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<h1 class="title toc-ignore">Gaussian mean 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> 2018-12-04</p>
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2018-12-04
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<p></details></p>
<hr />
<p>In this analysis, we assess the ability of different signal denoising methods to recover the true signal after being provided with Gaussian-distributed observations of the signal. We consider scenarios in which the data have homoskedastic errors (constant variance) and heteroskedastic errors (non-constant variance).</p>
<p>Since the simulation experiments are computationally intensive, they were implemented separately (see the “dsc” directory in this git repository), and here we only create plots to summarize the results of these experiments.</p>
<div id="set-up-environment" class="section level2">
<h2>Set up environment</h2>
<p>Load the ggplot2 and cowplot packages, and the functions definining the mean and variances used to simulate the data.</p>
<pre class="r"><code>library(ggplot2)
library(cowplot)
source("../code/signals.R")</code></pre>
</div>
<div id="load-results" class="section level2">
<h2>Load results</h2>
<p>Load the results of the simulation experiments.</p>
<pre class="r"><code>load("../output/dscr.RData")</code></pre>
</div>
<div id="simulated-data-with-constant-variances" class="section level2">
<h2>Simulated data with constant variances</h2>
<p>This plot reproduces Fig. 2 of the manuscript, which compares the accuracy of the mean curves estimated from the data sets that were simulated using the “Spikes” mean function with constant variance.</p>
<p>First, extract the results used to generate this plot.</p>
<pre class="r"><code>homo.data.smash <-
res[res$.id == "sp.3.v1" &
res$method == "smash.s8",]
homo.data.smash.homo <-
res[res$.id == "sp.3.v1" &
res$method == "smash.homo.s8",]
homo.data.tithresh <-
res[res$.id == "sp.3.v1" &
res$method == "tithresh.homo.s8",]
homo.data.ebayes <-
res[res$.id == "sp.3.v1" &
res$method == "ebayesthresh",]
homo.data.smash.true <-
res[res$.id == "sp.3.v1" &
res$method == "smash.true.s8",]
homo.data <-
res[res$.id == "sp.3.v1" &
(res$method == "smash.s8" |
res$method == "ebayesthresh" |
res$method == "tithresh.homo.s8"),]</code></pre>
<p>Transform these results into a data frame suitable for ggplot2.</p>
<pre class="r"><code>pdat <-
rbind(data.frame(method = "smash",
method.type = "est",
mise = homo.data.smash$mise),
data.frame(method = "smash.homo",
method.type = "homo",
mise = homo.data.smash.homo$mise),
data.frame(method = "tithresh",
method.type = "homo",
mise = homo.data.tithresh$mise),
data.frame(method = "ebayesthresh",
method.type = "homo",
mise = homo.data.ebayes$mise),
data.frame(method = "smash.true",
method.type = "true",
mise = homo.data.smash.true$mise))
pdat <-
transform(pdat,
method = factor(method,
names(sort(tapply(pdat$mise,pdat$method,mean),
decreasing = TRUE))))</code></pre>
<p>Create the combined boxplot and violin plot using ggplot2.</p>
<pre class="r"><code>p <- ggplot(pdat,aes(x = method,y = mise,fill = method.type)) +
geom_violin(fill = "skyblue",color = "skyblue") +
geom_boxplot(width = 0.15,outlier.shape = NA) +
scale_y_continuous(breaks = seq(6,16,2)) +
scale_fill_manual(values = c("darkorange","dodgerblue","gold"),
guide = FALSE) +
coord_flip() +
labs(x = "",y = "MISE") +
theme(axis.line = element_blank(),
axis.ticks.y = element_blank())
print(p)</code></pre>
