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<h1 id="Introduction">Introduction<a class="anchor-link" href="#Introduction">&#182;</a></h1><p>We begin by looking at a data set to make things concrete.</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="n">df</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s">&quot;/Users/jteitelbaum/Dropbox/CancerData.csv&quot;</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="n">fig</span><span class="p">,</span><span class="n">axes</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">ncols</span><span class="o">=</span><span class="mi">2</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="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">],</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Lung&#39;</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&#39;Lung&#39;</span><span class="p">)</span>
<span class="c">#plt.xlabel(&#39;Cigarettes Sold per Capita&#39;)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1500</span><span class="p">,</span><span class="mi">3000</span><span class="p">,</span><span class="mi">4500</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">4500</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">30</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">40</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">],</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Bladder&#39;</span><span class="p">],</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;black&#39;</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&#39;Bladder&#39;</span><span class="p">)</span>
<span class="c">#plt.xlabel(&#39;Cigarettes Sold per Capita&#39;)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1500</span><span class="p">,</span><span class="mi">3000</span><span class="p">,</span><span class="mi">4500</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">40</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">axes</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="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">],</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Leukemia&#39;</span><span class="p">],</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;green&#39;</span><span class="p">)</span>
<span class="n">axes</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="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&#39;Leukemia&#39;</span><span class="p">)</span>
<span class="c">#plt.xlabel(&#39;Cigarettes Sold per Capita&#39;)</span>
<span class="n">axes</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="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1500</span><span class="p">,</span><span class="mi">3000</span><span class="p">,</span><span class="mi">4500</span><span class="p">])</span>
<span class="n">axes</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="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">40</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">],</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Kidney&#39;</span><span class="p">],</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;red&#39;</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&#39;Kidney&#39;</span><span class="p">)</span>
<span class="c">#plt.xlabel(&#39;Cigarettes Sold per Capita&#39;)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1500</span><span class="p">,</span><span class="mi">3000</span><span class="p">,</span><span class="mi">4500</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">40</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s">&quot;Deaths from Cancer per 100K Population</span><span class="se">\n</span><span class="s"> vs </span><span class="se">\n</span><span class="s"> Cigarette sales per capita&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
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"
>
</div>
</div>
</div>
</div>
</div>
<div class="cell border-box-sizing code_cell rendered">
<div class="input">
<div class="prompt input_prompt">In&nbsp;[6]:</div>
<div class="inner_cell">
<div class="input_area">
<div class=" highlight hl-ipython3"><pre><span class="n">fig</span><span class="p">,</span><span class="n">axes</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">ncols</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
<span class="n">axes</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">],</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Lung&#39;</span><span class="p">])</span>
<span class="n">axes</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&#39;Lung Cancer per 100K Population</span><span class="se">\n</span><span class="s"> vs </span><span class="se">\n</span><span class="s"> Cigarette Sales per Capita&#39;</span><span class="p">)</span>
<span class="c">#plt.xlabel(&#39;Cigarettes Sold per Capita&#39;)</span>
<span class="n">axes</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1500</span><span class="p">,</span><span class="mi">3000</span><span class="p">,</span><span class="mi">4500</span><span class="p">])</span>
