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Classification Lecture.html
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Classification Lecture.html
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<div id="ipython-notebook">
<a class="interact-button" href="http://data8.berkeley.edu/hub/interact?repo=textbook&path=notebooks/ckd.csv&path=notebooks/banknote.csv&path=notebooks/Classification Lecture.ipynb">Interact</a>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {
inlineMath: [['$','$']],
processEscapes: true
}
});
</script>
<div class="inner_cell">
<div class="text_cell_render border-box-sizing rendered_html">
<h3 id="Chronic-kidney-disease">Chronic kidney disease<a class="anchor-link" href="#Chronic-kidney-disease">¶</a></h3><p>We're going to work with a data set that was collected to help doctors diagnose chronic kidney disease (CKD). Each row in the data set represents a single patient who was treated in the past and whose diagnosis is known. For each patient, we have a bunch of measurements from a blood test.</p></div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ckd</span> <span class="o">=</span> <span class="n">Table</span><span class="o">.</span><span class="n">read_table</span><span class="p">(</span><span class="s1">'ckd.csv'</span><span class="p">)</span><span class="o">.</span><span class="n">relabeled</span><span class="p">(</span><span class="s1">'Blood Glucose Random'</span><span class="p">,</span> <span class="s1">'Glucose'</span><span class="p">)</span>
<span class="n">ckd</span><span class="o">.</span><span class="n">show</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
</pre></div></div></div>
<div class="output_html rendered_html output_subarea ">
<table border="1" class="dataframe">
<thead>
<tr>
<th>Age</th> <th>Blood Pressure</th> <th>Specific Gravity</th> <th>Albumin</th> <th>Sugar</th> <th>Red Blood Cells</th> <th>Pus Cell</th> <th>Pus Cell clumps</th> <th>Bacteria</th> <th>Glucose</th> <th>Blood Urea</th> <th>Serum Creatinine</th> <th>Sodium</th> <th>Potassium</th> <th>Hemoglobin</th> <th>Packed Cell Volume</th> <th>White Blood Cell Count</th> <th>Red Blood Cell Count</th> <th>Hypertension</th> <th>Diabetes Mellitus</th> <th>Coronary Artery Disease</th> <th>Appetite</th> <th>Pedal Edema</th> <th>Anemia</th> <th>Class</th>
</tr>
</thead>
<tbody>
<tr>
<td>48 </td> <td>70 </td> <td>1.005 </td> <td>4 </td> <td>0 </td> <td>normal </td> <td>abnormal</td> <td>present </td> <td>notpresent</td> <td>117 </td> <td>56 </td> <td>3.8 </td> <td>111 </td> <td>2.5 </td> <td>11.2 </td> <td>32 </td> <td>6700 </td> <td>3.9 </td> <td>yes </td> <td>no </td> <td>no </td> <td>poor </td> <td>yes </td> <td>yes </td> <td>1 </td>
</tr>
</tbody>
<tbody><tr>
<td>53 </td> <td>90 </td> <td>1.02 </td> <td>2 </td> <td>0 </td> <td>abnormal </td> <td>abnormal</td> <td>present </td> <td>notpresent</td> <td>70 </td> <td>107 </td> <td>7.2 </td> <td>114 </td> <td>3.7 </td> <td>9.5 </td> <td>29 </td> <td>12100 </td> <td>3.7 </td> <td>yes </td> <td>yes </td> <td>no </td> <td>poor </td> <td>no </td> <td>yes </td> <td>1 </td>
</tr>
</tbody>
<tbody><tr>
<td>63 </td> <td>70 </td> <td>1.01 </td> <td>3 </td> <td>0 </td> <td>abnormal </td> <td>abnormal</td> <td>present </td> <td>notpresent</td> <td>380 </td> <td>60 </td> <td>2.7 </td> <td>131 </td> <td>4.2 </td> <td>10.8 </td> <td>32 </td> <td>4500 </td> <td>3.8 </td> <td>yes </td> <td>yes </td> <td>no </td> <td>poor </td> <td>yes </td> <td>no </td> <td>1 </td>
</tr>
</tbody>
</table>
<p>... (155 rows omitted)</p></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ckd</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="s1">'Hemoglobin'</span><span class="p">,</span> <span class="s1">'Glucose'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">ckd</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">))</span>
</pre></div></div></div>
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<img src="../notebooks-images/Classification Lecture_3_0.png"/></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ckd</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="s1">'White Blood Cell Count'</span><span class="p">,</span> <span class="s1">'Glucose'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">ckd</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">))</span>
</pre></div></div></div>
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<img src="../notebooks-images/Classification Lecture_4_0.png"/></div>