<p><img src="figure/gaussmeanest.Rmd/plot-1-create-1.png" width="480" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of plot-1-create-1.png:</em></summary>
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<a href="https://github.com/stephenslab/smash-paper/blob/05684ba3b0a6729c632d0127b82d8b563aa8a51c/docs/figure/gaussmeanest.Rmd/plot-1-create-1.png" target="_blank">05684ba</a>
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<p></details></p>
<p>From this plot, we see that the three variations of SMASH all outperformed EbayesThresh and TI thresholding in this setting.</p>
<p>Next, we compare the same methods in simulated data sets with heteroskedastic errors.</p>
</div>
<div id="simulated-data-with-heteroskedastic-errors-spikes-mean-signal-and-clipped-blocks-variance" class="section level2">
<h2>Simulated data with heteroskedastic errors: “Spikes” mean signal and “Clipped Blocks” variance</h2>
<p>In this scenario, the data sets were simulated using the “Spikes” mean function and the “Clipped Blocks” variance function. The next two plots reproduce part of Fig. 3 in the manuscript.</p>
<p>This plot shows the mean function as a block line, and the +/- 2 standard deviations as orange lines:</p>
<pre class="r"><code>t <- (1:1024)/1024
mu <- spikes.fn(t,"mean")
sigma.ini <- sqrt(cblocks.fn(t,"var"))
sd.fn <- sigma.ini/mean(sigma.ini) * sd(mu)/3
par(cex.axis = 1,cex.lab = 1.25)
plot(mu,type = "l", ylim = c(-0.05,1),xlab = "position",ylab = "",
lwd = 1.75,xaxp = c(0,1024,4),yaxp = c(0,1,4))
lines(mu + 2*sd.fn,col = "darkorange",lty = 5,lwd = 1.75)
lines(mu - 2*sd.fn,col = "darkorange",lty = 5,lwd = 1.75)</code></pre>
<p><img src="figure/gaussmeanest.Rmd/spikes-signal-1.png" width="576" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of spikes-signal-1.png:</em></summary>
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<a href="https://github.com/stephenslab/smash-paper/blob/05684ba3b0a6729c632d0127b82d8b563aa8a51c/docs/figure/gaussmeanest.Rmd/spikes-signal-1.png" target="_blank">05684ba</a>
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Peter Carbonetto
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2018-12-04
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<p></details></p>
<p>Extract the results from running the simulations.</p>
<pre class="r"><code>hetero.data.smash <-
res[res$.id == "sp.3.v5" & res$method == "smash.s8",]
hetero.data.smash.homo <-
res[res$.id == "sp.3.v5" & res$method == "smash.homo.s8",]
hetero.data.tithresh.homo <-
res[res$.id == "sp.3.v5" & res$method == "tithresh.homo.s8",]
hetero.data.tithresh.rmad <-
res[res$.id == "sp.3.v5" & res$method == "tithresh.rmad.s8",]
hetero.data.tithresh.smash <-
res[res$.id == "sp.3.v5" & res$method == "tithresh.smash.s8",]
hetero.data.tithresh.true <-
res[res$.id == "sp.3.v5" & res$method == "tithresh.true.s8",]
hetero.data.ebayes <-
res[res$.id == "sp.3.v5" & res$method == "ebayesthresh",]
hetero.data.smash.true <-
res[res$.id == "sp.3.v5" & res$method == "smash.true.s8",]</code></pre>
<p>Transform these results into a data frame suitable for ggplot2.</p>
<pre class="r"><code>pdat <-
rbind(data.frame(method = "smash",
method.type = "est",
mise = hetero.data.smash$mise),
data.frame(method = "smash.homo",
method.type = "homo",
mise = hetero.data.smash.homo$mise),
data.frame(method = "tithresh.rmad",
method.type = "tithresh",
mise = hetero.data.tithresh.rmad$mise),
data.frame(method = "tithresh.smash",
method.type = "tithresh",
mise = hetero.data.tithresh.smash$mise),
data.frame(method = "tithresh.true",
method.type = "tithresh",
