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<title>Cars dataset</title>
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<div id="header">
<h1 class="title">Cars dataset</h1>
</div>
<div id="table-of-contents">
<h2><a href="#content">Cars dataset</a></h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#preprocessing">Preprocessing</a></li>
<li><a href="#modeling">Modeling</a></li>
<li><a href="#error-analysis">Error Analysis</a></li>
</ul>
</div>
</div>
<div id="main">
<p><img style="float: right;" src="https://media.timtul.com/media/network22/ubiqum.png"></p>
<p><strong>Example of solution</strong></p>
<div id="preprocessing" class="section level2">
<h2>Preprocessing</h2>
<div id="load-data-and-change-variable-types" class="section level3">
<h3>Load data and change variable types</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">pacman<span class="op">::</span><span class="kw">p_load</span>(ggplot2, prettydoc)
cars <-<span class="st"> </span><span class="kw">read.csv</span>(<span class="st">"C:/Users/gabri/Desktop/Ubiqum/R/Data_Analytics_Predicting_Customer_Preference/Task_1/cars.csv"</span>)
<span class="co">#cars <- read.csv("C:/SARA/Ubiqum/CodeAcademy/Section2/Task1/cars.csv")</span>
cars<span class="op">$</span>speed.of.car <-<span class="st"> </span><span class="kw">as.numeric</span>(cars<span class="op">$</span>speed.of.car)
cars<span class="op">$</span>distance.of.car <-<span class="st"> </span><span class="kw">as.numeric</span>(cars<span class="op">$</span>distance.of.car)</code></pre></div>
</div>
<div id="exploratory-analysis" class="section level3">
<h3>Exploratory analysis</h3>
<p>In the initial exploration you can already estimate an outlier</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">plot</span>(cars<span class="op">$</span>speed.of.car, cars<span class="op">$</span>distance.of.car)</code></pre></div>
<p><img src="cars_v3_files/figure-html/plot%201-1.png" width="768" /></p>
<div id="check-for-distributions" class="section level4">
<h4>Check for distributions</h4>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">box_plot <-<span class="st"> </span><span class="kw">boxplot</span>(cars[, <span class="kw">c</span>(<span class="st">"distance.of.car"</span>,<span class="st">"speed.of.car"</span>)])</code></pre></div>
<p><img src="cars_v3_files/figure-html/boxplot-1.png" width="768" /></p>
</div>
<div id="exclude-outlier" class="section level4">
<h4>Exclude outlier</h4>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">cars <-<span class="st"> </span>cars[<span class="kw">which</span>(cars<span class="op">$</span>distance.of.car <span class="op">!=</span><span class="st"> </span>box_plot<span class="op">$</span>out),]</code></pre></div>
</div>
</div>
</div>
<div id="modeling" class="section level2">
<h2>Modeling</h2>
<div id="split-data" class="section level3">
<h3>Split data</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">set.seed</span>(<span class="dv">314</span>)
train_size <-<span class="st"> </span><span class="kw">round</span>(<span class="kw">nrow</span>(cars)<span class="op">*</span><span class="fl">0.7</span>)
test_size <-<span class="st"> </span><span class="kw">nrow</span>(cars)<span class="op">-</span>train_size
training_indices <-<span class="st"> </span><span class="kw">sample</span>(<span class="kw">seq_len</span>(<span class="kw">nrow</span>(cars)), <span class="dt">size =</span> train_size)
train_set <-<span class="st"> </span>cars[training_indices,]
test_set <-<span class="st"> </span>cars[<span class="op">-</span>training_indices,]</code></pre></div>
</div>
<div id="train-a-model" class="section level3">
<h3>Train a model</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">lm <-<span class="st"> </span><span class="kw">lm</span>(distance.of.car<span class="op">~</span><span class="st"> </span>speed.of.car,train_set)
Pred_dist <-<span class="st"> </span><span class="kw">predict</span>(lm,test_set)
test_set<span class="op">$</span>Pred_dist <-<span class="st"> </span>Pred_dist</code></pre></div>
</div>
<div id="plot-model" class="section level3">