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<h3 id="Decision-boundary">Decision boundary<a class="anchor-link" href="#Decision-boundary">¶</a></h3><p>For each scatter plot above, how would you draw the boundary between regions recognized as class 0 and class 1?</p></div></div>
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<h3 id="Banknote-authentication">Banknote authentication<a class="anchor-link" href="#Banknote-authentication">¶</a></h3><p>Predicting whether a banknote (e.g., a \$20 bill) is counterfeit or legitimate. Researchers have put together a data set for us, based on photographs of many individual banknotes: some counterfeit, some legitimate.</p></div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">banknotes</span> <span class="o">=</span> <span class="n">Table</span><span class="o">.</span><span class="n">read_table</span><span class="p">(</span><span class="s1">'banknote.csv'</span><span class="p">)</span>
<span class="n">banknotes</span>
</pre></div></div></div>
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<table border="1" class="dataframe">
<thead>
<tr>
<th>WaveletVar</th> <th>WaveletSkew</th> <th>WaveletCurt</th> <th>Entropy</th> <th>Class</th>
</tr>
</thead>
<tbody>
<tr>
<td>3.6216 </td> <td>8.6661 </td> <td>-2.8073 </td> <td>-0.44699</td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>4.5459 </td> <td>8.1674 </td> <td>-2.4586 </td> <td>-1.4621 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>3.866 </td> <td>-2.6383 </td> <td>1.9242 </td> <td>0.10645 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>3.4566 </td> <td>9.5228 </td> <td>-4.0112 </td> <td>-3.5944 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>0.32924 </td> <td>-4.4552 </td> <td>4.5718 </td> <td>-0.9888 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>4.3684 </td> <td>9.6718 </td> <td>-3.9606 </td> <td>-3.1625 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>3.5912 </td> <td>3.0129 </td> <td>0.72888 </td> <td>0.56421 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>2.0922 </td> <td>-6.81 </td> <td>8.4636 </td> <td>-0.60216</td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>3.2032 </td> <td>5.7588 </td> <td>-0.75345 </td> <td>-0.61251</td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>1.5356 </td> <td>9.1772 </td> <td>-2.2718 </td> <td>-0.73535</td> <td>0 </td>
</tr>
</tbody>
</table>
<p>... (1362 rows omitted)</p></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">banknotes</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="s1">'WaveletVar'</span><span class="p">,</span> <span class="s1">'WaveletCurt'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">banknotes</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">))</span>
</pre></div></div></div>
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<img src="../notebooks-images/Classification Lecture_8_0.png"/></div>
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<p>Suppose we used $k=11$. What parts of the plot would the classifier get right, and what parts would it make errors on? What would the decision boundary look like?</p></div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">banknotes</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="s1">'WaveletSkew'</span><span class="p">,</span> <span class="s1">'Entropy'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">banknotes</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">))</span>
</pre></div></div></div>
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<img src="../notebooks-images/Classification Lecture_10_0.png"/></div>
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<h3 id="Multiple-attributes">Multiple attributes<a class="anchor-link" href="#Multiple-attributes">¶</a></h3></div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">banknotes</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'WaveletSkew'</span><span class="p">),</span>
<span class="n">banknotes</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'WaveletVar'</span><span class="p">),</span>
<span class="n">banknotes</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'WaveletCurt'</span><span class="p">),</span>
<span class="n">c</span><span class="o">=</span><span class="n">banknotes</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">))</span>
</pre></div></div></div>
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<pre><mpl_toolkits.mplot3d.art3d.Path3DCollection at 0x10953d5f8></pre></div>
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<img src="../notebooks-images/Classification Lecture_12_1.png"/></div>