mise = hetero.data.tithresh.true$mise),
data.frame(method = "ebayesthresh",
method.type = "homo",
mise = hetero.data.ebayes$mise),
data.frame(method = "smash.true",
method.type = "true",
mise = hetero.data.smash.true$mise))
pdat <-
transform(pdat,
method = factor(method,
names(sort(tapply(pdat$mise,pdat$method,mean),
decreasing = TRUE))))</code></pre>
<p>Create the combined boxplot and violin plot using ggplot2.</p>
<pre class="r"><code>p <- ggplot(pdat,aes(x = method,y = mise,fill = method.type)) +
geom_violin(fill = "skyblue",color = "skyblue") +
geom_boxplot(width = 0.15,outlier.shape = NA) +
scale_fill_manual(values=c("darkorange","dodgerblue","limegreen","gold"),
guide = FALSE) +
coord_flip() +
scale_y_continuous(breaks = seq(10,70,10)) +
labs(x = "",y = "MISE") +
theme(axis.line = element_blank(),
axis.ticks.y = element_blank())
print(p)</code></pre>
<p><img src="figure/gaussmeanest.Rmd/plot-2-create-1.png" width="480" style="display: block; margin: auto;" /></p>
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<a href="https://github.com/stephenslab/smash-paper/blob/05684ba3b0a6729c632d0127b82d8b563aa8a51c/docs/figure/gaussmeanest.Rmd/plot-2-create-1.png" target="_blank">05684ba</a>
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Peter Carbonetto
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2018-12-04
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<p></details></p>
<p>In this scenario, we see that SMASH, when allowing for heteroskedastic errors, outperforms EbayesThresh and all variants of TI thresholding (including TI thresholding when provided with the true variance). Further, SMASH performs almost as well when estimating the variance compared to when provided with the true variance.</p>
</div>
<div id="simulated-data-with-heteroskedastic-errors-corner-mean-signal-and-doppler-variance" class="section level2">
<h2>Simulated data with heteroskedastic errors: “Corner” mean signal and “Doppler” variance</h2>
<p>In this next scenario, the data sets were simulated using the “Corner” mean function and the “Doppler” variance function. These plots were also used in Fig. 3 of the manuscript.</p>
<p>This plot shows the mean function as a block line, and the +/- 2 standard deviations as orange lines:</p>
<pre class="r"><code>mu <- cor.fn(t,"mean")
sigma.ini <- sqrt(doppler.fn(t,"var"))
sd.fn <- sigma.ini/mean(sigma.ini) * sd(mu)/3
plot(mu,type = "l", ylim = c(-0.05,1),xlab = "position",ylab = "",
lwd = 1.75,xaxp = c(0,1024,4),yaxp = c(0,1,4))
lines(mu + 2*sd.fn,col = "darkorange",lty = 5,lwd = 1.75)
lines(mu - 2*sd.fn,col = "darkorange",lty = 5,lwd = 1.75)</code></pre>
<p><img src="figure/gaussmeanest.Rmd/corner-signal-1.png" width="576" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of corner-signal-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/stephenslab/smash-paper/blob/05684ba3b0a6729c632d0127b82d8b563aa8a51c/docs/figure/gaussmeanest.Rmd/corner-signal-1.png" target="_blank">05684ba</a>
</td>
<td style="text-align:left;">
Peter Carbonetto
</td>
<td style="text-align:left;">
2018-12-04
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>Extract the results from running these simulations.</p>
<pre class="r"><code>hetero.data.smash.2 <-
res[res$.id == "cor.3.v3" & res$method == "smash.s8",]
hetero.data.smash.homo.2 <-
res[res$.id == "cor.3.v3" & res$method == "smash.homo.s8",]
hetero.data.tithresh.homo.2 <-
res[res$.id == "cor.3.v3" & res$method == "tithresh.homo.s8",]
hetero.data.tithresh.rmad.2 <-