<h3>Plot model</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(cars, <span class="kw">aes</span>(<span class="dt">x=</span>speed.of.car, <span class="dt">y=</span>distance.of.car)) <span class="op">+</span><span class="st"> </span><span class="kw">geom_point</span>() <span class="op">+</span><span class="st"> </span>
<span class="kw">geom_abline</span>(<span class="kw">aes</span>(<span class="dt">intercept=</span>lm<span class="op">$</span>coefficients[<span class="dv">1</span>], <span class="dt">slope=</span>lm<span class="op">$</span>coefficients[<span class="dv">2</span>]), <span class="dt">colour=</span><span class="st">'red'</span>)<span class="op">+</span>
<span class="st"> </span><span class="kw">labs</span>(<span class="dt">title =</span> <span class="kw">paste</span>(<span class="st">"Model with normal intercept</span><span class="ch">\n</span><span class="st">"</span>,<span class="st">"Adj R2 = "</span>,<span class="kw">signif</span>(<span class="kw">summary</span>(lm)<span class="op">$</span>adj.r.squared, <span class="dv">5</span>),
<span class="st">"Intercept ="</span>,<span class="kw">signif</span>(lm<span class="op">$</span>coef[[<span class="dv">1</span>]],<span class="dv">5</span> ),
<span class="st">" Slope ="</span>,<span class="kw">signif</span>(lm<span class="op">$</span>coef[[<span class="dv">2</span>]], <span class="dv">5</span>),
<span class="st">" P ="</span>,<span class="kw">signif</span>(<span class="kw">summary</span>(lm)<span class="op">$</span>coef[<span class="dv">2</span>,<span class="dv">4</span>], <span class="dv">5</span>)))</code></pre></div>
<p><img src="cars_v3_files/figure-html/plot%20lm-1.png" width="768" /></p>
<p>The intercept in the model above is at negative levels, which doesn’t make logical sense. The car should be driving at a speed of -25 in order to achieve a distance of zero. Instead, we will try to fix the intercept at zero. Below, we can see the fit becomes worse when we fix the intercept at zero and keep the model linear.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">lm_intercept <-<span class="st"> </span><span class="kw">lm</span>(distance.of.car<span class="op">~</span><span class="st"> </span><span class="dv">0</span><span class="op">+</span>speed.of.car,train_set)
Pred_dist_intercept <-<span class="st"> </span><span class="kw">predict</span>(lm_intercept,test_set)
test_set <-<span class="st"> </span><span class="kw">cbind</span>(test_set, Pred_dist_intercept)
<span class="co"># Model with intercept = 0</span>
<span class="kw">ggplot</span>(cars, <span class="kw">aes</span>(<span class="dt">x=</span>speed.of.car, <span class="dt">y=</span>distance.of.car)) <span class="op">+</span><span class="st"> </span><span class="kw">geom_point</span>() <span class="op">+</span><span class="st"> </span><span class="kw">xlim</span>(<span class="op">-</span><span class="dv">2</span>, <span class="dv">30</span>) <span class="op">+</span>
<span class="kw">geom_abline</span>(<span class="kw">aes</span>(<span class="dt">intercept=</span><span class="dv">0</span>, <span class="dt">slope=</span>lm_intercept<span class="op">$</span>coefficients[<span class="dv">1</span>]), <span class="dt">colour=</span><span class="st">'turquoise2'</span>) <span class="op">+</span><span class="st"> </span>
<span class="st"> </span><span class="kw">labs</span>(<span class="dt">title =</span> <span class="kw">paste</span>(<span class="st">"Model with intercept = 0</span><span class="ch">\n</span><span class="st">"</span>,<span class="st">"Adj R2 ="</span> ,<span class="kw">signif</span>(<span class="kw">summary</span>(lm_intercept)<span class="op">$</span>adj.r.squared, <span class="dv">5</span>),
<span class="st">"Intercept = 0 "</span>,
<span class="st">" Slope ="</span>,<span class="kw">signif</span>(lm_intercept<span class="op">$</span>coef[[<span class="dv">1</span>]], <span class="dv">5</span>),
<span class="st">" P ="</span>,<span class="kw">signif</span>(<span class="kw">summary</span>(lm_intercept)<span class="op">$</span>coef[<span class="dv">1</span>,<span class="dv">3</span>], <span class="dv">4</span>))) </code></pre></div>
