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<h1 class="title">Week 4</h1>
</div>
</div>
<div class="quarto-title-meta">
<div>
<div class="quarto-title-meta-heading">Author</div>
<div class="quarto-title-meta-contents">
<p>Carson Pedaci </p>
</div>
</div>
</div>
</header><div id="quarto-content" class="page-columns page-rows-contents page-layout-article">
<div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
<nav id="TOC" role="doc-toc" class="toc-active">
<h2 id="toc-title">Table of contents</h2>
<ul>
<li><a href="#tuesday-jan-31" id="toc-tuesday-jan-31" class="nav-link active" data-scroll-target="#tuesday-jan-31">Tuesday, Jan 31</a>
<ul class="collapse">
<li><a href="#statistical-learning" id="toc-statistical-learning" class="nav-link" data-scroll-target="#statistical-learning">Statistical Learning</a></li>
<li><a href="#different-flavors-statistical-learning" id="toc-different-flavors-statistical-learning" class="nav-link" data-scroll-target="#different-flavors-statistical-learning">Different Flavors: Statistical Learning</a></li>
<li><a href="#goal" id="toc-goal" class="nav-link" data-scroll-target="#goal">Goal</a></li>
<li><a href="#scatterplot" id="toc-scatterplot" class="nav-link" data-scroll-target="#scatterplot">Scatterplot</a></li>
</ul></li>
<li><a href="#least-squares-estimator" id="toc-least-squares-estimator" class="nav-link" data-scroll-target="#least-squares-estimator">Least Squares Estimator</a>
<ul class="collapse">
<li><a href="#best-fit-minimizes-residuals" id="toc-best-fit-minimizes-residuals" class="nav-link" data-scroll-target="#best-fit-minimizes-residuals">Best fit minimizes residuals</a></li>
</ul></li>
<li><a href="#thursday-feb-2" id="toc-thursday-feb-2" class="nav-link" data-scroll-target="#thursday-feb-2">Thursday, Feb 2</a>
<ul class="collapse">
<li><a href="#prediction" id="toc-prediction" class="nav-link" data-scroll-target="#prediction">Prediction</a></li>
</ul></li>
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<hr>
<section id="tuesday-jan-31" class="level2">
<h2 class="anchored" data-anchor-id="tuesday-jan-31">Tuesday, Jan 31</h2>
<div class="callout-important callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
TIL
</div>
</div>
<div class="callout-body-container callout-body">
<p>Today, I learnt the following concepts in class:</p>
<ol type="1">
<li><em>Intro to statistical learning</em></li>
<li>Simple Linear Regression</li>
</ol>
<ul>
<li>Motivation</li>
<li><span class="math inline">\(l_2\)</span> estimator</li>
<li>Inference</li>
<li>Prediction</li>
</ul>
<ol start="3" type="1">
<li>Multiple Regression</li>
</ol>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ISLR2)</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(dplyr)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(cowplot)</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(kableExtra)</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(htmlwidgets)</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<section id="statistical-learning" class="level3">
<h3 class="anchored" data-anchor-id="statistical-learning">Statistical Learning</h3>
<p>Suppose we are given dataset:</p>
<p><span class="math inline">\(X = [X_1, X_2,...,X_p]\)</span></p>
<p>Called predictor vars, indep vars, covariates</p>
<p><strong><em>y</em></strong></p>
<p>called response/outcome/dep vars</p>
<p>Goal: Find function f such that <span class="math inline">\(y = f(X)\)</span></p>
<p><span class="math inline">\(y_i = f(X_i) = f(X_i,1, X_i,2,..., X_i,p)\)</span></p>
</section>
<section id="different-flavors-statistical-learning" class="level3">
<h3 class="anchored" data-anchor-id="different-flavors-statistical-learning">Different Flavors: Statistical Learning</h3>
<ul>
<li>Supervised (y,x; quantitative responses)
<ul>
<li>Regression (y contained in R (all real))</li>
<li>Classification</li>
</ul></li>
<li>Unsupervised Learning (x)</li>
<li>Semi-supervised Learning (y, x (# of observations corresponding to y very small))</li>
<li>Reinforcement Learning</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>url <span class="ot"><-</span> <span class="st">"https://online.stat.psu.edu/stat462/sites/onlinecourses.science.psu.edu.stat462/files/data/poverty/index.txt"</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>df <span class="ot"><-</span> <span class="fu">read_tsv</span>(url)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stderr">
<pre><code>Rows: 51 Columns: 6
-- Column specification --------------------------------------------------------
Delimiter: "\t"
chr (1): Location
dbl (5): PovPct, Brth15to17, Brth18to19, ViolCrime, TeenBrth
i Use `spec()` to retrieve the full column specification for this data.
