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-<title>3&nbsp; Regression – Machine Learning</title>
+<title>3&nbsp; Regression (work in progress) – Machine Learning</title>
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-          <h1 class="quarto-secondary-nav-title"><span class="chapter-number">3</span>&nbsp; <span class="chapter-title">Regression</span></h1>
+          <h1 class="quarto-secondary-nav-title"><span class="chapter-number">3</span>&nbsp; <span class="chapter-title">Regression (work in progress)</span></h1>
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@@ -195,7 +195,7 @@ <h1 class="quarto-secondary-nav-title"><span class="chapter-number">3</span>&nbs
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   <a href="./02-Regression.html" class="sidebar-item-text sidebar-link active">
- <span class="menu-text"><span class="chapter-number">3</span>&nbsp; <span class="chapter-title">Regression</span></span></a>
+ <span class="menu-text"><span class="chapter-number">3</span>&nbsp; <span class="chapter-title">Regression (work in progress)</span></span></a>
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@@ -249,7 +249,7 @@ <h2 id="toc-title">Table of contents</h2>
 
 <header id="title-block-header" class="quarto-title-block default">
 <div class="quarto-title">
-<h1 class="title d-none d-lg-block"><span class="chapter-number">3</span>&nbsp; <span class="chapter-title">Regression</span></h1>
+<h1 class="title d-none d-lg-block"><span class="chapter-number">3</span>&nbsp; <span class="chapter-title">Regression (work in progress)</span></h1>
 </div>
 
 
@@ -277,7 +277,7 @@ <h3 data-number="3.1.1" class="anchored" data-anchor-id="one-predictor"><span cl
 MeanBloodPressure= \beta_0 + \beta_1 \cdot Age
 \]</span></p>
 <p>Our model would look like the following like the red line from our Training data:</p>
-<div id="ad9e4955" class="cell" data-execution_count="1">
+<div id="696bbdfc" class="cell" data-execution_count="1">
 <div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> pandas <span class="im">as</span> pd</span>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> seaborn <span class="im">as</span> sns</span>
 <span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
@@ -353,7 +353,7 @@ <h3 data-number="3.3.1" class="anchored" data-anchor-id="underfitting-and-overfi
 <p><span class="math display">\[
 MeanBloodPressure= \beta_0 + \beta_1 \cdot Age
 \]</span></p>
-<div id="c59662c3" class="cell" data-execution_count="2">
+<div id="1872e11e" class="cell" data-execution_count="2">
 <div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> pandas <span class="im">as</span> pd</span>
 <span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> seaborn <span class="im">as</span> sns</span>
 <span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
@@ -416,7 +416,7 @@ <h3 data-number="3.3.1" class="anchored" data-anchor-id="underfitting-and-overfi
 <li><p>The relationship between a predictor and response is fit via a higher order polynomial or smooth function</p></li>
 </ul>
 <p>Let’s look at what happens if we increase the complexity of the model by fitting it with a more smooth function. We use a polynomial function of order 2.</p>
-<div id="dbf103b9" class="cell" data-execution_count="3">
+<div id="4ae213a5" class="cell" data-execution_count="3">
 <div class="sourceCode cell-code" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>p_degree <span class="op">=</span> <span class="dv">2</span></span>
 <span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>y, X <span class="op">=</span> model_matrix(<span class="st">"BloodPressure ~ BMI + poly(BMI, degree="</span> <span class="op">+</span> <span class="bu">str</span>(p_degree) <span class="op">+</span> <span class="st">")"</span>, nhanes_tiny)</span>
 <span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a></span>
@@ -465,7 +465,7 @@ <h3 data-number="3.3.1" class="anchored" data-anchor-id="underfitting-and-overfi
 </div>
 <p>We see that both Training and Testing error both decreased slightly!</p>
 <p>What happens if we keep increasing the model complexity?</p>
-<div id="13bc74c9" class="cell" data-execution_count="4">
+<div id="54be1bb4" class="cell" data-execution_count="4">
 <div class="sourceCode cell-code" id="cb6"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> p_degree <span class="kw">in</span> [<span class="dv">4</span>, <span class="dv">10</span>]:</span>
 <span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>  y, X <span class="op">=</span> model_matrix(<span class="st">"BloodPressure ~ BMI + poly(BMI, degree="</span> <span class="op">+</span> <span class="bu">str</span>(p_degree) <span class="op">+</span> <span class="st">")"</span>, nhanes_tiny)</span>
 <span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a>  </span>
@@ -523,7 +523,7 @@ <h3 data-number="3.3.1" class="anchored" data-anchor-id="underfitting-and-overfi
 </div>
 </div>
 <p>Let’s summarize it:</p>
-<div id="93f3ed51" class="cell" data-execution_count="5">
+<div id="af04bf84" class="cell" data-execution_count="5">
 <div class="sourceCode cell-code" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>train_err <span class="op">=</span> []</span>