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<h3 id="Accuracy">Accuracy<a class="anchor-link" href="#Accuracy">¶</a></h3><p>How would you summarize the accuracy of a classifier for a dataset?</p></div></div>
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<h2 id="Breast-cancer-diagnosis">Breast cancer diagnosis<a class="anchor-link" href="#Breast-cancer-diagnosis">¶</a></h2><p>Brittany Wenger won the Google national science fair three years ago as a 17-year old high school student. Here's Brittany:</p>
<p><img alt="Brittany Wenger" src="http://i.huffpost.com/gen/701499/thumbs/o-GSF83-570.jpg?3"/></p>
<p>Brittany's science fair project was to build a classification algorithm to diagnose breast cancer. She won grand prize for building an algorithm whose accuracy was almost 99%.</p>
<p>Let's see how well we can do, with the ideas we've learned in this course.</p>
<p>So, let me tell you a little bit about the data set. Basically, if a woman has a lump in her breast, the doctors may want to take a biopsy to see if it is cancerous. There are several different procedures for doing that. Brittany focused on fine needle aspiration (FNA), because it is less invasive than the alternatives. The doctor gets a sample of the mass, puts it under a microscope, takes a picture, and a trained lab tech analyzes the picture to determine whether it is cancer or not. We get a picture like one of the following:</p>
<p><img alt="benign" src="https://lh5.googleusercontent.com/sYFBBiw6XB2uEkQBTLCDqQvfi1vzId7q-EFvGIkeEqgaq-c7Q7HEaT5tdUIM8rU7l5-a9E_8gZzqDhnFEu7xV8MnXAeez41Ckq9DN0wO_S8nEY0rqek"/></p>
<p><img alt="cancer" src="https://lh5.googleusercontent.com/OpQSE0LmsWmYTahY3XAwb0RTPUluMhwT_FEbKhF7OU27iVxHk6on9VTruCW2loeks6HICe3Chjg4zXZxp9ko0rQhC3X_QeThTZFyaQc87RTZaGzoc7Y"/></p>
<p>Unfortunately, distinguishing between benign vs malignant can be tricky. So, researchers have studied using machine learning to help with this task. The idea is that we'll ask the lab tech to analyze the image and compute various attributes: things like the typical size of a cell, how much variation there is among the cell sizes, and so on. Then, we'll try to use this information to predict (classify) whether the sample is malignant or not. We have a training set of past samples from women where the correct diagnosis is known, and we'll hope that our machine learning algorithm can use those to learn how to predict the diagnosis for future samples.</p>
<p>We end up with the following data set. For the "Class" column, 1 means malignant (cancer); 0 means benign (not cancer).</p></div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">patients</span> <span class="o">=</span> <span class="n">Table</span><span class="o">.</span><span class="n">read_table</span><span class="p">(</span><span class="s1">'breast-cancer.csv'</span><span class="p">)</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="s1">'ID'</span><span class="p">)</span>
<span class="n">patients</span>
</pre></div></div></div>
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<table border="1" class="dataframe">
<thead>
<tr>
<th>Clump Thickness</th> <th>Uniformity of Cell Size</th> <th>Uniformity of Cell Shape</th> <th>Marginal Adhesion</th> <th>Single Epithelial Cell Size</th> <th>Bare Nuclei</th> <th>Bland Chromatin</th> <th>Normal Nucleoli</th> <th>Mitoses</th> <th>Class</th>
</tr>
</thead>
<tbody>
<tr>
<td>5 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>2 </td> <td>1 </td> <td>3 </td> <td>1 </td> <td>1 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>5 </td> <td>4 </td> <td>4 </td> <td>5 </td> <td>7 </td> <td>10 </td> <td>3 </td> <td>2 </td> <td>1 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>3 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>2 </td> <td>2 </td> <td>3 </td> <td>1 </td> <td>1 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>6 </td> <td>8 </td> <td>8 </td> <td>1 </td> <td>3 </td> <td>4 </td> <td>3 </td> <td>7 </td> <td>1 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>4 </td> <td>1 </td> <td>1 </td> <td>3 </td> <td>2 </td> <td>1 </td> <td>3 </td> <td>1 </td> <td>1 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>8 </td> <td>10 </td> <td>10 </td> <td>8 </td> <td>7 </td> <td>10 </td> <td>9 </td> <td>7 </td> <td>1 </td> <td>1 </td>
</tr>
</tbody>
<tbody><tr>
<td>1 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>2 </td> <td>10 </td> <td>3 </td> <td>1 </td> <td>1 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>2 </td> <td>1 </td> <td>2 </td> <td>1 </td> <td>2 </td> <td>1 </td> <td>3 </td> <td>1 </td> <td>1 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>2 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>2 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>5 </td> <td>0 </td>