i Specify the column types or set `show_col_types = FALSE` to quiet this message.</code></pre>
</div>
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>df <span class="sc">%>%</span> <span class="fu">head</span>(., <span class="dv">20</span>) <span class="sc">%>%</span> kable</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<table>
<thead>
<tr>
<th style="text-align:left;"> Location </th>
<th style="text-align:right;"> PovPct </th>
<th style="text-align:right;"> Brth15to17 </th>
<th style="text-align:right;"> Brth18to19 </th>
<th style="text-align:right;"> ViolCrime </th>
<th style="text-align:right;"> TeenBrth </th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;"> Alabama </td>
<td style="text-align:right;"> 20.1 </td>
<td style="text-align:right;"> 31.5 </td>
<td style="text-align:right;"> 88.7 </td>
<td style="text-align:right;"> 11.2 </td>
<td style="text-align:right;"> 54.5 </td>
</tr>
<tr>
<td style="text-align:left;"> Alaska </td>
<td style="text-align:right;"> 7.1 </td>
<td style="text-align:right;"> 18.9 </td>
<td style="text-align:right;"> 73.7 </td>
<td style="text-align:right;"> 9.1 </td>
<td style="text-align:right;"> 39.5 </td>
</tr>
<tr>
<td style="text-align:left;"> Arizona </td>
<td style="text-align:right;"> 16.1 </td>
<td style="text-align:right;"> 35.0 </td>
<td style="text-align:right;"> 102.5 </td>
<td style="text-align:right;"> 10.4 </td>
<td style="text-align:right;"> 61.2 </td>
</tr>
<tr>
<td style="text-align:left;"> Arkansas </td>
<td style="text-align:right;"> 14.9 </td>
<td style="text-align:right;"> 31.6 </td>
<td style="text-align:right;"> 101.7 </td>
<td style="text-align:right;"> 10.4 </td>
<td style="text-align:right;"> 59.9 </td>
</tr>
<tr>
<td style="text-align:left;"> California </td>
<td style="text-align:right;"> 16.7 </td>
<td style="text-align:right;"> 22.6 </td>
<td style="text-align:right;"> 69.1 </td>
<td style="text-align:right;"> 11.2 </td>
<td style="text-align:right;"> 41.1 </td>
</tr>
<tr>
<td style="text-align:left;"> Colorado </td>
<td style="text-align:right;"> 8.8 </td>
<td style="text-align:right;"> 26.2 </td>
<td style="text-align:right;"> 79.1 </td>
<td style="text-align:right;"> 5.8 </td>
<td style="text-align:right;"> 47.0 </td>
</tr>
<tr>
<td style="text-align:left;"> Connecticut </td>
<td style="text-align:right;"> 9.7 </td>
<td style="text-align:right;"> 14.1 </td>
<td style="text-align:right;"> 45.1 </td>
<td style="text-align:right;"> 4.6 </td>
<td style="text-align:right;"> 25.8 </td>
</tr>
<tr>
<td style="text-align:left;"> Delaware </td>
<td style="text-align:right;"> 10.3 </td>
<td style="text-align:right;"> 24.7 </td>
<td style="text-align:right;"> 77.8 </td>
<td style="text-align:right;"> 3.5 </td>
<td style="text-align:right;"> 46.3 </td>
</tr>
<tr>
<td style="text-align:left;"> District_of_Columbia </td>
<td style="text-align:right;"> 22.0 </td>
<td style="text-align:right;"> 44.8 </td>
<td style="text-align:right;"> 101.5 </td>
<td style="text-align:right;"> 65.0 </td>
<td style="text-align:right;"> 69.1 </td>
</tr>
<tr>
<td style="text-align:left;"> Florida </td>
<td style="text-align:right;"> 16.2 </td>
<td style="text-align:right;"> 23.2 </td>
<td style="text-align:right;"> 78.4 </td>
<td style="text-align:right;"> 7.3 </td>
<td style="text-align:right;"> 44.5 </td>
</tr>
<tr>
<td style="text-align:left;"> Georgia </td>
<td style="text-align:right;"> 12.1 </td>
<td style="text-align:right;"> 31.4 </td>
<td style="text-align:right;"> 92.8 </td>
<td style="text-align:right;"> 9.5 </td>
<td style="text-align:right;"> 55.7 </td>
</tr>
<tr>
<td style="text-align:left;"> Hawaii </td>
<td style="text-align:right;"> 10.3 </td>
<td style="text-align:right;"> 17.7 </td>
<td style="text-align:right;"> 66.4 </td>
<td style="text-align:right;"> 4.7 </td>
<td style="text-align:right;"> 38.2 </td>
</tr>
<tr>
<td style="text-align:left;"> Idaho </td>
<td style="text-align:right;"> 14.5 </td>
<td style="text-align:right;"> 18.4 </td>
<td style="text-align:right;"> 69.1 </td>
<td style="text-align:right;"> 4.1 </td>
<td style="text-align:right;"> 39.1 </td>
</tr>
<tr>
<td style="text-align:left;"> Illinois </td>
<td style="text-align:right;"> 12.4 </td>