 <span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>test_err <span class="op">=</span> []</span>
 <span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>polynomials <span class="op">=</span> <span class="bu">list</span>(<span class="bu">range</span>(<span class="dv">1</span>, <span class="dv">10</span>))</span>
@@ -588,14 +588,14 @@ <h3 data-number="3.4.1" class="anchored" data-anchor-id="linear-model"><span cla
 <p><span class="math inline">\(\beta_0\)</span> is a parameter describing the intercept of the line, and <span class="math inline">\(\beta_1\)</span> is a parameter describing the slope of the line.</p>
 <p>Suppose that from fitting the model on the Training Set, <span class="math inline">\(\beta_1=2\)</span>. That means increasing <span class="math inline">\(BMI\)</span> by 1 will lead to an increase of <span class="math inline">\(BloodPressure\)</span> by 2. This measures the strength of association between a variable and the outcome.</p>
 <p>Let’s see this in practice:</p>
-<div id="78cfd7ec" class="cell" data-execution_count="6">
+<div id="368e9426" class="cell" data-execution_count="6">
 <div class="sourceCode cell-code" id="cb10"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>y, X <span class="op">=</span> model_matrix(<span class="st">"BloodPressure ~ BMI"</span>, nhanes_tiny)</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>X_train, X_test, y_train, y_test <span class="op">=</span> train_test_split(X, y, test_size<span class="op">=</span><span class="fl">0.5</span>, random_state<span class="op">=</span><span class="dv">42</span>)</span>
 <span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a>linear_model <span class="op">=</span> sm.OLS(y_train, X_train).fit()</span>
 <span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a></span>
 <span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a>linear_model.summary()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<div class="cell-output cell-output-display" data-execution_count="32">
+<div class="cell-output cell-output-display" data-execution_count="6">
 <table class="simpletable caption-top table table-sm table-striped small" data-quarto-postprocess="true">
 <caption>OLS Regression Results</caption>
 <tbody>
@@ -619,13 +619,13 @@ <h3 data-number="3.4.1" class="anchored" data-anchor-id="linear-model"><span cla
 </tr>
 <tr class="even">
 <td data-quarto-table-cell-role="th">Date:</td>
-<td>Wed, 21 Jan 2026</td>
+<td>Fri, 23 Jan 2026</td>
 <td data-quarto-table-cell-role="th">Prob (F-statistic):</td>
 <td>0.00118</td>
 </tr>
 <tr class="odd">
 <td data-quarto-table-cell-role="th">Time:</td>
-<td>15:25:31</td>
+<td>20:04:22</td>
 <td data-quarto-table-cell-role="th">Log-Likelihood:</td>
 <td>-520.69</td>
 </tr>
@@ -747,7 +747,7 @@ <h4 data-number="3.6.0.1" class="anchored" data-anchor-id="correlations-and-inte
 </section>
 <section id="exercise-correlations" class="level4" data-number="3.6.0.2">
 <h4 data-number="3.6.0.2" class="anchored" data-anchor-id="exercise-correlations"><span class="header-section-number">3.6.0.2</span> Exercise: Correlations</h4>
-<div id="6ce1ed28" class="cell" data-execution_count="7">
+<div id="2310c641" class="cell" data-execution_count="7">
 <div class="sourceCode cell-code" id="cb11"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>plt.clf()</span>
 <span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>ax <span class="op">=</span> sns.regplot(y<span class="op">=</span><span class="st">"Age"</span>, x<span class="op">=</span><span class="st">"BMI"</span>, data<span class="op">=</span>nhanes_train, lowess<span class="op">=</span><span class="va">True</span>, scatter_kws<span class="op">=</span>{<span class="st">'alpha'</span>:<span class="fl">0.1</span>}, line_kws<span class="op">=</span>{<span class="st">'color'</span>:<span class="st">"r"</span>})</span>
 <span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a>ax.set_xlim([<span class="dv">10</span>, <span class="dv">50</span>])</span>
@@ -763,7 +763,7 @@ <h4 data-number="3.6.0.2" class="anchored" data-anchor-id="exercise-correlations
 </section>
 <section id="exercise-interactions" class="level4" data-number="3.6.0.3">
 <h4 data-number="3.6.0.3" class="anchored" data-anchor-id="exercise-interactions"><span class="header-section-number">3.6.0.3</span> Exercise: Interactions</h4>
-<div id="43e93275" class="cell" data-execution_count="8">
+<div id="b8bbfdab" class="cell" data-execution_count="8">
 <div class="sourceCode cell-code" id="cb12"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>plt.clf()</span>
 <span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>ax <span class="op">=</span> sns.lmplot(y<span class="op">=</span><span class="st">"MeanBloodPressure"</span>, x<span class="op">=</span><span class="st">"BMI"</span>, hue<span class="op">=</span><span class="st">"Gender"</span>, data<span class="op">=</span>nhanes_train, lowess<span class="op">=</span><span class="va">True</span>, scatter_kws<span class="op">=</span>{<span class="st">'alpha'</span>:<span class="fl">0.1</span>})</span>
 <span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a>ax.<span class="bu">set</span>(xlim<span class="op">=</span>(<span class="dv">10</span>, <span class="dv">50</span>)) </span>