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<p>Okay, great, it looks like when someone’s BMI is higher, then it is more likely that the person has Hypertension.</p>
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<p>Now, let’s build the model <spanclass="math inline">\(Hypertension = f(BMI)\)</span> to make a prediction of <spanclass="math inline">\(Hyptertension\)</span> given <spanclass="math inline">\(BMI\)</span>.</p>
<p>Instead of boxplots, we plotted the data just using points, with “Hypertension” having a probability of 1 and “No Hypertension” having a probability of 0. We see that we have a fitted line in blue for every value of BMI, which represents our machine learning model <spanclass="math inline">\(f(BMI)\)</span>. This model is called <strong>Logistic Regression</strong>.</p>
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<p>The first thing we want to investigate about this model is how well it performs in terms of Classification. Just using <spanclass="math inline">\(BMI\)</span> as a variable, what is the Accuracy of <spanclass="math inline">\(f(BMI)\)</span> classifying whether a person has <spanclass="math inline">\(Hypertension\)</span>? Notice that first <spanclass="math inline">\(f(BMI)\)</span> gives us continuous probability values, such as given a BMI of 30, there is a 20% chance the person has Hypertension. We need a discrete cutoff of this model to decide whether the person has Hypertension.</p>
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<p>A reasonable cutoff to start is 50%: if the probability of having Hypertension is >=50%, then classify that person having Hypertension. Same for < 50%. This is called the <strong>Decision Boundary</strong>.</p>
<spanid="cb5-3"><ahref="#cb5-3" aria-hidden="true" tabindex="-1"></a>prediction_cut <spanclass="op">=</span> [<spanclass="dv">1</span><spanclass="cf">if</span> x <spanclass="op">>=</span><spanclass="fl">.5</span><spanclass="cf">else</span><spanclass="dv">0</span><spanclass="cf">for</span> x <spanclass="kw">in</span> logit_model.predict()]</span>
<p>Suppose we try to use the single variable <spanclass="math inline">\(BMI\)</span> to predict <spanclass="math inline">\(BloodPressure\)</span> using a linear model.</p>
@@ -517,27 +517,27 @@ <h3 data-number="2.4.1" class="anchored" data-anchor-id="prediction"><span class
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<p>We examine how well our model performs in terms of prediction by seeing how close our model’s predicted <spanclass="math inline">\(BloodPressure\)</span> is to the Training Set’s true <spanclass="math inline">\(BloodPressure\)</span>: the <strong>Training Error</strong>. We also take the model to the Testing Set to predict <spanclass="math inline">\(BloodPressure\)</span> using predictors from the Test Set and compare to the true <spanclass="math inline">\(BloodPressure\)</span> in the Test Set: the <strong>Testing Error.</strong> We want the model’s Training Error to be adequately small on the Training Set, but what we really care about is the Testing Error, because it is a true test of how the model performs on unseen, new data, and allows us to see how generalizeable the model is.</p>
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<p>Okay, let’s how it does on the Training Set:</p>
<spanid="cb18-3"><ahref="#cb18-3" aria-hidden="true" tabindex="-1"></a>plt.legend()<spanclass="op">;</span></span></code><buttontitle="Copy to Clipboard" class="code-copy-button"><iclass="bi"></i></button></pre></div>
@@ -551,7 +551,7 @@ <h3 data-number="2.4.1" class="anchored" data-anchor-id="prediction"><span class
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<p>We see that the Training Error is fairly high, and the Testing Error is even higher. This is an example of <strong>Underfitting</strong>, where our model failed to capture the complexity of the data in both the Training and Testing Set.</p>
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<p>Let’s return to the drawing board and fit a new type of model that has more flexibility around complicated patterns of data. Let’s see how it does on the Training Set:</p>
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