><img src="figure/mashvflash.Rmd/lfsr3-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of lfsr3-1.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/willwerscheid/MASHvFLASH/blob/a1180b51f9cae5996e43a7bb368317bad76cf981/docs/figure/mashvflash.Rmd/lfsr3-1.png" target="_blank">a1180b5</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/624de1dfb3db11bdc40a4a3d946a8fbd7abf40ce/docs/figure/mashvflash.Rmd/lfsr3-1.png" target="_blank">624de1d</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/9f81f0649020df46c440e91d61356176321aa930/docs/figure/mashvflash.Rmd/lfsr3-1.png" target="_blank">9f81f06</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-11
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/6508f2be3666427c3f8bd6c348e59ab7b9355565/docs/figure/mashvflash.Rmd/lfsr3-1.png" target="_blank">6508f2b</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-09
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/861d4b3595125a0b8a304aa13375835acf4c8505/docs/figure/mashvflash.Rmd/lfsr3-1.png" target="_blank">861d4b3</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-09
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/MASHvFLASH/blob/a76f3cf9f3dad726abb36bc409df06e9c31f57d2/docs/figure/mashvflash.Rmd/lfsr3-1.png" target="_blank">a76f3cf</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-06-09
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>The FLASH fit took 27.86 s. The MASH fit took 3.38 s.</p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Sierra 10.12.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/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] MASS_7.3-48 mashr_0.2-7 ashr_2.2-7 flashr_0.5-8
loaded via a namespace (and not attached):
[1] Rcpp_0.12.16 pillar_1.2.1
[3] plyr_1.8.4 compiler_3.4.3
[5] git2r_0.21.0 workflowr_1.0.1
[7] R.methodsS3_1.7.1 R.utils_2.6.0
[9] iterators_1.0.9 tools_3.4.3
[11] testthat_2.0.0 digest_0.6.15
[13] tibble_1.4.2 evaluate_0.10.1
[15] memoise_1.1.0 gtable_0.2.0
[17] lattice_0.20-35 rlang_0.2.0
[19] Matrix_1.2-12 foreach_1.4.4
[21] commonmark_1.4 yaml_2.1.17
[23] parallel_3.4.3 mvtnorm_1.0-7
[25] ebnm_0.1-11 withr_2.1.1.9000
[27] stringr_1.3.0 roxygen2_6.0.1.9000
[29] xml2_1.2.0 knitr_1.20
[31] REBayes_1.2 devtools_1.13.4
[33] rprojroot_1.3-2 grid_3.4.3
[35] R6_2.2.2 rmarkdown_1.8
[37] rmeta_3.0 ggplot2_2.2.1
[39] magrittr_1.5 whisker_0.3-2
[41] backports_1.1.2 scales_0.5.0
[43] codetools_0.2-15 htmltools_0.3.6
[45] assertthat_0.2.0 softImpute_1.4
[47] colorspace_1.3-2 stringi_1.1.6
[49] Rmosek_7.1.3 lazyeval_0.2.1
[51] munsell_0.4.3 doParallel_1.0.11
[53] pscl_1.5.2 truncnorm_1.0-8
[55] SQUAREM_2017.10-1 ExtremeDeconvolution_1.3
[57] R.oo_1.21.0 </code></pre>
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