<span class="n">axes</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">4500</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">30</span><span class="p">])</span>
<span class="n">axes</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">40</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<div class="output_wrapper">
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"
>
</div>
</div>
</div>
</div>
</div>
<div class="cell border-box-sizing text_cell rendered">
<div class="prompt input_prompt">
</div>
<div class="inner_cell">
<div class="text_cell_render border-box-sizing rendered_html">
<p>There are 44 data points in the set (each corresponds to a state.)</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">df</span><span class="p">[[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">,</span><span class="s">&#39;Lung&#39;</span><span class="p">]][</span><span class="mi">0</span><span class="p">:</span><span class="mi">10</span><span class="p">]</span>
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<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>Cigarettes</th>
<th>Lung</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td> 1820</td>
<td> 17.05</td>
</tr>
<tr>
<th>1</th>
<td> 3034</td>
<td> 25.88</td>
</tr>
<tr>
<th>2</th>
<td> 2582</td>
<td> 19.80</td>
</tr>
<tr>
<th>3</th>
<td> 1824</td>
<td> 15.98</td>
</tr>
<tr>
<th>4</th>
<td> 2860</td>
<td> 22.07</td>
</tr>
<tr>
<th>5</th>
<td> 3110</td>
<td> 22.83</td>
</tr>
<tr>
<th>6</th>
<td> 3360</td>
<td> 24.55</td>
</tr>
<tr>
<th>7</th>
<td> 4046</td>
<td> 27.27</td>
</tr>
<tr>
<th>8</th>
<td> 2827</td>
<td> 23.57</td>
</tr>
<tr>
<th>9</th>
<td> 2010</td>
<td> 13.58</td>
</tr>
</tbody>
</table>
<p>10 rows × 2 columns</p>
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<p>The basic problem of simple linear regression is to find the slope $m$ and intercept $b$ so that the equation
of the line
$$
y=mx+b
$$
is the <em>best fit</em> to the data above.</p>
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<p>To be specific, we look at the function
$$
E(m,b)=\sum_{i=1}^{N} (y_i-mx_i-b)^2
$$
and we want to find $m$ and $b$ so that this is as small as possible. (For our example, $N=44$.)</p>
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<h1 id="Calculus">Calculus<a class="anchor-link" href="#Calculus">&#182;</a></h1>
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<p>Although the equation for $E$ looks complicated, it is really a function of two variables and so we can use calculus to find
the minimum value. We compute:
$$
\frac{\partial E}{\partial m}=\sum_{i=1}^{N} 2(y_i-mx_i-b)x_i
$$
and
$$
\frac{\partial E}{\partial b}=\sum_{i=1}^{N} 2(y_i-mx_i-b).
$$</p>
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<p>Remembering that the variables are $m$ and $b$, set these two equations to zero and find the following two equations:
$$S_{yx}=S_{xx}m+S_{x}b$$
and
$$
S<em>{y}=S</em>{x}m+Nb
$$
where I've written $S_{x}=\sum_{i=1}^{N} x_i$, $S_{xy}=\sum_{i=1}^{N} x_iy_i$, and so on.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">Sx</span><span class="o">=</span><span class="nb">sum</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">])</span>
<span class="n">Sy</span><span class="o">=</span><span class="nb">sum</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Lung&#39;</span><span class="p">])</span>
<span class="n">Sxy</span><span class="o">=</span><span class="nb">sum</span><span class="p">([</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Lung&#39;</span><span class="p">][</span><span class="n">i</span><span class="p">]</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="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">]))])</span>
<span class="n">Sxx</span><span class="o">=</span><span class="nb">sum</span><span class="p">([</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">][</span><span class="n">i</span><span class="p">]</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="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">]))])</span>
<span class="n">N</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">])</span>
<span class="nb">print</span> <span class="s">&#39;N=&#39;</span><span class="p">,</span><span class="n">N</span><span class="p">,</span><span class="s">&#39;Sx=&#39;</span><span class="p">,</span><span class="n">Sx</span><span class="p">,</span><span class="s">&#39;Sy=&#39;</span><span class="p">,</span><span class="n">Sy</span><span class="p">,</span><span class="s">&#39;Sxx=&#39;</span><span class="p">,</span><span class="n">Sxx</span><span class="p">,</span><span class="s">&#39;Sxy=&#39;</span><span class="p">,</span><span class="n">Sxy</span>
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<pre>N= 44 Sx= 109622 Sy= 874.74 Sxx= 286469700 Sxy= 2248867.14
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<p>These are two equations in two unknowns that you can solve "by hand" or, for the general solution, you can use
Cramer's rule. Let
$$
M=(NS<em>{yx}-S</em>{x}S_{y})
$$
$$
B=(S<em>{xx}S_y-S</em>{yx}S_x)
$$
$$
D=(S<em>{xx}N-S</em>{x}^2)
$$
Then
$$
m=M/D
$$
and $$
b=B/D.