<td style="text-align:right;"> 23.4 </td>
<td style="text-align:right;"> 70.5 </td>
<td style="text-align:right;"> 10.3 </td>
<td style="text-align:right;"> 42.2 </td>
</tr>
<tr>
<td style="text-align:left;"> Indiana </td>
<td style="text-align:right;"> 9.6 </td>
<td style="text-align:right;"> 22.6 </td>
<td style="text-align:right;"> 78.5 </td>
<td style="text-align:right;"> 8.0 </td>
<td style="text-align:right;"> 44.6 </td>
</tr>
<tr>
<td style="text-align:left;"> Iowa </td>
<td style="text-align:right;"> 12.2 </td>
<td style="text-align:right;"> 16.4 </td>
<td style="text-align:right;"> 55.4 </td>
<td style="text-align:right;"> 1.8 </td>
<td style="text-align:right;"> 32.5 </td>
</tr>
<tr>
<td style="text-align:left;"> Kansas </td>
<td style="text-align:right;"> 10.8 </td>
<td style="text-align:right;"> 21.4 </td>
<td style="text-align:right;"> 74.2 </td>
<td style="text-align:right;"> 6.2 </td>
<td style="text-align:right;"> 43.0 </td>
</tr>
<tr>
<td style="text-align:left;"> Kentucky </td>
<td style="text-align:right;"> 14.7 </td>
<td style="text-align:right;"> 26.5 </td>
<td style="text-align:right;"> 84.8 </td>
<td style="text-align:right;"> 7.2 </td>
<td style="text-align:right;"> 51.0 </td>
</tr>
<tr>
<td style="text-align:left;"> Louisiana </td>
<td style="text-align:right;"> 19.7 </td>
<td style="text-align:right;"> 31.7 </td>
<td style="text-align:right;"> 96.1 </td>
<td style="text-align:right;"> 17.0 </td>
<td style="text-align:right;"> 58.1 </td>
</tr>
<tr>
<td style="text-align:left;"> Maine </td>
<td style="text-align:right;"> 11.2 </td>
<td style="text-align:right;"> 11.9 </td>
<td style="text-align:right;"> 45.2 </td>
<td style="text-align:right;"> 2.0 </td>
<td style="text-align:right;"> 25.4 </td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="goal" class="level3">
<h3 class="anchored" data-anchor-id="goal">Goal</h3>
<p>Predict the birth rate as function of poverty rate</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fu">colnames</span>(df) <span class="ot"><-</span> <span class="fu">tolower</span>(<span class="fu">colnames</span>(df))</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>x <span class="ot"><-</span> df<span class="sc">$</span>povpct</span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>y <span class="ot"><-</span> df<span class="sc">$</span>brth15to17</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="scatterplot" class="level3">
<h3 class="anchored" data-anchor-id="scatterplot">Scatterplot</h3>
<p>Visualize the relationship between <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> vars</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co"># fig-height = 5</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>plt <span class="ot"><-</span> <span class="cf">function</span>(){</span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">plot</span>(</span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a> x,</span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a> y,</span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a> <span class="at">pch=</span><span class="dv">20</span>,</span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a> <span class="at">xlab =</span> <span class="st">'Pov %'</span>,</span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a> <span class="at">ylab =</span> <span class="st">'Birth rate (15 - 17)'</span></span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a><span class="fu">plt</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-4-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p><span class="math display">\[y = Beta_1 + Beta_1x\]</span> Always gives straight line</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>b0 <span class="ot"><-</span> <span class="dv">1</span></span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>b1 <span class="ot"><-</span> <span class="dv">2</span></span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plt</span>()</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>(b0 <span class="sc">+</span> b1 <span class="sc">*</span>x, <span class="dv">0</span>, <span class="dv">30</span>, <span class="at">add=</span>T, <span