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<h1 class="title toc-ignore">MASH v FLASH</h1>
</div>
<p><strong>Last updated:</strong> 2018-06-15</p>
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<p><details> <summary> <strong style="color:blue;">✔</strong> <strong>Repository version:</strong> <a href="https://github.com/willwerscheid/MASHvFLASH/tree/0c0d6b22487b32387084dd31b4353c782cc56c3b" target="_blank">0c0d6b2</a> </summary></p>
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<a href="https://github.com/willwerscheid/MASHvFLASH/blob/0c0d6b22487b32387084dd31b4353c782cc56c3b/analysis/MASHvFLASHsims.Rmd" target="_blank">0c0d6b2</a>
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<p></details></p>
<hr />
<p>All simulated datasets <span class="math inline">\(Y\)</span> were of dimension 25 x 1000.</p>
<div id="null-model" class="section level2">
<h2>Null model</h2>
<p>Here the entries of <span class="math inline">\(Y\)</span> are just independent <span class="math inline">\(N(0, 1)\)</span> draws.</p>
<table>
<thead>
<tr class="header">
<th></th>
<th align="right">MASH</th>
<th align="right">FLASH_OHL</th>
<th align="right">FLASH_OHF</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>MSE</td>
<td align="right">0.002209</td>
<td align="right">0.0025228</td>
<td align="right">0.0025228</td>
</tr>
<tr class="even">
<td>95% CI cov</td>
<td align="right">0.663880</td>
<td align="right">0.9999920</td>
<td align="right">0.9999880</td>
</tr>
</tbody>
</table>
<p><img src="./output/sim1time.png" style="display: block; margin: auto;" /></p>
</div>
<div id="code" class="section level2">
<h2>Code</h2>
<p>for simulating datasets…</p>
<pre class="r"><code>## SIMULATION FUNCTIONS -------------------------------------------------
# n is number of conditions, p is number of genes
# Noise is i.i.d. N(0, 1)
get_E <- function(n, p, sd = 1) {
matrix(rnorm(n * p, 0, sd), n, p)
}
# Simulate from null model ----------------------------------------------
null_sim <- function(n, p, seed = NULL) {
set.seed(seed)
Y <- get_E(n, p)
true_Y <- matrix(0, n, p)
list(Y = Y, true_Y = true_Y)
}
# Simulate from MASH model ----------------------------------------------
# Sigma is list of covariance matrices
# pi[j] is probability that effect j has covariance Sigma[[j]]
# s is sparsity (percentage of null effects)
mash_sim <- function(n, p, Sigma, pi = NULL, s = 0.8, seed = NULL) {
set.seed(NULL)
if (is.null(pi)) {
pi = rep(1, length(Sigma)) # default to uniform distribution
}
assertthat::are_equal(length(pi), length(Sigma))
for (j in length(Sigma)) {
assertthat::are_equal(dim(Sigma[j]), c(n, n))
}
pi <- pi / sum(pi) # normalize pi to sum to one
which_sigma <- sample(1:length(pi), p, replace=TRUE, prob=pi)
nonnull_fx <- sample(1:p, floor((1 - s)*p), replace=FALSE)
X <- matrix(0, n, p)
for (j in nonnull_fx) {
X[, j] <- MASS::mvrnorm(1, rep(0, n), Sigma[[which_sigma[j]]])
}
Y <- X + get_E(n, p)
list(Y = Y, true_Y = X)
}
# Simulate from FLASH model ---------------------------------------------
# fs is sparsity of factors (percentage of null effects)
# fvar is variance of effects (generated from normal distribution)
# ls is sparsity of loadings
# lvar is variance of loadings
# UVvar is variance of dense rank-one matrix included to mimic something
# like unwanted variation (set it to 0 to ignore it)
flash_sim <- function(n, p, k, fs, fvar, ls, lvar, UVvar = 0, seed = NULL) {
set.seed(seed)
nonnull_ll <- matrix(sample(c(0, 1), n*k, TRUE, c(ls, 1 - ls)), n, k)
LL <- nonnull_ll * matrix(rnorm(n*k, 0, sqrt(lvar)), n, k)
nonnull_ff <- matrix(sample(c(0, 1), k*p, TRUE, c(fs, 1 - fs)), k, p)
FF <- nonnull_ff * matrix(rnorm(k*p, 0, sqrt(fvar)), k, p)
X <- LL %*% FF
Y <- X + get_E(n, p)
# add unwanted variation
Y <- Y + outer(rnorm(n, 0, sqrt(UVvar)), rnorm(p, 0, sqrt(UVvar)))
list(Y = Y, true_Y = X)
