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<h1 class="title toc-ignore">Smoothing with covariate</h1>
<h4 class="author"><em>Dongyue Xie</em></h4>
<h4 class="date"><em>May 24, 2018</em></h4>
</div>
<p><strong>Last updated:</strong> 2018-05-25</p>
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<p><details> <summary> <strong style="color:blue;">✔</strong> <strong>Repository version:</strong> <a href="https://github.com/DongyueXie/smash-gen/tree/c89829a317d646eaf7d14d49c2f6a3118dbdd882" target="_blank">c89829a</a> </summary></p>
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<a href="https://github.com/DongyueXie/smash-gen/blob/c89829a317d646eaf7d14d49c2f6a3118dbdd882/analysis/covariate.Rmd" target="_blank">c89829a</a>
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covariate smooth
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<p></details></p>
<hr />
<p>Now suppose at each <span class="math inline">\(t\)</span>, <span class="math inline">\(Y_t=X_t\beta+\mu_t+\epsilon_t\)</span>, where <span class="math inline">\(\mu\)</span> has smooth structure and <span class="math inline">\(\epsilon_t\sim N(0,\sigma^2_t)\)</span>.</p>
<div id="method" class="section level1">
<h1>Method</h1>
<ol style="list-style-type: decimal">
<li>Fit <span class="math inline">\(Y=X\gamma+\epsilon\)</span> using ordinary least square and compute residual <span class="math inline">\(e=Y-X\hat{\gamma}\)</span>.</li>
<li>Apply <code>smash.gaus</code> to <span class="math inline">\(e\)</span> and obtain <span class="math inline">\(\hat\mu_t, \hat\sigma_t\)</span>, <span class="math inline">\(t=1,2,\dots,T\)</span>.</li>
<li>Estimate <span class="math inline">\(\beta\)</span> by ordinary least square or weighted least square: <span class="math inline">\(Y-\hat\mu=X\beta+\hat\epsilon\)</span>, where <span class="math inline">\(\hat\epsilon_t\sim N(0,\hat\sigma_t^2)\)</span>.</li>
</ol>
<p>Rationale: the stucture of <span class="math inline">\(\mu\)</span> cannot be explained by the ordinary least square in step 1 so it is contained in the residual <span class="math inline">\(e\)</span>. Thus <span class="math inline">\(e\)</span> consists of <span class="math inline">\(\mu\)</span> and noises. Using <code>smash.gaus</code> recovers <span class="math inline">\(\mu\)</span> and estimates <span class="math inline">\(\sigma^2\)</span>.</p>
</div>
<div id="experiments" class="section level1">
<h1>Experiments</h1>
<p>We now show the performance of smash when covariates exist. The signal-to-noise ratio(SNR) is fixed at 2.</p>
<p>(Note: The SNR is the ratio of the sample standard deviation of the signal (although it is not random) to the standard deviation of the added noise. If the signal is constant, the SNR is mean(signal) to standard deviation of the added noise)</p>
<p>The length of sequence is <span class="math inline">\(n=256\)</span>.</p>
<pre class="r"><code>simu_study_x=function(mu,beta,snr=2,nsimu=100,filter.number=1,family='DaubExPhase',seed=1234){
set.seed(1234)
n=length(mu)
p=length(beta)
X=matrix(rnorm(n*p,0,1),nrow=n,byrow = T)
cte=X%*%beta
sd.noise=sd(mu)/snr
sd.noise=ifelse(sd.noise==0,mean(mu),sd.noise)
mse.mu=c()
mse.beta=c()
for(i in 1:nsimu){
y=cte+mu+rnorm(n,0,sd.noise)
s.out=smash.gaus.x(X,y,filter.number,family)
mu.hat=s.out$mu.hat
beta.hat=s.out$beta.hat
mse.mu[i]=mse(mu,mu.hat)
mse.beta[i]=mse(beta,beta.hat)
}
return(list(mse.mu=mse.mu,mse.beta=mse.beta,mu.hat=mu.hat,beta.hat=beta.hat,y=y))
}</code></pre>
<div id="step-trend" class="section level2">
<h2>Step trend</h2>
<pre class="r"><code>library(smashrgen)
library(ggplot2)
n=256
mu=c(rep(1,64),rep(2,64),rep(5,64),rep(1,64))
beta=c(1,2,3,4,5)
beta=beta/norm(beta,'2')
result=simu_study_x(mu,beta)