</tr>
</tbody>
<tbody><tr>
<td>4 </td> <td>2 </td> <td>1 </td> <td>1 </td> <td>2 </td> <td>1 </td> <td>2 </td> <td>1 </td> <td>1 </td> <td>0 </td>
</tr>
</tbody>
</table>
<p>... (673 rows omitted)</p></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">patients</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="s1">'Bland Chromatin'</span><span class="p">,</span> <span class="s1">'Single Epithelial Cell Size'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">patients</span><span class="p">[</span><span class="s1">'Class'</span><span class="p">])</span>
</pre></div></div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">randomize_column</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
<span class="k">return</span> <span class="n">a</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.09</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
<span class="n">Table</span><span class="p">()</span><span class="o">.</span><span class="n">with_columns</span><span class="p">([</span>
<span class="s1">'Bland Chromatin (jittered)'</span><span class="p">,</span>
<span class="n">randomize_column</span><span class="p">(</span><span class="n">patients</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'Bland Chromatin'</span><span class="p">)),</span>
<span class="s1">'Single Epithelial Cell Size (jittered)'</span><span class="p">,</span>
<span class="n">randomize_column</span><span class="p">(</span><span class="n">patients</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'Single Epithelial Cell Size'</span><span class="p">)),</span>
<span class="p">])</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">patients</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">))</span>
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<h3 id="Applying-the-k-nearest-neighbor-classifier-to-breast-cancer-diagnosis">Applying the k-nearest neighbor classifier to breast cancer diagnosis<a class="anchor-link" href="#Applying-the-k-nearest-neighbor-classifier-to-breast-cancer-diagnosis">¶</a></h3><p>The first thing we need is a way to compute the distance between two points. How do we do this? In 2-dimensional space, it's pretty easy. If we have a point at coordinates $(x_0,y_0)$ and another at $(x_1,y_1)$, the distance between them is</p>
$$D = \sqrt{(x_0-x_1)^2 + (y_0-y_1)^2}.$$<p>In 3-dimensional space, the formula is</p>
$$D = \sqrt{x_0-x_1)^2 + (y_0-y_1)^2 + (z_0-z_1)^2}.$$</div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">distance</span><span class="p">(</span><span class="n">pt1</span><span class="p">,</span> <span class="n">pt2</span><span class="p">):</span>
<span class="n">total</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">pt1</span><span class="p">)):</span>
<span class="n">total</span> <span class="o">=</span> <span class="n">total</span> <span class="o">+</span> <span class="p">(</span><span class="n">pt1</span><span class="o">.</span><span class="n">item</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">-</span> <span class="n">pt2</span><span class="o">.</span><span class="n">item</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span>
<span class="k">return</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">total</span><span class="p">)</span>
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<h3 id="Implementing-a-Classifier">Implementing a Classifier<a class="anchor-link" href="#Implementing-a-Classifier">¶</a></h3></div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">closest</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="o">...</span>
<span class="k">def</span> <span class="nf">majority</span><span class="p">(</span><span class="n">topkclasses</span><span class="p">):</span>
<span class="o">...</span>
<span class="k">def</span> <span class="nf">classify</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="n">kclosest</span> <span class="o">=</span> <span class="n">closest</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
<span class="n">kclosest</span><span class="o">.</span><span class="n">classes</span> <span class="o">=</span> <span class="n">kclosest</span><span class="o">.</span><span class="n">select</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">)</span>
<span class="k">return</span> <span class="n">majority</span><span class="p">(</span><span class="n">kclosest</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">computetablewithdists</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
