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<!DOCTYPE html>
<html>
<head>
<title>Predicting with regression</title>
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<meta name="description" content="Predicting with regression">
<meta name="author" content="Jeffrey Leek">
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<slide class="title-slide segue nobackground">
<aside class="gdbar">
<img src="../../assets/img/bloomberg_shield.png">
</aside>
<hgroup class="auto-fadein">
<h1>Predicting with regression</h1>
<h2></h2>
<p>Jeffrey Leek<br/>Johns Hopkins Bloomberg School of Public Health</p>
</hgroup>
</slide>
<!-- SLIDES -->
<slide class="" id="slide-1" style="background:;">
<hgroup>
<h2>Key ideas</h2>
</hgroup>
<article>
<ul>
<li>Fit a simple regression model</li>
<li>Plug in new covariates and multiply by the coefficients</li>
<li>Useful when the linear model is (nearly) correct</li>
</ul>
<p><strong>Pros</strong>:</p>
<ul>
<li>Easy to implement</li>
<li>Easy to interpret</li>
</ul>
<p><strong>Cons</strong>:</p>
<ul>
<li>Often poor performance in nonlinear settings</li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-2" style="background:;">
<hgroup>
<h2>Example: Old faithful eruptions</h2>
</hgroup>
<article>
<p><img class=center src=../../assets/img/08_PredictionAndMachineLearning/yellowstone.png height=400></p>
<p>Image Credit/Copyright Wally Pacholka <a href="http://www.astropics.com/">http://www.astropics.com/</a></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-3" style="background:;">
<hgroup>
<h2>Example: Old faithful eruptions</h2>
</hgroup>
<article>
<pre><code class="r">library(caret);data(faithful); set.seed(333)
inTrain <- createDataPartition(y=faithful$waiting,
p=0.5, list=FALSE)
trainFaith <- faithful[inTrain,]; testFaith <- faithful[-inTrain,]
head(trainFaith)
</code></pre>
<pre><code> eruptions waiting
6 2.883 55
11 1.833 54
16 2.167 52
19 1.600 52
22 1.750 47
27 1.967 55
</code></pre>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-4" style="background:;">
<hgroup>
<h2>Eruption duration versus waiting time</h2>
</hgroup>
<article>
<pre><code class="r">plot(trainFaith$waiting,trainFaith$eruptions,pch=19,col="blue",xlab="Waiting",ylab="Duration")
</code></pre>
<div class="rimage center"><img src="fig/unnamed-chunk-1.png" title="plot of chunk unnamed-chunk-1" alt="plot of chunk unnamed-chunk-1" class="plot" /></div>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-5" style="background:;">
<hgroup>
<h2>Fit a linear model</h2>
</hgroup>
<article>
<p>\[ ED_i = b_0 + b_1 WT_i + e_i \]</p>
<pre><code class="r">lm1 <- lm(eruptions ~ waiting,data=trainFaith)
summary(lm1)
</code></pre>
<pre><code>
Call:
lm(formula = eruptions ~ waiting, data = trainFaith)
Residuals:
Min 1Q Median 3Q Max
-1.2699 -0.3479 0.0398 0.3659 1.0502
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.79274 0.22787 -7.87 1e-12 ***
waiting 0.07390 0.00315 23.47 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.495 on 135 degrees of freedom
Multiple R-squared: 0.803, Adjusted R-squared: 0.802
F-statistic: 551 on 1 and 135 DF, p-value: <2e-16
</code></pre>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-6" style="background:;">
<hgroup>
<h2>Model fit</h2>
</hgroup>
<article>
<pre><code class="r">plot(trainFaith$waiting,trainFaith$eruptions,pch=19,col="blue",xlab="Waiting",ylab="Duration")
lines(trainFaith$waiting,lm1$fitted,lwd=3)
</code></pre>
<div class="rimage center"><img src="fig/unnamed-chunk-2.png" title="plot of chunk unnamed-chunk-2" alt="plot of chunk unnamed-chunk-2" class="plot" /></div>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-7" style="background:;">