}
## SIMULATIONS ----------------------------------------------------------
# Functions to generate seven types of datasets. One is null; three are
# from the MASH model; three are from the FLASH model.
sim_fns <- function(n, p, s, mashvar, fvar, lvar, UVvar) {
# 1. Everything is null
sim_null <- function(){ null_sim(n, p) }
Sigma <- list()
Sigma[[1]] <- diag(rep(mashvar, n))
# 2. Effects are independent across conditions
sim_ind <- function(){ mash_sim(n, p, Sigma) }
Sigma[[2]] <- matrix(mashvar, n, n)
# 3. Effects are either independent or shared
sim_indsh <- function(){ mash_sim(n, p, Sigma) }
for (j in 1:n) {
Sigma[[2 + j]] <- matrix(0, n, n)
Sigma[[2 + j]][j, j] <- mashvar
}
pi <- c(n, n, rep(1, n))
# 4. Effects are independent, shared, or unique to a single condition
sim_mash <- function(){ mash_sim(n, p, Sigma) }
# 5. Rank one model
sim_rank1 <- function(){ flash_sim(n, p, 1, s, fvar, 0.5, lvar) }
# 6. Rank 5 model
sim_rank5 <- function(){ flash_sim(n, p, 5, s, fvar, 0.2, lvar) }
# 7. Rank 3 model with unwanted variation
sim_UV <- function(){ flash_sim(n, p, 3, s, fvar, 0.3, lvar, UVvar) }
c(sim_null, sim_ind, sim_indsh, sim_mash, sim_rank1, sim_rank5, sim_UV)
}
sim_names <- c("Null simulation", "All independent effects",
"Independent and shared", "Independent, shared, and unique",
"Rank 1 FLASH model", "Rank 5 FLASH model",
"Rank 3 FLASH with UV")</code></pre>
<p>…for fitting MASH and FLASH objects…</p>
<pre class="r"><code># Fit using FLASH -------------------------------------------------------
fit_flash <- function(Y, Kmax, add_onehots_first=T) {
n <- nrow(Y)
data <- flash_set_data(Y, S = 1)
timing <- list()
t0 <- Sys.time()
if (add_onehots_first) {
fl <- flash_add_fixed_l(data, diag(rep(1, n)))
fl <- flash_backfit(data, fl, nullcheck = F, var_type = "zero")
t1 <- Sys.time()
timing$backfit <- t1 - t0
fl <- flash_add_greedy(data, Kmax, fl, var_type = "zero")
timing$greedy <- Sys.time() - t1
} else {
fl <- flash_add_greedy(data, Kmax, var_type = "zero")
t1 <- Sys.time()
timing$greedy <- t1 - t0
fl <- flash_add_fixed_l(data, diag(rep(1, n)), fl)
fl <- flash_backfit(data, fl, nullcheck = F, var_type = "zero")
timing$backfit <- Sys.time() - t1
}
timing$total <- Reduce(`+`, timing)
list(fl = fl, timing = timing)
}
# Fit using MASH -------------------------------------------------------
fit_mash <- function(Y, ed=T) {
data <- mash_set_data(t(Y))
timing <- list()
# time to create canonical matrices is negligible
U = cov_canonical(data)
if (ed) {
t0 <- Sys.time()
m.1by1 <- mash_1by1(data)
strong <- get_significant_results(m.1by1, 0.05)
U.pca <- cov_pca(data, 5, strong)
U.ed <- cov_ed(data, U.pca, strong)
U <- c(U, U.ed)
timing$ed <- Sys.time() - t0
}
t0 <- Sys.time()
m <- mash(data, U)
timing$mash <- Sys.time() - t0
timing$total <- Reduce(`+`, timing)
list(m = m, timing = timing)
}</code></pre>
<p>…for evaluating performance…</p>
<pre class="r"><code># Evaluate methods based on MSE, CI coverage, and TPR vs. FPR -----------
flash_diagnostics <- function(fl, Y, true_Y, nsamp) {
MSE <- flash_mse(fl, true_Y)
# Sample from FLASH fit to estimate CI coverage and TPR vs. FPR
fl_sampler <- flash_lf_sampler(Y, fl, ebnm_fn=ebnm_pn, fixed="loadings")
fl_samp <- fl_sampler(nsamp)
CI <- flash_ci(fl_samp, true_Y)