<span class="n">dists</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">training</span><span class="o">.</span><span class="n">num_rows</span><span class="p">)</span>
<span class="n">attributes</span> <span class="o">=</span> <span class="n">training</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">training</span><span class="o">.</span><span class="n">num_rows</span><span class="p">):</span>
<span class="n">dists</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">distance</span><span class="p">(</span><span class="n">attributes</span><span class="o">.</span><span class="n">row</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">p</span><span class="p">)</span>
<span class="k">return</span> <span class="n">training</span><span class="o">.</span><span class="n">with_column</span><span class="p">(</span><span class="s1">'Distance'</span><span class="p">,</span> <span class="n">dists</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">closest</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="n">withdists</span> <span class="o">=</span> <span class="n">computetablewithdists</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
<span class="n">sortedbydist</span> <span class="o">=</span> <span class="n">withdists</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="s1">'Distance'</span><span class="p">)</span>
<span class="n">topk</span> <span class="o">=</span> <span class="n">sortedbydist</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">k</span><span class="p">))</span>
<span class="k">return</span> <span class="n">topk</span>
<span class="k">def</span> <span class="nf">majority</span><span class="p">(</span><span class="n">topkclasses</span><span class="p">):</span>
<span class="k">if</span> <span class="n">topkclasses</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">num_rows</span> <span class="o">></span> <span class="n">topkclasses</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">num_rows</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">1</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="k">def</span> <span class="nf">classify</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="n">closestk</span> <span class="o">=</span> <span class="n">closest</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
<span class="n">topkclasses</span> <span class="o">=</span> <span class="n">closestk</span><span class="o">.</span><span class="n">select</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">)</span>
<span class="k">return</span> <span class="n">majority</span><span class="p">(</span><span class="n">topkclasses</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">patients</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span>
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<table border="1" class="dataframe">
<thead>
<tr>
<th>Clump Thickness</th> <th>Uniformity of Cell Size</th> <th>Uniformity of Cell Shape</th> <th>Marginal Adhesion</th> <th>Single Epithelial Cell Size</th> <th>Bare Nuclei</th> <th>Bland Chromatin</th> <th>Normal Nucleoli</th> <th>Mitoses</th> <th>Class</th>
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<tr>
<td>5 </td> <td>3 </td> <td>3 </td> <td>3 </td> <td>2 </td> <td>3 </td> <td>4 </td> <td>4 </td> <td>1 </td> <td>1 </td>
</tr>
</tbody>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">example</span> <span class="o">=</span> <span class="n">patients</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">)</span><span class="o">.</span><span class="n">row</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span>
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<p>Let's take $k=5$. We can find the 5 nearest neighbors:</p></div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">closest</span><span class="p">(</span><span class="n">patients</span><span class="o">.</span><span class="n">exclude</span><span class="p">(</span><span class="mi">12</span><span class="p">),</span> <span class="n">example</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
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<table border="1" class="dataframe">
<thead>
<tr>
<th>Clump Thickness</th> <th>Uniformity of Cell Size</th> <th>Uniformity of Cell Shape</th> <th>Marginal Adhesion</th> <th>Single Epithelial Cell Size</th> <th>Bare Nuclei</th> <th>Bland Chromatin</th> <th>Normal Nucleoli</th> <th>Mitoses</th> <th>Class</th> <th>Distance</th>