<p><img src="cars_v3_files/figure-html/model%20intercept-1.png" width="768" /></p>
<p><strong>Let’s compare both models</strong></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="co"># Comparing both linear regressions</span>
<span class="kw">ggplot</span>(cars, <span class="kw">aes</span>(<span class="dt">x=</span>speed.of.car, <span class="dt">y=</span>distance.of.car)) <span class="op">+</span><span class="st"> </span><span class="kw">geom_point</span>() <span class="op">+</span><span class="st"> </span>
<span class="kw">geom_abline</span>(<span class="kw">aes</span>(<span class="dt">intercept=</span>lm<span class="op">$</span>coefficients[<span class="dv">1</span>], <span class="dt">slope=</span>lm<span class="op">$</span>coefficients[<span class="dv">2</span>], <span class="dt">colour=</span><span class="st">'LM'</span>)) <span class="op">+</span>
<span class="kw">geom_abline</span>(<span class="kw">aes</span>(<span class="dt">intercept=</span><span class="dv">0</span>, <span class="dt">slope=</span>lm_intercept<span class="op">$</span>coefficients[<span class="dv">1</span>], <span class="dt">colour=</span><span class="st">'LM with intercept = 0 '</span>)) </code></pre></div>
<p><img src="cars_v3_files/figure-html/Comparison-1.png" width="768" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">test_set<span class="op">$</span>error_lm <-<span class="st"> </span>test_set<span class="op">$</span>distance.of.car <span class="op">-</span><span class="st"> </span>test_set<span class="op">$</span>Pred_dist
test_set<span class="op">$</span>abs_error_lm <-<span class="st"> </span><span class="kw">abs</span>(test_set<span class="op">$</span>distance.of.car <span class="op">-</span><span class="st"> </span>test_set<span class="op">$</span>Pred_dist)
test_set<span class="op">$</span>rel_error_lm <-<span class="st"> </span>test_set<span class="op">$</span>abs_error_lm<span class="op">/</span>test_set<span class="op">$</span>distance.of.car
test_set<span class="op">$</span>abs_error_model_intercept <-<span class="st"> </span><span class="kw">abs</span>(test_set<span class="op">$</span>distance.of.car <span class="op">-</span><span class="st"> </span>test_set<span class="op">$</span>Pred_dist_intercept)
test_set<span class="op">$</span>rel_error_model_intercept <-<span class="st"> </span>test_set<span class="op">$</span>abs_error_model_intercept<span class="op">/</span>test_set<span class="op">$</span>distance.of.car
MAE_lm <-<span class="st"> </span><span class="kw">mean</span>(test_set<span class="op">$</span>abs_error_lm)
MAE_model_intercept <-<span class="st"> </span><span class="kw">mean</span>(test_set<span class="op">$</span>abs_error_model_intercept)</code></pre></div>
</div>
<div id="log-model" class="section level3">
<h3>Log model</h3>
<p>We can try to log it:</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">lm_log <-<span class="st"> </span><span class="kw">lm</span>(<span class="kw">log</span>(distance.of.car) <span class="op">~</span><span class="st"> </span>speed.of.car,train_set)
test_set<span class="op">$</span>pred_dist_log <-<span class="st"> </span><span class="kw">predict</span>(lm_log,test_set)
test_set<span class="op">$</span>pred_dist_log <-<span class="st"> </span><span class="kw">exp</span>(test_set<span class="op">$</span>pred_dist_log)
<span class="kw">ggplot</span>(cars, <span class="kw">aes</span>(<span class="dt">x=</span>speed.of.car, <span class="dt">y=</span><span class="kw">log</span>(distance.of.car))) <span class="op">+</span><span class="st"> </span><span class="kw">geom_point</span>() <span class="op">+</span><span class="st"> </span>
<span class="kw">geom_abline</span>(<span class="kw">aes</span>(<span class="dt">intercept=</span>lm_log<span class="op">$</span>coefficients[<span class="dv">1</span>], <span class="dt">slope=</span>lm_log<span class="op">$</span>coefficients[<span class="dv">2</span>]), <span class="dt">colour=</span><span class="st">'red'</span>)<span class="op">+</span>
