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<h1 class="title toc-ignore">Gaussian variance estimation in simulated data sets</h1>
<h4 class="author"><em>Zhengrong Xing, Peter Carbonetto and Matthew Stephens</em></h4>
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<p><strong>Last updated:</strong> 2018-11-09</p>
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<pre><code>
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</code></pre>
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<p></details></p>
<hr />
<p>This analysis implements the “Gaussian variance estimation” simulation experiments in the paper. In particular, we compare the Mean Field Variational Bayes (MFVB) method against SMASH in two scenarios. The figure and table generated at the end of this script should match up with the figure and table shown in the paper.</p>
<p>Running the code could take several hours to complete as it runs the two methods on 100 simulated data sets for each of the two scenarios.</p>
<p>We thank M. Menictas & M. Wand for generously sharing code that was used to implement these experiments.</p>
<div id="initial-setup-instructions" class="section level2">
<h2>Initial setup instructions</h2>
<p>To run this example on your own computer, please follow these setup instructions. These instructions assume you already have R and/or RStudio installed on your computer.</p>
<p>First, download or clone the <a href="https://github.com/stephenslab/smash-paper">git repository</a> on your computer.</p>
<p>Launch R, and change the working directory to be the “analysis” folder inside your local copy of the git repository.</p>
<p>Finally, install the smashr package from GitHub:</p>
<pre class="r"><code>devtools::install_github("stephenslab/smashr")</code></pre>
<p>See the “Session Info” at the bottom for the versions of the software and R packages that were used to generate the results shown below.</p>
</div>
<div id="set-up-r-environment" class="section level2">
<h2>Set up R environment</h2>
<p>Load the smashr package, as well as some functions used in the analysis below.</p>
<pre class="r"><code>library(smashr)
source("../code/mfvb.R")</code></pre>
</div>
<div id="analysis-settings" class="section level2">
<h2>Analysis settings</h2>
<p>Specify the number of data sets simulated in the first and second simulation scenarios.</p>
<pre class="r"><code>nsim1 <- 100
nsim2 <- 100</code></pre>
<p>Next, specify the hyperparameters used in running the MFVB method.</p>
<pre class="r"><code>Au.hyp <- 1e5
Av.hyp <- 1e5
sigsq.gamma <- 1e10
sigsq.beta <- 1e10</code></pre>
<p>These variables specify some colours used in the plots.</p>
<pre class="r"><code>mainCol <- "darkslateblue"
ptCol <- "paleturquoise3"
lineCol <- "skyblue"
axisCol <- "black"</code></pre>
<p>These are additional plotting parameters.</p>
<pre class="r"><code>cex.pt <- 0.75
cex.mainVal <- 1.7
cex.labVal <- 1.3
xlabVal <- "x"</code></pre>
</div>
<div id="plot-mean-and-variance-functions-used-to-simulate-data" class="section level2">
<h2>Plot mean and variance functions used to simulate data</h2>