$$</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">M</span><span class="o">=</span><span class="n">N</span><span class="o">*</span><span class="n">Sxy</span><span class="o">-</span><span class="n">Sx</span><span class="o">*</span><span class="n">Sy</span>
<span class="n">B</span><span class="o">=</span><span class="n">Sxx</span><span class="o">*</span><span class="n">Sy</span><span class="o">-</span><span class="n">Sxy</span><span class="o">*</span><span class="n">Sx</span>
<span class="n">D</span><span class="o">=</span><span class="n">N</span><span class="o">*</span><span class="n">Sxx</span><span class="o">-</span><span class="n">Sx</span><span class="o">*</span><span class="n">Sx</span>
<span class="n">m</span><span class="o">=</span><span class="n">M</span><span class="o">/</span><span class="n">D</span>
<span class="n">b</span><span class="o">=</span><span class="n">B</span><span class="o">/</span><span class="n">D</span>
<span class="nb">print</span> <span class="s">&#39;m=&#39;</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="s">&#39;b=&#39;</span><span class="p">,</span><span class="n">b</span>
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<pre>m= 0.00520586968046 b= 6.91050349746
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<div class=" highlight hl-ipython3"><pre><span class="n">fig</span><span class="p">,</span><span class="n">axes</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">ncols</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
<span class="n">axes</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">],</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Lung&#39;</span><span class="p">])</span>
<span class="n">axes</span><span class="o">.</span><span class="n">plot</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="mi">0</span><span class="p">,</span><span class="mi">4500</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="n">m</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">4500</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">+</span><span class="n">b</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</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="mi">0</span><span class="p">,</span><span class="mi">4500</span><span class="p">,</span><span class="mi">1</span><span class="p">))),</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;red&quot;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s">&#39;Regression Line&#39;</span><span class="p">)</span>
<span class="n">axes</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&#39;Lung Cancer per 100K Population</span><span class="se">\n</span><span class="s"> vs </span><span class="se">\n</span><span class="s"> Cigarette Sales per Capita&#39;</span><span class="p">)</span>
<span class="c">#plt.xlabel(&#39;Cigarettes Sold per Capita&#39;)</span>
<span class="n">axes</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1500</span><span class="p">,</span><span class="mi">3000</span><span class="p">,</span><span class="mi">4500</span><span class="p">])</span>
<span class="n">axes</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">4500</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">30</span><span class="p">])</span>
<span class="n">axes</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">40</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<h1 id="Linear-Algebra">Linear Algebra<a class="anchor-link" href="#Linear-Algebra">&#182;</a></h1><p>Let's look at the problem from a different point of view. Let's consider three column vectors:</p>
<ul>
<li>$Y=\left[\begin{matrix} y_1 \cr y_2 \cr \vdots \cr y_N\end{matrix}\right]$</li>
<li>$X=\left[\begin{matrix} x_1 \cr x_2 \cr \vdots \cr x_N\end{matrix}\right]$</li>
<li>$E=\left[\begin{matrix} 1 \cr 1 \cr \vdots \cr 1 \end{matrix}\right]$</li>
</ul>
<p>If things were "perfect", meaning that the points all belonged to the line $y=mx+b$, we would have
$$
Y=mX+bE.
$$</p>
<p>But, of course, we don't. In linear algebra terms, the vector $Y$ <strong>does not lie in the plane spanned by $X$ and $E$ in $\mathbb{R}^{44}$</strong></p>
<p>So let's try to find the point in the plane spanned by $E$ and $X$ which is closest to $Y$.</p>
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<div class="prompt input_prompt">In&nbsp;[11]:</div>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="k">import</span> <span class="n">Axes3D</span>
<span class="c">#plt.subplots(nrows=1,ncols=1,figsize=(6,10))</span>
<span class="n">fig</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">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</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="s">&#39;3d&#39;</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="n">Z</span><span class="o">=</span><span class="mi">2</span><span class="o">*</span><span class="n">Y</span><span class="o">-</span><span class="n">X</span>
<span class="n">t</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="o">.</span><span class="mi">1</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot_wireframe</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y</span><span class="p">,</span><span class="n">Z</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">t</span><span class="p">,</span><span class="mi">6</span><span class="o">+</span><span class="n">t</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;red&quot;</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="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">6</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;red&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&quot;Closest point lying in a given plane from a point outside the plane&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>We can use linear algebra to solve the problem of finding the point in the plane $P$ spanned by $X$ and $E$ that is closest to $Y$. Essentially, we need to take the perpendicular projection of the vector $Y$ into the plane spanned by $X$ and $E$. For that, we can construct an orthonormal basis for the plane $P$ by using the Gram-Schmidt process on the two vectors $E$ and $X$.</p>
<p>First, we normalize $E$ to obtain our first unit vector $\mathbf{e}=E/\sqrt{N}$, since $E\cdot E=N$.</p>
<p>Next, we subtract the projection of $X$ onto the $E$ direction and normalize. Let $x=X-(X\cdot \mathbf{e})\mathbf{e}$
and
$$
\mathbf{x}=\frac{x}{\sqrt{x\cdot x}}
$$</p>
<p>Then the point we are interested in is
$$
\mathbf{y}=(Y\cdot\mathbf{x})\mathbf{x}+(Y\cdot\mathbf{e})\mathbf{e}
$$</p>
<p>This point lies in the plane $P$, and $Y-\mathbf{y}$ is perpendicular to both $E$ and $X$.</p>
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<p>Let's make a few observations about this. First of all, $(Y\cdot\mathbf{e})\mathbf{e}=\overline{y}E$ and $(X\cdot \mathbf{e})\mathbf{e}=\overline{x}E$ where $\overline{x}$ and $\overline{y}$ are the averages (means) of the $x$ and $y$ values respetively.</p>