ROC <- flash_roc(fl, fl_samp, true_Y)
list(MSE = MSE, CI = CI, TP = ROC$TP, FP = ROC$FP,
n_nulls = ROC$n_nulls, n_nonnulls = ROC$n_nonnulls)
}
mash_diagnostics <- function(m, true_Y) {
MSE <- mash_mse(m, true_Y)
CI <- mash_ci(m, true_Y)
ROC <- mash_roc(m, true_Y)
list(MSE = MSE, CI = CI, TP = ROC$TP, FP = ROC$FP,
n_nulls = ROC$n_nulls, n_nonnulls = ROC$n_nonnulls)
}
# MSE of posterior means (FLASH) ----------------------------------------
flash_mse <- function(fl, true_Y) {
mean((flash_get_lf(fl) - true_Y)^2)
}
# MSE for MASH ----------------------------------------------------------
mash_mse <- function(m, true_Y) {
mean((get_pm(m) - t(true_Y))^2)
}
# 95% CI coverage for FLASH ---------------------------------------------
flash_ci <- 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))}))
mean((as.vector(true_Y) >= CI[, 1]) & (as.vector(true_Y) <= CI[, 2]))
}
# 95% CI coverage for MASH ----------------------------------------------
mash_ci <- function(m, true_Y) {
Y <- t(true_Y)
mean((Y > get_pm(m) - 1.96 * get_psd(m))
& (Y < get_pm(m) + 1.96 * get_psd(m)))
}
# LFSR for FLASH --------------------------------------------------------
flash_lfsr <- function(fl_samp) {
nsamp <- length(fl_samp)
n <- nrow(fl_samp[[1]])
p <- ncol(fl_samp[[1]])
pp <- matrix(0, nrow=n, ncol=p)
pn <- matrix(0, nrow=n, ncol=p)
for (i in 1:nsamp) {
pp <- pp + (fl_samp[[i]] > 0)
pn <- pn + (fl_samp[[i]] < 0)
}
1 - pmax(pp, pn) / nsamp
}
# Quantities for plotting ROC curves -----------------------------------
flash_roc <- function(fl, fl_samp, true_Y, step=0.01) {
roc_data(flash_get_lf(fl), true_Y, flash_lfsr(fl_samp), step)
}
mash_roc <- function(m, true_Y, step=0.01) {
roc_data(get_pm(m), t(true_Y), get_lfsr(m), step)
}
roc_data <- function(pm, true_Y, lfsr, step) {
correct_sign <- pm * true_Y > 0
is_null <- true_Y == 0
n_nulls <- sum(is_null)
n_nonnulls <- length(true_Y) - n_nulls
ts <- seq(0, 1, by=step)
tp <- rep(0, length(ts))
fp <- rep(0, length(ts))
for (t in 1:length(ts)) {
signif <- lfsr <= ts[t]
tp[t] <- sum(signif & correct_sign)
fp[t] <- sum(signif & is_null)
}
list(ts = ts, TP = tp, FP = fp, n_nulls = n_nulls, n_nonnulls = n_nonnulls)
}
# empirical false sign rate vs. local false sign rate
# efsr_by_lfsr <- function(pm, true_Y, lfsr, step) {
# pred_signs <- sign(pm)
# pred_zeros <- pred_signs == 0
# pred_signs[pred_zeros] <- sample(c(0, 1), length(pred_zeros), replace=T)
#
# gotitright <- (pred_signs == sign(true_Y))
#
# nsteps <- floor(.5 / step)
# efsr_by_lfsr <- rep(0, nsteps)
# for (k in 1:nsteps) {
# idx <- (lfsr >= (step * (k - 1)) & lfsr < (step * k))
# efsr_by_lfsr[k] <- ifelse(sum(idx) == 0, NA,
# 1 - sum(gotitright[idx]) / sum(idx))
# }
# efsr_by_lfsr
# }</code></pre>
<p>…and some ugly functions that run everything and plot results.</p>
<pre class="r"><code>run_sims <- function(sim_fn, nsims, plot_title, fpath) {
#suppressMessages(
#suppressWarnings(
#capture.output(
if (nsims == 1) {
res = run_one_sim(sim_fn)
} else {
res = run_many_sims(sim_fn, nsims)
}
#)
#)
#)
saveRDS(output_res_mat(res, plot_title), paste0(fpath, "res.rds"))
if (!(plot_title == "Null simulation")) {
png(paste0(fpath, "ROC.png"))
plot_ROC(res, plot_title)
dev.off()
}
png(paste0(fpath, "time.png"))
plot_timing(res)
dev.off()
}