<p>Compare this plot against the one shown in Fig. 4 of the paper.</p>
<pre class="r"><code>xgrid <- (0:10000)/10000
plot(xgrid,fTrue(xgrid),type = "l",ylim = c(-5,5),ylab = "y",xlab = "X",
lwd = 2)
lines(xgrid,fTrue(xgrid) + 2*sqrt(gTrue(xgrid)),col = "darkorange",lwd = 2)
lines(xgrid,fTrue(xgrid) - 2*sqrt(gTrue(xgrid)),col = "darkorange",lwd = 2)</code></pre>
<p><img src="figure/gaussvarest.Rmd/plot-mean-and-variance-1.png" width="576" style="display: block; margin: auto;" /></p>
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<p></details></p>
</div>
<div id="first-simulation-scenario-unevenly-spaced-data" class="section level2">
<h2>First simulation scenario: unevenly spaced data</h2>
<p>In the first scenario, we simulate data sets with 500 unevenly spaced data points, and assess accuracy, separately for the mean and variance estimates) by computing the mean of the squared errors (MSE) evaluated at 201 equally spaced points.</p>
<pre class="r"><code>mse.mu.uneven.mfvb <- 0
mse.mu.uneven.smash <- 0
mse.sd.uneven.mfvb <- 0
mse.sd.uneven.smash <- 0</code></pre>
<p>Run the SMASH and MFVB methods for each simulated data set.</p>
<pre class="r"><code>cat(sprintf("Running %d simulations: ",nsim1))
for (j in 1:nsim1) {
cat(sprintf("%d ",j))
# SIMULATE DATA
set.seed(3*j)
n <- 500
xOrig <- runif(n)
set.seed(3*j)
yOrig <- fTrue(xOrig) + sqrt(exp(loggTrue(xOrig)))*rnorm(n)
aOrig <- min(xOrig)
bOrig <- max(xOrig)
mean.x <- mean(xOrig)
sd.x <- sd(xOrig)
mean.y <- mean(yOrig)
sd.y <- sd(yOrig)
a <- (aOrig - mean.x)/sd.x
b <- (bOrig - mean.x)/sd.x
x <- (xOrig - mean.x)/sd.x
y <- (yOrig - mean.y)/sd.y
numIntKnotsU <- 17
intKnotsU <- quantile(x,seq(0,1,length=numIntKnotsU+2)[-c(1,numIntKnotsU+2)])
Zu <- ZOSull(x,intKnots=intKnotsU,range.x=c(a,b))
numKnotsU <- ncol(Zu)
numIntKnotsV <- numIntKnotsU
intKnotsV <-
quantile(x,seq(0,1,length = numIntKnotsV + 2)[-c(1,numIntKnotsV+2)])
Zv <- ZOSull(x,intKnots=intKnotsV,range.x=c(a,b))
numKnotsV <- ncol(Zv)
# RUN MEAN FIELD VARIATIONAL BAYES
X <- cbind(rep(1,n),x)
Cumat <- cbind(X,Zu)
Cvmat <- cbind(X,Zv)
ncX <- ncol(X)
ncZu <- ncol(Zu)
ncZv <- ncol(Zv)
ncCu <- ncol(Cumat)
ncCv <- ncol(Cvmat)
MFVBfit <- meanVarMFVB(y,X,ncZu,ncZv,Au.hyp,Av.hyp,
sigsq.gamma,sigsq.beta)
ng <- 201
xgOrig <- seq(aOrig,bOrig,length=ng)
xg <- (xgOrig - mean.x)/sd.x
Xg <- cbind(rep(1,ng),xg)
Zug <- ZOSull(xg,intKnots=intKnotsU,range.x=c(a,b))
Cug <- cbind(Xg,Zug)
Zvg <- ZOSull(xg,intKnots=intKnotsV,range.x=c(a,b))
Cvg <- cbind(Xg,Zvg)
mu.q.nu <- MFVBfit$mu.q.nu
mu.q.omega <- MFVBfit$mu.q.omega
Sigma.q.nu <- MFVBfit$Sigma.q.nu
Sigma.q.omega <- MFVBfit$Sigma.q.omega
fhatMFVBg <- Cug%*%mu.q.nu
fhatMFVBgOrig <- fhatMFVBg*sd.y + mean.y
logghatMFVBg <- Cvg%*%mu.q.omega
logghatMFVBgOrig <- logghatMFVBg + 2*log(sd.y)
sdloggMFVBgOrig <- sqrt(diag(Cvg%*%Sigma.q.omega%*%t(Cvg)))
credLowloggMFVBgOrig <- logghatMFVBgOrig - qnorm(0.975)*sdloggMFVBgOrig
credUpploggMFVBgOrig <- logghatMFVBgOrig + qnorm(0.975)*sdloggMFVBgOrig
sqrtghatMFVBg <- exp(0.5*Cvg %*% mu.q.omega
+ 0.125*diag(Cvg%*%Sigma.q.omega%*%t(Cvg)))
sqrtghatMFVBgOrig <- sqrtghatMFVBg*sd.y
