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<h1 class="title toc-ignore">Mean and variance functions used to simulate Gaussian data</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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Peter Carbonetto
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I now have plots of all the mean functions in gaussian_signals.Rmd.
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Peter Carbonetto
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Made a few small adjustments to the text in the “gaussianmeanest” analysis.
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<p></details></p>
<hr />
<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="generate-the-ground-truth-signals" class="section level2">
<h2>Generate the ground-truth signals</h2>
<p>Here, <code>n</code> specifies the length of the signals.</p>
<pre class="r"><code>n = 1024
t = 1:n/n</code></pre>
<p>Define the Spikes mean function.</p>
<pre class="r"><code>mu.s = spike.f(t)</code></pre>
<p>Define the Bumps variance function.</p>
<pre class="r"><code>pos = c(0.1, 0.13, 0.15, 0.23, 0.25, 0.4, 0.44, 0.65, 0.76, 0.78, 0.81)
hgt = 2.97/5 * c(4, 5, 3, 4, 5, 4.2, 2.1, 4.3, 3.1, 5.1, 4.2)
wth = c(0.005, 0.005, 0.006, 0.01, 0.01, 0.03, 0.01, 0.01, 0.005, 0.008, 0.005)
mu.b = rep(0, n)
for (j in 1:length(pos))
mu.b = mu.b + hgt[j]/((1 + (abs(t - pos[j])/wth[j]))^4)</code></pre>
<p>Define the Doppler mean function.</p>
<pre class="r"><code>mu.dop = dop.f(t)
mu.dop = 3/(max(mu.dop) - min(mu.dop)) * (mu.dop - min(mu.dop))
mu.dop.var = 10 * dop.f(t)
mu.dop.var = mu.dop.var - min(mu.dop.var)</code></pre>
<p>Define the Angle mean function.</p>
<pre class="r"><code>sig = ((2 * t + 0.5) * (t <= 0.15)) +
((-12 * (t - 0.15) + 0.8) * (t > 0.15 & t <= 0.2)) +
0.2 * (t > 0.2 & t <= 0.5) +
((6 * (t - 0.5) + 0.2) * (t > 0.5 & t <= 0.6)) +
((-10 * (t - 0.6) + 0.8) * (t > 0.6 & t <= 0.65)) +
((-0.5 * (t - 0.65) + 0.3) * (t > 0.65 & t <= 0.85)) +
((2 * (t - 0.85) + 0.2) * (t > 0.85))
mu.ang = 3/5 * ((5/(max(sig) - min(sig))) * sig - 1.6) - 0.0419569</code></pre>
<p>Define the Block mean and variance functions.</p>
<pre class="r"><code>pos = c(0.1, 0.13, 0.15, 0.23, 0.25, 0.4, 0.44, 0.65, 0.76, 0.78, 0.81)
hgt = 2.88/5 * c(4, (-5), 3, (-4), 5, (-4.2), 2.1, 4.3, (-3.1), 2.1, (-4.2))
mu.blk = rep(0, n)
for (j in 1:length(pos))
mu.blk = mu.blk + (1 + sign(t - pos[j])) * (hgt[j]/2)
mu.cblk = mu.blk
mu.cblk[mu.cblk < 0] = 0</code></pre>
<p>Define the Blip mean function.</p>
<pre class="r"><code>mu.blip = (0.32 + 0.6 * t +
0.3 * exp(-100 * (t - 0.3)^2)) * (t >= 0 & t <= 0.8) +
(-0.28 + 0.6 * t + 0.3 * exp(-100 * (t - 1.3)^2)) * (t > 0.8 & t <= 1)</code></pre>
<p>Define the Corner mean function.</p>
<pre class="r"><code>mu.cor = 623.87 * t^3 * (1 - 2 * t) * (t >= 0 & t <= 0.5) +
187.161 * (0.125 - t^3) * t^4 * (t > 0.5 & t <= 0.8) +
3708.470441 * (t - 1)^3 * (t > 0.8 & t <= 1)
mu.cor = (0.6/(max(mu.cor) - min(mu.cor))) * mu.cor
mu.cor = mu.cor - min(mu.cor) + 0.2</code></pre>
<p>Define the rest of the mean functions.</p>
<pre class="r"><code>mu.sp = (1 + mu.s)/5
mu.bump = (1 + mu.b)/5
mu.blk = 0.2 + 0.6 * (mu.blk - min(mu.blk))/max(mu.blk - min(mu.blk))
mu.ang = (1 + mu.ang)/5
mu.dop = (1 + mu.dop)/5</code></pre>
<p>Define the variance functions.</p>