class="at">col=</span><span class="st">'firebrick'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-5-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>b0 <span class="ot"><-</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">2</span>, <span class="dv">0</span>, <span class="dv">2</span>)</span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>b1 <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">2</span>)</span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a><span class="fu">par</span>(<span class="at">mfrow=</span><span class="fu">c</span>(<span class="dv">3</span>,<span class="dv">3</span>))</span>
<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span>(B0 <span class="cf">in</span> b0){</span>
<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(B1 <span class="cf">in</span> b1){</span>
<span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a> <span class="fu">plt</span>()</span>
<span id="cb8-9"><a href="#cb8-9" aria-hidden="true" tabindex="-1"></a> <span class="fu">curve</span>(B0 <span class="sc">+</span> B1 <span class="sc">*</span>x, <span class="dv">0</span>, <span class="dv">30</span>, <span class="at">add=</span>T, <span class="at">col=</span><span class="st">'firebrick'</span>)</span>
<span id="cb8-10"><a href="#cb8-10" aria-hidden="true" tabindex="-1"></a> <span class="fu">title</span>(<span class="at">main =</span> <span class="fu">paste</span>(<span class="st">'b0 = '</span>, B0, <span class="st">' and b1 = '</span>, B1))</span>
<span id="cb8-11"><a href="#cb8-11" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb8-12"><a href="#cb8-12" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-6-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
</section>
<section id="least-squares-estimator" class="level2">
<h2 class="anchored" data-anchor-id="least-squares-estimator">Least Squares Estimator</h2>
<div class="cell">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>b0 <span class="ot"><-</span> <span class="dv">10</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>b1 <span class="ot"><-</span> <span class="fl">1.1</span></span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a>yhat <span class="ot"><-</span> b0 <span class="sc">+</span> b1 <span class="sc">*</span> x</span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plt</span>()</span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>(b0 <span class="sc">+</span> b1 <span class="sc">*</span> x, <span class="dv">0</span>, <span class="dv">30</span>, <span class="at">add=</span>T, <span class="at">col=</span><span class="st">'firebrick'</span>)</span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a><span class="fu">title</span>(<span class="at">main =</span> <span class="fu">paste</span>(<span class="st">'b0 = '</span>, b0, <span class="st">' and b1 = '</span>, b1))</span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a><span class="fu">segments</span>(x, y, x, yhat)</span>
<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a>resids <span class="ot"><-</span> <span class="fu">abs</span>(y <span class="sc">-</span> yhat)<span class="sc">^</span><span class="dv">2</span></span>
<span id="cb9-11"><a href="#cb9-11" aria-hidden="true" tabindex="-1"></a>ss_resids <span class="ot"><-</span> <span class="fu">sum</span>(resids)</span>
<span id="cb9-12"><a href="#cb9-12" aria-hidden="true" tabindex="-1"></a><span class="fu">title</span>(<span class="at">main =</span> <span class="fu">paste</span>(<span class="st">'b0, b1, ss_residuals = '</span>, b0, b1, ss_resids, <span class="at">sep=</span><span class="st">','</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-7-1.png" class="img-fluid" width="1152"></p>
</div>
</div>
<p>Error = <span class="math inline">\(y - \hat{y}\)</span></p>
<section id="best-fit-minimizes-residuals" class="level3">
<h3 class="anchored" data-anchor-id="best-fit-minimizes-residuals">Best fit minimizes residuals</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>model <span class="ot"><-</span> <span class="fu">lm</span>(y <span class="sc">~</span> x)</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(<span class="fu">residuals</span>(model)<span class="sc">^</span><span class="dv">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 1509.635</code></pre>