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<td>5 </td> <td>3 </td> <td>3 </td> <td>4 </td> <td>2 </td> <td>4 </td> <td>3 </td> <td>4 </td> <td>1 </td> <td>1 </td> <td>1.73205 </td>
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<td>5 </td> <td>1 </td> <td>3 </td> <td>3 </td> <td>2 </td> <td>2 </td> <td>2 </td> <td>3 </td> <td>1 </td> <td>0 </td> <td>3.16228 </td>
</tr>
</tbody>
<tbody><tr>
<td>5 </td> <td>2 </td> <td>2 </td> <td>2 </td> <td>2 </td> <td>2 </td> <td>3 </td> <td>2 </td> <td>2 </td> <td>0 </td> <td>3.16228 </td>
</tr>
</tbody>
<tbody><tr>
<td>5 </td> <td>3 </td> <td>3 </td> <td>1 </td> <td>3 </td> <td>3 </td> <td>3 </td> <td>3 </td> <td>3 </td> <td>1 </td> <td>3.31662 </td>
</tr>
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<td>4 </td> <td>3 </td> <td>3 </td> <td>1 </td> <td>2 </td> <td>1 </td> <td>3 </td> <td>3 </td> <td>1 </td> <td>0 </td> <td>3.31662 </td>
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<p>3 out of the 5 nearest neighbors have class 0, so the majority is 0 (no cancer) -- and that is the output of our classifier for this patient:</p></div></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">classify</span><span class="p">(</span><span class="n">patients</span><span class="o">.</span><span class="n">exclude</span><span class="p">(</span><span class="mi">12</span><span class="p">),</span> <span class="n">example</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
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<pre>0</pre></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">patients</span> <span class="o">=</span> <span class="n">patients</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="mi">683</span><span class="p">)</span> <span class="c1"># Randomly permute the rows</span>
<span class="n">trainset</span> <span class="o">=</span> <span class="n">patients</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">342</span><span class="p">))</span>
<span class="n">testset</span> <span class="o">=</span> <span class="n">patients</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">342</span><span class="p">,</span> <span class="mi">683</span><span class="p">))</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">evaluate_accuracy</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">test</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="n">testattrs</span> <span class="o">=</span> <span class="n">test</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">)</span>
<span class="n">numcorrect</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">test</span><span class="o">.</span><span class="n">num_rows</span><span class="p">):</span>
<span class="c1"># Run the classifier on the ith patient in the test set</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">classify</span><span class="p">(</span><span class="n">training</span><span class="p">,</span> <span class="n">testattrs</span><span class="o">.</span><span class="n">rows</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">k</span><span class="p">)</span>
<span class="c1"># Was the classifier's prediction correct?</span>
<span class="k">if</span> <span class="n">c</span> <span class="o">==</span> <span class="n">test</span><span class="o">.</span><span class="n">column</span><span class="p">(</span><span class="s1">'Class'</span><span class="p">)</span><span class="o">.</span><span class="n">item</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
<span class="n">numcorrect</span> <span class="o">=</span> <span class="n">numcorrect</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">numcorrect</span> <span class="o">/</span> <span class="n">test</span><span class="o">.</span><span class="n">num_rows</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">evaluate_accuracy</span><span class="p">(</span><span class="n">trainset</span><span class="p">,</span> <span class="n">testset</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
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<pre>0.9706744868035191</pre></div>
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<p>As a footnote, you might have noticed that Brittany Wenger did even better. What techniques did she use? One key innovation is that she incorporated a confidence score into her results: her algorithm had a way to determine when it was not able to make a confident prediction, and for those patients, it didn't even try to predict their diagnosis. Her algorithm was 99% accurate on the patients where it made a prediction -- so that extension seemed to help quite a bit.</p></div></div></div>