# RUN SMASH
x.mod <- unique(sort(xOrig))
y.mod <- 0
for(i in 1:length(x.mod))
y.mod[i] <- median(yOrig[xOrig == x.mod[i]])
y.exp <- c(y.mod,y.mod[length(y.mod):(2*length(y.mod)-2^9+1)])
y.final <- c(y.exp,y.exp[length(y.exp):1])
mu.est <- smash.gaus(y.final,filter.number=1,family="DaubExPhase")
var.est <- smash.gaus(y.final,v.est=TRUE)
mu.est <- mu.est[1:500]
var.est <- var.est[1:500]
mu.est.inter <- approx(x.mod,mu.est,xgOrig,'linear')$y
var.est.inter <- approx(x.mod,var.est,xgOrig,'linear')$y
mse.mu.uneven.mfvb[j]<-mean((fhatMFVBgOrig - fTrue(xgOrig))^2)
mse.sd.uneven.mfvb[j]<-mean((sqrtghatMFVBgOrig-exp((loggTrue(xgOrig))/2))^2)
mu.est <- smash.gaus(y.final,filter.number=8,family="DaubLeAsymm")
var.est <- smash.gaus(y.final,v.est=TRUE,v.basis=TRUE,filter.number=8,
family="DaubLeAsymm")
mu.est <- mu.est[1:500]
var.est <- var.est[1:500]
mu.est.inter <- approx(x.mod,mu.est,xgOrig,'linear')$y
var.est.inter <- approx(x.mod,var.est,xgOrig,'linear')$y
mse.mu.uneven.smash[j] <- mean((mu.est.inter-fTrue(xgOrig))^2)
mse.sd.uneven.smash[j] <-
mean((sqrt(var.est.inter)-exp((loggTrue(xgOrig))/2))^2)
}
# Running 100 simulations: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100</code></pre>
</div>
<div id="second-simulation-scenario-evenly-spaced-points" class="section level2">
<h2>Second simulation scenario: evenly spaced points</h2>
<p>In this scenario, we simulate data sets with 1,024 evenly spaced data points. We assess accuracy separately for the mean and standard deviation as the mean of the MSEs evaluated at each of the locations.</p>
<pre class="r"><code>mse.mu.even.mfvb <- 0
mse.mu.even.smash <- 0
mse.sd.even.mfvb <- 0
mse.sd.even.smash <- 0</code></pre>
<p>Run the SMASH and MFVB methods for each simulated data set.</p>
<pre class="r"><code>cat(sprintf("Running %d simulations: ",nsim2))
for (j in 1:nsim2) {
cat(sprintf("%d ",j))
# SIMULATE DATA
n <- 2^10
xOrig <- (1:n)/n
set.seed(30*j)
yOrig <- fTrue(xOrig) + sqrt(exp(loggTrue(xOrig)))*rnorm(n)
aOrig <- min(xOrig)
bOrig <- max(xOrig)
mean.x <- mean(xOrig)
sd.x <- sd(xOrig)
mean.y <- mean(yOrig)
sd.y <- sd(yOrig)
a <- (aOrig - mean.x)/sd.x
b <- (bOrig - mean.x)/sd.x
x <- (xOrig - mean.x)/sd.x
y <- (yOrig - mean.y)/sd.y
numIntKnotsU <- 17
intKnotsU <- quantile(x,seq(0,1,length=numIntKnotsU+2)[-c(1,numIntKnotsU+2)])
Zu <- ZOSull(x,intKnots=intKnotsU,range.x=c(a,b))
numKnotsU <- ncol(Zu)
numIntKnotsV <- numIntKnotsU
intKnotsV <- quantile(x,seq(0,1,length=numIntKnotsV+2)[-c(1,numIntKnotsV+2)])
Zv <- ZOSull(x,intKnots=intKnotsV,range.x=c(a,b))
numKnotsV <- ncol(Zv)
# RUN MEAN FIELD VARIATIONAL BAYES
X <- cbind(rep(1,n),x)
Cumat <- cbind(X,Zu)
Cvmat <- cbind(X,Zv)
ncX <- ncol(X)
ncZu <- ncol(Zu)
ncZv <- ncol(Zv)
ncCu <- ncol(Cumat)
ncCv <- ncol(Cvmat)
MFVBfit <- meanVarMFVB(y,X,ncZu,ncZv,Au.hyp,Av.hyp,
sigsq.gamma,sigsq.beta)
ng <- 2^10
xgOrig <- seq(aOrig,bOrig,length=ng)
xg <- (xgOrig - mean.x)/sd.x
Xg <- cbind(rep(1,ng),xg)
Zug <- ZOSull(xg,intKnots=intKnotsU,range.x=c(a,b))
Cug <- cbind(Xg,Zug)
Zvg <- ZOSull(xg,intKnots=intKnotsV,range.x=c(a,b))
Cvg <- cbind(Xg,Zvg)