</div>
</div>
</section>
</section>
<section id="thursday-feb-2" class="level2">
<h2 class="anchored" data-anchor-id="thursday-feb-2">Thursday, Feb 2</h2>
<div class="callout-important callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
TIL
</div>
</div>
<div class="callout-body-container callout-body">
<p>Today, I learnt the following concepts in class:</p>
<ol type="1">
<li>Linear Regression Model</li>
<li><span class="math inline">\(p\)</span>-values</li>
<li><span class="math inline">\(R^2\)</span> values</li>
<li>Prediction</li>
</ol>
</div>
</div>
<p>In our case we want to model <span class="math inline">\(y\)</span> as function of <span class="math inline">\(x\)</span>. IN <code>R</code> the formula looks like:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="fu">typeof</span>(<span class="fu">formula</span>(y <span class="sc">~</span> x))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] "language"</code></pre>
</div>
</div>
<p>Linear regression model in <code>R</code> is called using the <strong>L</strong>inear <strong>M</strong>odel, i.e., ‘lm()’</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>model <span class="ot"><-</span> <span class="fu">lm</span>(y <span class="sc">~</span> x)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(model)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-11.2275 -3.6554 -0.0407 2.4972 10.5152
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.2673 2.5297 1.687 0.098 .
x 1.3733 0.1835 7.483 1.19e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.551 on 49 degrees of freedom
Multiple R-squared: 0.5333, Adjusted R-squared: 0.5238
F-statistic: 56 on 1 and 49 DF, p-value: 1.188e-09</code></pre>
</div>
</div>
<p><strong>What are null and alternate hypotheses for regression model?</strong></p>
<p>Objective: We want to find the best linear model to fit <span class="math inline">\(y \sim x\)</span></p>
<p>Null Hypothesis: >There is no linear relationship between <span class="math inline">\(y\)</span> an <span class="math inline">\(x\)</span></p>
<p>In terms of <span class="math inline">\(\beta_9\)</span> and <span class="math inline">\(\beta_1\)</span>?</p>
<p><span class="math inline">\(\beta_1 = 0\)</span> in <span class="math inline">\(H_0\)</span></p>
<p>Alternate: <span class="math inline">\(\beta_1 \neq 0\)</span></p>
<p>Summary:</p>
<p><span class="math display">\[
\begin{align}
H_0: \beta_1 = 0 && H_1: \beta_1 \neq 0
\end{align}
\]</span> When we see small <span class="math inline">\(p\)</span>-val, reject null hypothesis in favor of alternate hypothesis.</p>
<p>Implication of this w.r.t the original model objective?</p>
<blockquote class="blockquote">
<p><strong>There is a significant relationship between <span class="math inline">\(y\)</span> and $x4. In mathematical terms, there is significant evidence in favor of a correlation between <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span></strong></p>
</blockquote>
<p>This is what the <span class="math inline">\(p\)</span>-values in model output are capturing. We can use the <code>kable</code> function to print results nicely.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(broom)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stderr">
<pre><code>Warning: package 'broom' was built under R version 4.1.3</code></pre>
</div>
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(model)<span class="sc">%>%</span></span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a> broom<span class="sc">::</span><span class="fu">tidy</span>()<span class="sc">%>%</span></span>
<span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a> knitr<span class="sc">::</span><span class="fu">kable</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<table>
<thead>
<tr>
<th style="text-align:left;"> term </th>
<th style="text-align:right;"> estimate </th>
<th style="text-align:right;"> std.error </th>
<th style="text-align:right;"> statistic </th>
<th style="text-align:right;"> p.value </th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;"> (Intercept) </td>