<p>We can arrange for our data to have $\overline{x}=0$ and $\overline{y}=0$ by subtracting $\overline{x}$ and $\overline{y}$ each of the points. This amounts to a coordinate change that moves the origin of our coordinate system to $(\overline{x},\overline{y})$.</p>
<p>In these coordinates, we get a huge simplification. If both means are zero, we get
$$
\mathbf{x}=\frac{X}{\sqrt{X\cdot X}}
$$
and
$$
\mathbf{y}=\frac{Y\cdot X}{X\cdot X}X
$$
In other words, in these coordinates, $b=0$ and $m=\frac{Y\cdot X}{X\cdot X}$.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">Xnorm</span><span class="o">=</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">]</span><span class="o">-</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Cigarettes&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
<span class="n">Ynorm</span><span class="o">=</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Lung&#39;</span><span class="p">]</span><span class="o">-</span><span class="n">df</span><span class="p">[</span><span class="s">&#39;Lung&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">Xnorm</span><span class="p">,</span><span class="n">Ynorm</span><span class="p">)</span>
<span class="n">m</span><span class="o">=</span><span class="n">Xnorm</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Ynorm</span><span class="p">)</span><span class="o">/</span><span class="n">Xnorm</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Xnorm</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Xnorm</span><span class="p">,</span> <span class="n">m</span><span class="o">*</span><span class="n">Xnorm</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<h1 id="Statistics">Statistics<a class="anchor-link" href="#Statistics">&#182;</a></h1><p>What accounts for the variation of the points around the "true value"? Statisticians approach this through the idea of a <strong>statistical model</strong>. Let's leave the lung cancer data behind for the moment, and imagine an abstract problem. Suppose
that we make measurements of a dependent variable $Y$ that is related to an independent variable $X$ by a linear equation.
Suppose, however, that the $Y$ values include a certain amount of random error $\epsilon$, so that
$$
Y=mX+b+\epsilon.
$$</p>
<p>We will assume that the error term $\epsilon$ is a <em>normally distributed</em> error, with <em>mean</em> zero, variance $\sigma$, and that the errors we obtain from separate measurements of $Y$ are independent of one another.</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="k">import</span> <span class="n">norm</span>
<span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="o">.</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">norm</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">scale</span><span class="o">=.</span><span class="mi">7</span><span class="p">,</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">),</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;red&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s">&#39;v=.5&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">norm</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;blue&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s">&#39;v=1&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">norm</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="mf">1.5</span><span class="p">),</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;orange&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s">&#39;v=2.25&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">&#39;Normal Distributions&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<p>Suppose for the sake of concreteness that the $X$ values are 0.0,0.1,...,9.9 and that $Y=X+1+\epsilon$ where $\epsilon$ has variance $1$. Different measurements create different scatter plots.</p>
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<div class="prompt input_prompt">In&nbsp;[14]:</div>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">MB</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y</span><span class="p">):</span>
<span class="n">Sxx</span><span class="o">=</span><span class="nb">sum</span><span class="p">([</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">]</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="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">))])</span>
<span class="n">Sxy</span><span class="o">=</span><span class="nb">sum</span><span class="p">([</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">Y</span><span class="p">[</span><span class="n">i</span><span class="p">]</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="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">))])</span>
<span class="n">Sx</span><span class="o">=</span><span class="nb">sum</span><span class="p">([</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">]</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="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">))])</span>
<span class="n">Sy</span><span class="o">=</span><span class="nb">sum</span><span class="p">([</span><span class="n">Y</span><span class="p">[</span><span class="n">i</span><span class="p">]</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="nb">len</span><span class="p">(</span><span class="n">Y</span><span class="p">))])</span>
<span class="n">N</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">M</span><span class="o">=</span><span class="n">N</span><span class="o">*</span><span class="n">Sxy</span><span class="o">-</span><span class="n">Sx</span><span class="o">*</span><span class="n">Sy</span>
<span class="n">B</span><span class="o">=</span><span class="n">Sxx</span><span class="o">*</span><span class="n">Sy</span><span class="o">-</span><span class="n">Sxy</span><span class="o">*</span><span class="n">Sx</span>
<span class="n">D</span><span class="o">=</span><span class="n">N</span><span class="o">*</span><span class="n">Sxx</span><span class="o">-</span><span class="n">Sx</span><span class="o">*</span><span class="n">Sx</span>
<span class="n">m</span><span class="o">=</span><span class="n">M</span><span class="o">/</span><span class="n">D</span>
<span class="n">b</span><span class="o">=</span><span class="n">B</span><span class="o">/</span><span class="n">D</span>
<span class="k">return</span> <span class="n">m</span><span class="p">,</span><span class="n">b</span>
<span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="o">.</span><span class="mi">1</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span><span class="n">ax</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