mu.q.nu <- MFVBfit$mu.q.nu
mu.q.omega <- MFVBfit$mu.q.omega
Sigma.q.nu <- MFVBfit$Sigma.q.nu
Sigma.q.omega <- MFVBfit$Sigma.q.omega
# Get the mean function estimate.
fhatMFVBg <- Cug %*% mu.q.nu
fhatMFVBgOrig <- fhatMFVBg*sd.y + mean.y
logghatMFVBg <- Cvg%*%mu.q.omega
logghatMFVBgOrig <- logghatMFVBg + 2*log(sd.y)
sdloggMFVBgOrig <- sqrt(diag(Cvg%*%Sigma.q.omega%*%t(Cvg)))
credLowloggMFVBgOrig <- logghatMFVBgOrig - qnorm(0.975)*sdloggMFVBgOrig
credUpploggMFVBgOrig <- logghatMFVBgOrig + qnorm(0.975)*sdloggMFVBgOrig
sqrtghatMFVBg <- exp(0.5*Cvg%*%mu.q.omega
+ 0.125*diag(Cvg%*%Sigma.q.omega%*%t(Cvg)))
sqrtghatMFVBgOrig <- sqrtghatMFVBg*sd.y
# RUN SMASH
mu.est <- smash.gaus(yOrig,filter.number=1,family="DaubExPhase")
var.est <- smash.gaus(yOrig,v.est=TRUE)
mse.mu.even.mfvb[j] <- mean((fhatMFVBgOrig-fTrue(xgOrig))^2)
mse.sd.even.mfvb[j] <- mean((sqrtghatMFVBgOrig-exp((loggTrue(xgOrig))/2))^2)
mu.est <- smash.gaus(yOrig,filter.number=8,family="DaubLeAsymm")
var.est <- smash.gaus(yOrig,v.est=TRUE,v.basis=TRUE,filter.number=8,
family = "DaubLeAsymm")
mse.mu.even.smash[j] <- mean((mu.est - fTrue(xgOrig))^2)
mse.sd.even.smash[j] <- mean((sqrt(var.est)-exp((loggTrue(xgOrig))/2))^2)
}
# Running 100 simulations: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100</code></pre>
</div>
<div id="summarize-results-of-simulations" class="section level2">
<h2>Summarize results of simulations</h2>
<p>The following two tables show the mean squared error (MSE) averaged over the 100 simulations in each of the scenarios. Compare these results with Table 1 in the paper.</p>
<pre class="r"><code>mse.table1 <- rbind(c(mean(mse.mu.uneven.mfvb),mean(mse.sd.uneven.mfvb)),
c(mean(mse.mu.uneven.smash),mean(mse.sd.uneven.smash)))
mse.table2 <- rbind(c(mean(mse.mu.even.mfvb),mean(mse.sd.even.mfvb)),
c(mean(mse.mu.even.smash),mean(mse.sd.even.smash)))
rownames(mse.table1) <- c("MFVB","SMASH")
colnames(mse.table1) <- c("mean","sd")
rownames(mse.table2) <- c("MFVB","SMASH")
colnames(mse.table2) <- c("mean","sd")
cat(sprintf("MSE averaged across %d simulations in Scenario 1:\n",nsim1))
print(mse.table1)
cat("\n")
cat(sprintf("MSE averaged across %d simulations in Scenario 2:\n",nsim2))
print(mse.table2)
# MSE averaged across 100 simulations in Scenario 1:
# mean sd
# MFVB 0.03302778 0.01991272
# SMASH 0.03343833 0.01869602
#
# MSE averaged across 100 simulations in Scenario 2:
# mean sd
# MFVB 0.01721438 0.008471983
# SMASH 0.01582035 0.006513889</code></pre>
<p>In Scenario 1, the data are not equally spaced, and the number of data points is not a power of 2; in this setting, SMASH is more accurate in estimating both the mean and s.d.</p>
<p>In Scenario 2, the data are equally spaced, and the number of data points is a power of 2; SMASH again outperforms MFVB in both mean and s.d. estimation.</p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()
# R version 3.4.3 (2017-11-30)
# 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/3.4/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] splines stats graphics grDevices utils datasets methods
# [8] base
#
# other attached packages:
# [1] smashr_1.2-0
#
# loaded via a namespace (and not attached):
# [1] Rcpp_0.12.19 compiler_3.4.3 git2r_0.23.0
# [4] workflowr_1.1.1 R.methodsS3_1.7.1 R.utils_2.6.0
# [7] bitops_1.0-6 iterators_1.0.9 tools_3.4.3
# [10] digest_0.6.17 evaluate_0.11 lattice_0.20-35
# [13] Matrix_1.2-12 foreach_1.4.4 yaml_2.2.0
# [16] parallel_3.4.3 stringr_1.3.1 knitr_1.20
# [19] caTools_1.17.1 REBayes_1.3 rprojroot_1.3-2
# [22] grid_3.4.3 data.table_1.11.4 rmarkdown_1.10
# [25] ashr_2.2-23 magrittr_1.5 whisker_0.3-2
# [28] backports_1.1.2 codetools_0.2-15 htmltools_0.3.6
# [31] MASS_7.3-48 assertthat_0.2.0 wavethresh_4.6.8
# [34] stringi_1.2.4 Rmosek_8.0.69 doParallel_1.0.11
# [37] pscl_1.5.2 truncnorm_1.0-8 SQUAREM_2017.10-1
# [40] R.oo_1.21.0</code></pre>
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