<td style="text-align:right;"> 4.267293 </td>
<td style="text-align:right;"> 2.529747 </td>
<td style="text-align:right;"> 1.686846 </td>
<td style="text-align:right;"> 0.0979904 </td>
</tr>
<tr>
<td style="text-align:left;"> x </td>
<td style="text-align:right;"> 1.373345 </td>
<td style="text-align:right;"> 0.183523 </td>
<td style="text-align:right;"> 7.483234 </td>
<td style="text-align:right;"> 0.0000000 </td>
</tr>
</tbody>
</table>
</div>
</div>
<p>We have the following terminology for different components of the model.</p>
<ol type="1">
<li>Covariate: <span class="math inline">\(x\)</span></li>
</ol>
<div class="cell">
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span>(x)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 20.1 7.1 16.1 14.9 16.7 8.8</code></pre>
</div>
</div>
<ol start="2" type="1">
<li>Response: <span class="math inline">\(y\)</span></li>
</ol>
<div class="cell">
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span>(y)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 31.5 18.9 35.0 31.6 22.6 26.2</code></pre>
</div>
</div>
<ol start="3" type="1">
<li>Fitted values: <span class="math inline">\(\hat{y}\)</span></li>
</ol>
<div class="cell">
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a>yhat <span class="ot"><-</span> <span class="fu">fitted</span>(model)</span>
<span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a>yhat</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> 1 2 3 4 5 6 7 8
31.87154 14.01805 26.37815 24.73014 27.20216 16.35273 17.58874 18.41275
9 10 11 12 13 14 15 16
34.48089 26.51549 20.88477 18.41275 24.18080 21.29678 17.45141 21.02211
17 18 19 20 21 22 23 24
19.09942 24.45547 31.32220 19.64876 18.13808 19.37409 21.02211 16.90207
25 26 27 28 29 30 31 32
36.54091 17.17674 25.27948 17.45141 19.51143 11.54602 14.97939 39.01293
33 34 35 36 37 38 39 40
26.92749 21.57144 20.74744 20.06076 27.75150 19.64876 21.02211 18.82475
41 42 43 44 45 46 47 48
31.59687 24.18080 25.55415 28.16350 15.80339 18.41275 18.27542 21.43411
49 50 51
27.20216 15.94073 21.02211 </code></pre>
</div>
</div>
<ol start="4" type="1">
<li>Residuals: <span class="math inline">\(e = y - \hat{y}\)</span></li>
</ol>
<div class="cell">
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a>res <span class="ot"><-</span> <span class="fu">residuals</span>(model)</span>
<span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span>(res)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> 1 2 3 4 5 6
-0.3715352 4.8819549 8.6218464 6.8698609 -4.6021608 9.8472677 </code></pre>
</div>
</div>
<p>Another important summary in model output is <span class="math inline">\(R^2\)</span>, given as follows:</p>
<p><span class="math inline">\(R^2\)</span> =</p>
<ol type="1">
<li>Sum of square for residuals:</li>
</ol>
<p><span class="math inline">\(SS_{Res} = \sum_{i=1}^n e_i^2 = \sum_{i=1}^n (y_i - \hat{y}_i^2)\)</span></p>
<ol start="2" type="1">
<li><p>Sum of squares for regression: <span class="math inline">\(SS_{Reg} = \sum_{i=1}^n (\hat{y}_i - \bar{y})^2\)</span></p></li>
<li><p>Sum of squares Total: $SS_{Tot} = _{i=1}^n (y_i - {y})^2</p></li>
</ol>
<div class="cell">
<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a>x <span class="ot"><-</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">5</span>, <span class="at">length =</span> <span class="dv">100</span>)</span>
<span id="cb28-2"><a href="#cb28-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb28-3"><a href="#cb28-3" aria-hidden="true" tabindex="-1"></a>b0 <span class="ot"><-</span> <span class="dv">2</span></span>
<span id="cb28-4"><a href="#cb28-4" aria-hidden="true" tabindex="-1"></a>b1 <span class="ot"><-</span> <span class="dv">3</span></span>
<span id="cb28-5"><a href="#cb28-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb28-6"><a href="#cb28-6" aria-hidden="true" tabindex="-1"></a>y1 <span class="ot"><-</span> b0 <span class="sc">+</span> b1 <span class="sc">*</span> x <span class="sc">+</span> <span class="fu">rnorm</span>(<span class="dv">100</span>)</span>