boxplot(result$mse.mu,main='Estimate of mu',ylab='MSE')</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-2-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>boxplot(result$mse.beta,main='Estimate of beta',ylab='MSE')</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-2-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(result$y,col='gray80')
lines(mu)
lines(result$mu.hat,col=4)</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-2-3.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(beta,result$beta.hat,xlab = 'True beta', ylab = 'Beta hat')
abline(0,1)</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-2-4.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="wave" class="section level2">
<h2>Wave</h2>
<pre class="r"><code>f=function(x){return(0.5 + 0.2*cos(4*pi*x) + 0.1*cos(24*pi*x))}
mu=f((1:n)/n)
result=simu_study_x(mu,beta,filter.number = 8,family='DaubLeAsymm')
boxplot(result$mse.mu,main='Estimate of mu',ylab='MSE')</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-3-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>boxplot(result$mse.beta,main='Estimate of beta',ylab='MSE')</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-3-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(result$y,col='gray80')
lines(mu)
lines(result$mu.hat,col=4)</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-3-3.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(beta,result$beta.hat,xlab = 'True beta', ylab = 'Beta hat')
abline(0,1)</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-3-4.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="parabola" class="section level2">
<h2>Parabola</h2>
<pre class="r"><code>r=function(x,c){return((x-c)^2*(x>c)*(x<=1))}
f=function(x){return(0.8 − 30*r(x,0.1) + 60*r(x, 0.2) − 30*r(x, 0.3) +
500*r(x, 0.35) − 1000*r(x, 0.37) + 1000*r(x, 0.41) − 500*r(x, 0.43) +
7.5*r(x, 0.5) − 15*r(x, 0.7) + 7.5*r(x, 0.9))}
mu=f(1:n/n)
result=simu_study_x(mu,beta,filter.number = 8,family='DaubLeAsymm')
boxplot(result$mse.mu,main='Estimate of mu',ylab='MSE')</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>boxplot(result$mse.beta,main='Estimate of beta',ylab='MSE')</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(result$y,col='gray80')
lines(mu)
lines(result$mu.hat,col=4)</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-4-3.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(beta,result$beta.hat,xlab = 'True beta', ylab = 'Beta hat')
abline(0,1)</code></pre>
<p><img src="figure/covariate.Rmd/unnamed-chunk-4-4.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.0 (2017-04-21)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 16299)
Matrix products: default
locale:
[1] LC_COLLATE=English_United States.1252
[2] LC_CTYPE=English_United States.1252
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C
[5] LC_TIME=English_United States.1252
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ggplot2_2.2.1 smashrgen_0.1.0 wavethresh_4.6.8 MASS_7.3-47
[5] caTools_1.17.1 ashr_2.2-7 smashr_1.1-5
loaded via a namespace (and not attached):
[1] Rcpp_0.12.16 plyr_1.8.4 compiler_3.4.0
[4] git2r_0.21.0 workflowr_1.0.1 R.methodsS3_1.7.1
[7] R.utils_2.6.0 bitops_1.0-6 iterators_1.0.8
[10] tools_3.4.0 digest_0.6.13 tibble_1.3.3
[13] evaluate_0.10 gtable_0.2.0 lattice_0.20-35
[16] rlang_0.1.2 Matrix_1.2-9 foreach_1.4.3
[19] yaml_2.1.19 parallel_3.4.0 stringr_1.3.0
[22] knitr_1.20 REBayes_1.3 rprojroot_1.3-2
[25] grid_3.4.0 data.table_1.10.4-3 rmarkdown_1.8
[28] magrittr_1.5 whisker_0.3-2 backports_1.0.5
[31] scales_0.4.1 codetools_0.2-15 htmltools_0.3.5
[34] assertthat_0.2.0 colorspace_1.3-2 stringi_1.1.6
[37] Rmosek_8.0.69 lazyeval_0.2.1 munsell_0.4.3
[40] doParallel_1.0.11 pscl_1.4.9 truncnorm_1.0-7
[43] SQUAREM_2017.10-1 R.oo_1.21.0 </code></pre>
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