res[res$.id == "cor.3.v3" & res$method == "tithresh.rmad.s8",]
hetero.data.tithresh.smash.2 <-
res[res$.id == "cor.3.v3" & res$method == "tithresh.smash.s8",]
hetero.data.tithresh.true.2 <-
res[res$.id == "cor.3.v3" & res$method == "tithresh.true.s8",]
hetero.data.ebayes.2 <-
res[res$.id == "cor.3.v3" & res$method == "ebayesthresh",]
hetero.data.smash.true.2 <-
res[res$.id == "cor.3.v3" & res$method == "smash.true.s8",]</code></pre>
<p>Transform these results into a data frame suitable for ggplot2.</p>
<pre class="r"><code>pdat <-
rbind(data.frame(method = "smash",
method.type = "est",
mise = hetero.data.smash.2$mise),
data.frame(method = "smash.homo",
method.type = "homo",
mise = hetero.data.smash.homo.2$mise),
data.frame(method = "tithresh.rmad",
method.type = "tithresh",
mise = hetero.data.tithresh.rmad.2$mise),
data.frame(method = "tithresh.smash",
method.type = "tithresh",
mise = hetero.data.tithresh.smash.2$mise),
data.frame(method = "tithresh.true",
method.type = "tithresh",
mise = hetero.data.tithresh.true.2$mise),
data.frame(method = "ebayesthresh",
method.type = "homo",
mise = hetero.data.ebayes.2$mise),
data.frame(method = "smash.true",
method.type = "true",
mise = hetero.data.smash.true.2$mise))
pdat <-
transform(pdat,
method = factor(method,
names(sort(tapply(pdat$mise,pdat$method,mean),
decreasing = TRUE))))</code></pre>
<p>Create the combined boxplot and violin plot using ggplot2.</p>
<pre class="r"><code>p <- ggplot(pdat,aes(x = method,y = mise,fill = method.type)) +
geom_violin(fill = "skyblue",color = "skyblue") +
geom_boxplot(width = 0.15,outlier.shape = NA) +
scale_fill_manual(values=c("darkorange","dodgerblue","limegreen","gold"),
guide = FALSE) +
coord_flip() +
scale_y_continuous(breaks = seq(1,5)) +
labs(x = "",y = "MISE") +
theme(axis.line = element_blank(),
axis.ticks.y = element_blank())
print(p)</code></pre>
<p><img src="figure/gaussmeanest.Rmd/plot-3-create-1.png" width="480" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of plot-3-create-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/stephenslab/smash-paper/blob/05684ba3b0a6729c632d0127b82d8b563aa8a51c/docs/figure/gaussmeanest.Rmd/plot-3-create-1.png" target="_blank">05684ba</a>
</td>
<td style="text-align:left;">
Peter Carbonetto
</td>
<td style="text-align:left;">
2018-12-04
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>Similar to the “Spikes” scenario, we see that the SMASH method, when allowing for heteroskedastic variances, outperforms both the TI thresholding and EbayesThresh approaches.</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] stats graphics grDevices utils datasets methods base
#
# other attached packages:
# [1] cowplot_0.9.3 ggplot2_3.1.0
#
# loaded via a namespace (and not attached):
# [1] Rcpp_1.0.0 later_0.7.2 dscr_0.1-7
# [4] compiler_3.4.3 pillar_1.2.1 git2r_0.23.0
# [7] plyr_1.8.4 workflowr_1.1.1 bindr_0.1.1
# [10] R.methodsS3_1.7.1 R.utils_2.6.0 tools_3.4.3
# [13] digest_0.6.17 evaluate_0.11 tibble_1.4.2
# [16] gtable_0.2.0 pkgconfig_2.0.2 rlang_0.2.2
# [19] shiny_1.1.0 yaml_2.2.0 bindrcpp_0.2.2
# [22] withr_2.1.2 stringr_1.3.1 dplyr_0.7.6
# [25] knitr_1.20 rprojroot_1.3-2 grid_3.4.3
# [28] tidyselect_0.2.4 glue_1.3.0 R6_2.2.2
# [31] rmarkdown_1.10 purrr_0.2.5 magrittr_1.5
# [34] whisker_0.3-2 promises_1.0.1 backports_1.1.2
# [37] scales_0.5.0 htmltools_0.3.6 assertthat_0.2.0
# [40] xtable_1.8-2 mime_0.5 colorspace_1.4-0
# [43] httpuv_1.4.3 stringi_1.2.4 lazyeval_0.2.1
# [46] munsell_0.4.3 R.oo_1.21.0</code></pre>
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