<span id="cb28-7"><a href="#cb28-7" aria-hidden="true" tabindex="-1"></a>y2 <span class="ot"><-</span> b0 <span class="sc">+</span> b1 <span class="sc">*</span> x <span class="sc">+</span> <span class="fu">rnorm</span>(<span class="dv">100</span>) <span class="sc">*</span> <span class="dv">3</span></span>
<span id="cb28-8"><a href="#cb28-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb28-9"><a href="#cb28-9" aria-hidden="true" tabindex="-1"></a><span class="fu">par</span>(<span class="at">mfrow=</span><span class="fu">c</span>(<span class="dv">1</span>,<span class="dv">2</span>))</span>
<span id="cb28-10"><a href="#cb28-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb28-11"><a href="#cb28-11" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(x, y1)</span>
<span id="cb28-12"><a href="#cb28-12" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(x, y2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-17-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a>model1 <span class="ot"><-</span> <span class="fu">lm</span>(y1 <span class="sc">~</span> x)</span>
<span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a>model2 <span class="ot"><-</span> <span class="fu">lm</span>(y2 <span class="sc">~</span> x)</span>
<span id="cb29-3"><a href="#cb29-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb29-4"><a href="#cb29-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(x, y1)</span>
<span id="cb29-5"><a href="#cb29-5" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>(<span class="fu">coef</span>(model1)[<span class="dv">1</span>] <span class="sc">+</span> <span class="fu">coef</span>(model1)[<span class="dv">2</span>] <span class="sc">*</span> x, <span class="at">add=</span>T, <span class="at">col=</span><span class="st">'red'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-18-1.png" class="img-fluid" width="672"></p>
</div>
<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(x, y2)</span>
<span id="cb30-2"><a href="#cb30-2" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>(<span class="fu">coef</span>(model2)[<span class="dv">1</span>] <span class="sc">+</span> <span class="fu">coef</span>(model2)[<span class="dv">2</span>] <span class="sc">*</span> x, <span class="at">add=</span>T, <span class="at">col=</span><span class="st">'red'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-18-2.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(model1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>
Call:
lm(formula = y1 ~ x)
Residuals:
Min 1Q Median 3Q Max
-2.13780 -0.90815 0.03056 0.71063 2.54635
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.23152 0.21353 10.45 <2e-16 ***
x 2.88860 0.07378 39.15 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.076 on 98 degrees of freedom
Multiple R-squared: 0.9399, Adjusted R-squared: 0.9393
F-statistic: 1533 on 1 and 98 DF, p-value: < 2.2e-16</code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb33"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(model2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>
Call:
lm(formula = y2 ~ x)
Residuals:
Min 1Q Median 3Q Max
-8.2194 -2.3514 -0.2461 2.2420 7.3651
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.2188 0.6084 2.003 0.0479 *
x 3.3734 0.2102 16.047 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.065 on 98 degrees of freedom
Multiple R-squared: 0.7243, Adjusted R-squared: 0.7215
F-statistic: 257.5 on 1 and 98 DF, p-value: < 2.2e-16</code></pre>
</div>
</div>
<section id="prediction" class="level3">
<h3 class="anchored" data-anchor-id="prediction">Prediction</h3>
<p>Ability of a model to predict vals for “unseen” data</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb35"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a>x <span class="ot"><-</span> df<span class="sc">$</span>povpct</span>
<span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a>y <span class="ot"><-</span> df<span class="sc">$</span>brth15to17</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>What is best guess for a given prediction? Look at regression line</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb36"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plt</span>()</span>
<span id="cb36-2"><a href="#cb36-2" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">v=</span><span class="dv">21</span>, <span class="at">col=</span><span class="st">'green'</span>)</span>