<hgroup>
<h2>Predict a new value</h2>
</hgroup>
<article>
<p>\[\hat{ED} = \hat{b}_0 + \hat{b}_1 WT\]</p>
<pre><code class="r">coef(lm1)[1] + coef(lm1)[2]*80
</code></pre>
<pre><code>(Intercept)
4.119
</code></pre>
<pre><code class="r">newdata <- data.frame(waiting=80)
predict(lm1,newdata)
</code></pre>
<pre><code> 1
4.119
</code></pre>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-8" style="background:;">
<hgroup>
<h2>Plot predictions - training and test</h2>
</hgroup>
<article>
<pre><code class="r">par(mfrow=c(1,2))
plot(trainFaith$waiting,trainFaith$eruptions,pch=19,col="blue",xlab="Waiting",ylab="Duration")
lines(trainFaith$waiting,predict(lm1),lwd=3)
plot(testFaith$waiting,testFaith$eruptions,pch=19,col="blue",xlab="Waiting",ylab="Duration")
lines(testFaith$waiting,predict(lm1,newdata=testFaith),lwd=3)
</code></pre>
<div class="rimage center"><img src="fig/unnamed-chunk-4.png" title="plot of chunk unnamed-chunk-4" alt="plot of chunk unnamed-chunk-4" class="plot" /></div>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-9" style="background:;">
<hgroup>
<h2>Get training set/test set errors</h2>
</hgroup>
<article>
<pre><code class="r"># Calculate RMSE on training
sqrt(sum((lm1$fitted-trainFaith$eruptions)^2))
</code></pre>
<pre><code>[1] 5.752
</code></pre>
<pre><code class="r">
# Calculate RMSE on test
sqrt(sum((predict(lm1,newdata=testFaith)-testFaith$eruptions)^2))
</code></pre>
<pre><code>[1] 5.839
</code></pre>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-10" style="background:;">
<hgroup>
<h2>Prediction intervals</h2>
</hgroup>
<article>
<pre><code class="r">pred1 <- predict(lm1,newdata=testFaith,interval="prediction")
ord <- order(testFaith$waiting)
plot(testFaith$waiting,testFaith$eruptions,pch=19,col="blue")
matlines(testFaith$waiting[ord],pred1[ord,],type="l",,col=c(1,2,2),lty = c(1,1,1), lwd=3)
</code></pre>
<div class="rimage center"><img src="fig/unnamed-chunk-6.png" title="plot of chunk unnamed-chunk-6" alt="plot of chunk unnamed-chunk-6" class="plot" /></div>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-11" style="background:;">
<hgroup>
<h2>Same process with caret</h2>
</hgroup>
<article>
<pre><code class="r">modFit <- train(eruptions ~ waiting,data=trainFaith,method="lm")
summary(modFit$finalModel)
</code></pre>
<pre><code>
Call:
lm(formula = modFormula, data = data)
Residuals:
Min 1Q Median 3Q Max
-1.2699 -0.3479 0.0398 0.3659 1.0502
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.79274 0.22787 -7.87 1e-12 ***
waiting 0.07390 0.00315 23.47 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.495 on 135 degrees of freedom
Multiple R-squared: 0.803, Adjusted R-squared: 0.802
F-statistic: 551 on 1 and 135 DF, p-value: <2e-16
</code></pre>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-12" style="background:;">
<hgroup>
<h2>Notes and further reading</h2>
</hgroup>
<article>
<ul>
<li>Regression models with multiple covariates can be included</li>
<li>Often useful in combination with other models </li>
<li><a href="http://www-stat.stanford.edu/%7Etibs/ElemStatLearn/">Elements of statistical learning</a></li>
<li><a href="http://www.amazon.com/Modern-Applied-Statistics-W-N-Venables/dp/0387954570">Modern applied statistics with S</a></li>
<li><a href="http://www-bcf.usc.edu/%7Egareth/ISL/">Introduction to statistical learning</a></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
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