<span class="st"> </span><span class="kw">labs</span>(<span class="dt">title =</span> <span class="kw">paste</span>(<span class="st">"Model with log </span><span class="ch">\n</span><span class="st">"</span>,<span class="st">"Adj R2 = "</span>,<span class="kw">signif</span>(<span class="kw">summary</span>(lm_log)<span class="op">$</span>adj.r.squared, <span class="dv">5</span>),
<span class="st">"Intercept ="</span>,<span class="kw">signif</span>(lm_log<span class="op">$</span>coef[[<span class="dv">1</span>]],<span class="dv">5</span> ),
<span class="st">" Slope ="</span>,<span class="kw">signif</span>(lm_log<span class="op">$</span>coef[[<span class="dv">2</span>]], <span class="dv">5</span>),
<span class="st">" P ="</span>,<span class="kw">signif</span>(<span class="kw">summary</span>(lm_log)<span class="op">$</span>coef[<span class="dv">2</span>,<span class="dv">4</span>], <span class="dv">5</span>)))</code></pre></div>
<p><img src="cars_v3_files/figure-html/log%20model-1.png" width="768" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">lm_sqr <-<span class="st"> </span><span class="kw">lm</span>(distance.of.car <span class="op">~</span><span class="st"> </span><span class="kw">I</span>(speed.of.car<span class="op">^</span><span class="dv">2</span>),train_set)
test_set<span class="op">$</span>pred_dist_sqr <-<span class="st"> </span><span class="kw">predict</span>(lm_log,test_set)
<span class="kw">ggplot</span>(cars, <span class="kw">aes</span>(<span class="dt">x=</span><span class="kw">I</span>(speed.of.car<span class="op">^</span><span class="dv">2</span>), <span class="dt">y=</span>distance.of.car)) <span class="op">+</span><span class="st"> </span><span class="kw">geom_point</span>() <span class="op">+</span><span class="st"> </span>
<span class="kw">geom_abline</span>(<span class="kw">aes</span>(<span class="dt">intercept=</span>lm_sqr<span class="op">$</span>coefficients[<span class="dv">1</span>], <span class="dt">slope=</span>lm_sqr<span class="op">$</span>coefficients[<span class="dv">2</span>]), <span class="dt">colour=</span><span class="st">'red'</span>)<span class="op">+</span>
<span class="st"> </span><span class="kw">labs</span>(<span class="dt">title =</span> <span class="kw">paste</span>(<span class="st">"Model with squared x</span><span class="ch">\n</span><span class="st">"</span>,<span class="st">"Adj R2 = "</span>,<span class="kw">signif</span>(<span class="kw">summary</span>(lm_sqr)<span class="op">$</span>adj.r.squared, <span class="dv">5</span>),
<span class="st">"Intercept ="</span>,<span class="kw">signif</span>(lm_sqr<span class="op">$</span>coef[[<span class="dv">1</span>]],<span class="dv">5</span> ),
<span class="st">" Slope ="</span>,<span class="kw">signif</span>(lm_sqr<span class="op">$</span>coef[[<span class="dv">2</span>]], <span class="dv">5</span>),
<span class="st">" P ="</span>,<span class="kw">signif</span>(<span class="kw">summary</span>(lm_sqr)<span class="op">$</span>coef[<span class="dv">2</span>,<span class="dv">4</span>], <span class="dv">5</span>)))</code></pre></div>
<p><img src="cars_v3_files/figure-html/squared%20model-1.png" width="768" /></p>
</div>
</div>
<div id="error-analysis" class="section level2">
<h2>Error Analysis</h2>
<div id="relative-error" class="section level3">
<h3>Relative error</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">plot</span>(test_set<span class="op">$</span>speed.of.car, test_set<span class="op">$</span>rel_error_lm, <span class="dt">main =</span> <span class="st">"Relative error"</span>)</code></pre></div>
<p><img src="cars_v3_files/figure-html/plot%202-1.png" width="768" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">plot</span>(test_set<span class="op">$</span>speed.of.car, test_set<span class="op">$</span>error_lm, <span class="dt">main =</span> <span class="st">"Error"</span>)</code></pre></div>
<p><img src="cars_v3_files/figure-html/plot%203-1.png" width="768" /></p>
</div>
</div>
</div>
</div>
<div id="postamble" data-toggle="wy-nav-shift" class="status">
<p class="date"><span class="glyphicon glyphicon-calendar"></span> 08/04/2019</p>
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