<span id="cb36-3"><a href="#cb36-3" aria-hidden="true" tabindex="-1"></a><span class="fu">lines</span>(x, <span class="fu">fitted</span>(<span class="fu">lm</span>(y<span class="sc">~</span>x)), <span class="at">col=</span><span class="st">'red'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-22-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Best prediction is intersection. In <code>R</code> we can use the <code>predict()</code> function to do this:<br>
</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb37"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb37-1"><a href="#cb37-1" aria-hidden="true" tabindex="-1"></a>new_x <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="dv">21</span>))</span>
<span id="cb37-2"><a href="#cb37-2" aria-hidden="true" tabindex="-1"></a>new_y <span class="ot"><-</span> <span class="fu">predict</span>(model, new_x)</span>
<span id="cb37-3"><a href="#cb37-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb37-4"><a href="#cb37-4" aria-hidden="true" tabindex="-1"></a>new_y</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> 1
33.10755 </code></pre>
</div>
</div>
<p>If we plot this new point we get:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb39"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb39-1"><a href="#cb39-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plt</span>()</span>
<span id="cb39-2"><a href="#cb39-2" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">v=</span><span class="dv">21</span>, <span class="at">col=</span><span class="st">'green'</span>)</span>
<span id="cb39-3"><a href="#cb39-3" aria-hidden="true" tabindex="-1"></a><span class="fu">lines</span>(x, <span class="fu">fitted</span>(<span class="fu">lm</span>(y<span class="sc">~</span>x)), <span class="at">col=</span><span class="st">'red'</span>)</span>
<span id="cb39-4"><a href="#cb39-4" aria-hidden="true" tabindex="-1"></a><span class="fu">points</span>(new_x, new_y, <span class="at">col =</span> <span class="st">'blue'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-24-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Make predictions not just for single observation but for collection.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb40"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a>new_x <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">21</span>))</span>
<span id="cb40-2"><a href="#cb40-2" aria-hidden="true" tabindex="-1"></a>new_y <span class="ot"><-</span> <span class="fu">predict</span>(model, new_x)</span>
<span id="cb40-3"><a href="#cb40-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb40-4"><a href="#cb40-4" aria-hidden="true" tabindex="-1"></a>new_y</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> 1 2 3 4 5 6 7 8
5.640638 7.013984 8.387329 9.760674 11.134020 12.507365 13.880711 15.254056
9 10 11 12 13 14 15 16
16.627401 18.000747 19.374092 20.747438 22.120783 23.494128 24.867474 26.240819
17 18 19 20 21
27.614164 28.987510 30.360855 31.734201 33.107546 </code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb42"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plt</span>()</span>
<span id="cb42-2"><a href="#cb42-2" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span>(a <span class="cf">in</span> new_x){<span class="fu">abline</span>(<span class="at">v=</span>a, <span class="at">col=</span><span class="st">'green'</span>)}</span>
<span id="cb42-3"><a href="#cb42-3" aria-hidden="true" tabindex="-1"></a><span class="fu">lines</span>(x, <span class="fu">fitted</span>(<span class="fu">lm</span>(y<span class="sc">~</span>x)), <span class="at">col=</span><span class="st">'red'</span>)</span>
<span id="cb42-4"><a href="#cb42-4" aria-hidden="true" tabindex="-1"></a><span class="fu">points</span>(new_x <span class="sc">%>%</span> <span class="fu">unlist</span>(), new_y <span class="sc">%>%</span> <span class="fu">unlist</span>(), <span class="at">col =</span> <span class="st">'blue'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-26-1.png" class="img-fluid" width="672"></p>
</div>
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