<pre class="r"><code>var1 = rep(1, n)
var2 = (1e-02 + 4 * (exp(-550 * (t - 0.2)^2) +
exp(-200 * (t - 0.5)^2) +
exp(-950 * (t - 0.8)^2)))
var3 = (1e-02 + 2 * mu.dop.var)
var4 = 1e-02 + mu.b
var5 = 1e-02 + 1 * (mu.cblk - min(mu.cblk))/max(mu.cblk)
var1 = var1/2
var2 = var2/max(var2)
var3 = var3/max(var3)
var4 = var4/max(var4)
var5 = var5/max(var5)</code></pre>
</div>
<div id="plot-the-signal-means" class="section level2">
<h2>Plot the signal means</h2>
<p>These plots show each of the mean functions used in generating the Gaussian data sets.</p>
<pre class="r"><code>plot_grid(qplot(t,mu.sp, geom="line",xlab="",ylab="",main="Spikes (sp)"),
qplot(t,mu.bump,geom="line",xlab="",ylab="",main="Bumps (bump)"),
qplot(t,mu.blk, geom="line",xlab="",ylab="",main="Blocks (blk)"),
qplot(t,mu.ang, geom="line",xlab="",ylab="",main="Angles (ang)"),
qplot(t,mu.dop, geom="line",xlab="",ylab="",main="Doppler (dop)"),
qplot(t,mu.blip,geom="line",xlab="",ylab="",main="Blip (blip)"),
qplot(t,mu.cor, geom="line",xlab="",ylab="",main="Corner (cor)"),
nrow = 4,ncol = 2)</code></pre>
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</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/stephenslab/smash-paper/blob/dc4c6cdeab52ee2f5a43f313c91fe24a4dff89bb/docs/figure/gaussian_signals.Rmd/plot-mean-functions-1.png" target="_blank">dc4c6cd</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>
</div>
<div id="plot-the-signal-variances" class="section level2">
<h2>Plot the signal variances</h2>
<p>These plots show the variance functions used in generating the Gaussian data sets. In practice, these functions are rescaled in the simulations to achieve the desired signal-to-noise ratios (see the paper for a more detailed explanation).</p>
<pre class="r"><code>plot_grid(
qplot(t,var1,geom="line",xlab="",ylab="",main="Constant variance (v1)"),
qplot(t,var2,geom="line",xlab="",ylab="",main="Triple exponential (v2)"),
qplot(t,var3,geom="line",xlab="",ylab="",main="Doppler (v3)"),
qplot(t,var4,geom="line",xlab="",ylab="",main="Bumps (v4)"),
qplot(t,var5,geom="line",xlab="",ylab="",main="Clipped (v5)"),
nrow = 3,ncol = 2)</code></pre>
<p><img src="figure/gaussian_signals.Rmd/plot-variance-functions-1.png" width="576" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of plot-variance-functions-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/abc74e5dfa683eba26cee0d09abfebbdf053413c/docs/figure/gaussian_signals.Rmd/plot-variance-functions-1.png" target="_blank">abc74e5</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>
</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 compiler_3.4.3 pillar_1.2.1
# [4] git2r_0.23.0 plyr_1.8.4 workflowr_1.1.1
# [7] bindr_0.1.1 R.methodsS3_1.7.1 R.utils_2.6.0
# [10] tools_3.4.3 digest_0.6.17 evaluate_0.11
# [13] tibble_1.4.2 gtable_0.2.0 pkgconfig_2.0.2
# [16] rlang_0.2.2 yaml_2.2.0 bindrcpp_0.2.2
# [19] withr_2.1.2 stringr_1.3.1 dplyr_0.7.6
# [22] knitr_1.20 rprojroot_1.3-2 grid_3.4.3
# [25] tidyselect_0.2.4 glue_1.3.0 R6_2.2.2
# [28] rmarkdown_1.10 purrr_0.2.5 magrittr_1.5
# [31] whisker_0.3-2 backports_1.1.2 scales_0.5.0
# [34] htmltools_0.3.6 assertthat_0.2.0 colorspace_1.4-0
# [37] labeling_0.3 stringi_1.2.4 lazyeval_0.2.1
# [40] munsell_0.4.3 R.oo_1.21.0</code></pre>
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