run_many_sims <- function(sim_fn, nsims) {
res <- list()
combined_res <- list()
for (i in 1:nsims) {
res[[i]] <- run_one_sim(sim_fn)
}
list_elem <- names(res[[1]])
for (elem in list_elem) {
combined_res[[elem]] <- list()
sub_elems <- names(res[[1]][[elem]])
for (sub_elem in sub_elems) {
tmp <- lapply(res, function(x) {x[[elem]][[sub_elem]]})
combined_res[[elem]][[sub_elem]] <- Reduce(`+`, tmp)
combined_res[[elem]][[sub_elem]] <- combined_res[[elem]][[sub_elem]] / nsims
}
}
combined_res
}
run_one_sim <- function(sim_fn, Kmax = 10, nsamp=200) {
data <- do.call(sim_fn, list())
# If there are no strong signals, trying to run ED throws an error, so
# we need to do some error handling to fit the MASH object
try(mfit <- fit_mash(data$Y))
if (!exists("mfit")) {
mfit <- fit_mash(data$Y, ed=F)
mfit$timing$ed = as.difftime(0, units="secs")
}
flfit1 <- fit_flash(data$Y, Kmax, add_onehots_first = T)
flfit2 <- fit_flash(data$Y, Kmax, add_onehots_first = F)
message("Running MASH diagnostics")
mres <- mash_diagnostics(mfit$m, data$true_Y)
message("Running FLASH diagnostics")
flres1 <- flash_diagnostics(flfit1$fl, data$Y, data$true_Y, nsamp)
flres2 <- flash_diagnostics(flfit2$fl, data$Y, data$true_Y, nsamp)
list(mash_timing = mfit$timing, mash_res = mres,
flash_OHF_timing = flfit1$timing, flash_OHF_res = flres1,
flash_OHL_timing = flfit2$timing, flash_OHL_res = flres2)
}
output_res_mat <- function(res, caption) {
data.frame(MASH = c(res$mash_res$MSE, res$mash_res$CI),
FLASH_OHL = c(res$flash_OHL_res$MSE, res$flash_OHL_res$CI),
FLASH_OHF = c(res$flash_OHF_res$MSE, res$flash_OHF_res$CI),
row.names = c("MSE", "95% CI cov"))
}
plot_timing <- function(res) {
data <- c(res$mash_timing$ed, res$mash_timing$mash,
res$flash_OHL_timing$greedy, res$flash_OHL_timing$backfit,
res$flash_OHF_timing$greedy, res$flash_OHF_timing$backfit)
time_units <- units(data)
data <- matrix(as.numeric(data), 2, 3)
barplot(data, axes=T,
main=paste("Average time to fit in", time_units),
names.arg = c("MASH", "FLASH-OHL", "FLASH-OHF"),
legend.text = c("ED/Greedy", "MASH/Backfit"),
ylim = c(0, max(colSums(data))*2))
# (increasing ylim is easiest way to deal with legend getting in way)
}
plot_ROC <- function(res, main="ROC curve") {
m_y <- res$mash_res$TP / res$mash_res$n_nonnulls
m_x <- res$mash_res$FP / res$mash_res$n_nulls
ohl_y <- res$flash_OHL_res$TP / res$flash_OHL_res$n_nonnulls
ohl_x <- res$flash_OHL_res$FP / res$flash_OHL_res$n_nulls
ohf_y <- res$flash_OHF_res$TP / res$flash_OHF_res$n_nonnulls
ohf_x <- res$flash_OHF_res$FP / res$flash_OHF_res$n_nulls
plot(m_x, m_y, xlim=c(0, 1), ylim=c(0, 1), type='l',
xlab='FPR', ylab='TPR', main=main)
lines(ohl_x, ohl_y, lty=2)
lines(ohf_x, ohf_y, lty=3)
legend("bottomright", c("MASH", "FLASH-OHL", "FLASH-OHF"), lty=1:3)
}</code></pre>
</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
loaded via a namespace (and not attached):
[1] workflowr_1.0.1 Rcpp_0.12.17 digest_0.6.15
[4] rprojroot_1.3-2 R.methodsS3_1.7.1 backports_1.1.2
[7] git2r_0.21.0 magrittr_1.5 evaluate_0.10.1
[10] highr_0.6 stringi_1.1.6 whisker_0.3-2
[13] R.oo_1.21.0 R.utils_2.6.0 rmarkdown_1.8
[16] tools_3.4.3 stringr_1.3.0 yaml_2.1.17
[19] compiler_3.4.3 htmltools_0.3.6 knitr_1.20 </code></pre>
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