<span class="n">epsilon</span><span class="o">=</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">Y0</span><span class="o">=</span><span class="n">X</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">))</span><span class="o">+</span><span class="n">epsilon</span>
<span class="n">epsilon</span><span class="o">=</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">Y1</span><span class="o">=</span><span class="n">X</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">))</span><span class="o">+</span><span class="n">epsilon</span>
<span class="n">epsilon</span><span class="o">=</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">Y2</span><span class="o">=</span><span class="n">X</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">))</span><span class="o">+</span><span class="n">epsilon</span>
<span class="n">epsilon</span><span class="o">=</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">Y3</span><span class="o">=</span><span class="n">X</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">))</span><span class="o">+</span><span class="n">epsilon</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</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="k">for</span> <span class="n">j</span> <span class="ow">in</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">ax</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">17</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y0</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;blue&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y1</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;red&#39;</span><span class="p">)</span>
<span class="n">ax</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="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y2</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;green&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y3</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;black&#39;</span><span class="p">)</span>
<span class="n">m0</span><span class="p">,</span><span class="n">b0</span><span class="o">=</span><span class="n">MB</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y0</span><span class="p">)</span>
<span class="n">m1</span><span class="p">,</span><span class="n">b1</span><span class="o">=</span><span class="n">MB</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y1</span><span class="p">)</span>
<span class="n">m2</span><span class="p">,</span><span class="n">b2</span><span class="o">=</span><span class="n">MB</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y2</span><span class="p">)</span>
<span class="n">m3</span><span class="p">,</span><span class="n">b3</span><span class="o">=</span><span class="n">MB</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y3</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">m0</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">b0</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;blue&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">m1</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">b1</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;red&quot;</span><span class="p">)</span>
<span class="n">ax</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="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">m2</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">b2</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;green&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">m3</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">b3</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;black&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>We can superimpose these different lines and you can see that they are slightly different -- with varying slope and intercept -- which come from different snapshots of the underlying random data.</p>
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<div class="prompt input_prompt">In&nbsp;[15]:</div>
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<div class=" highlight hl-ipython3"><pre><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">m0</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">b0</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;blue&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">m1</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">b1</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;red&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">m2</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">b2</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;green&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">m3</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">b3</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;black&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>Suppose we draw 5000 samples of 100 Y-values each, where each of the 100 $Y$ values from a given sample
satisfy $Y=X+1+\epsilon$. For each of the 5000 samples, we calculate the slope and intercept for those $100$ $Y$-values.
The picture below shows that these slope/intercept pairs lie in an ellipse centered at $(1,1)$.</p>
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<div class="prompt input_prompt">In&nbsp;[16]:</div>
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<div class=" highlight hl-ipython3"><pre><span class="n">Y</span><span class="o">=</span><span class="n">X</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
<span class="n">M</span><span class="p">,</span><span class="n">B</span><span class="o">=</span><span class="p">[],[]</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="mi">5000</span><span class="p">):</span>
<span class="n">Yn</span><span class="o">=</span><span class="n">Y</span><span class="o">+</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">m</span><span class="p">,</span><span class="n">b</span><span class="o">=</span><span class="n">MB</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Yn</span><span class="p">)</span>
<span class="n">M</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
<span class="n">B</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">&quot;Slope/Intercept Pairs for Different Samples&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">&quot;Slope&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&quot;Intercept&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">B</span><span class="p">,</span><span class="n">M</span><span class="p">,</span><span class="n">s</span><span class="o">=.</span><span class="mi">3</span><span class="p">,</span><span class="n">alpha</span><span class="o">=.</span><span class="mi">3</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;blue&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>To see where this ellipse comes from, let's recall the orthonormal vectors $e$ and $x$ introduced earlier. We saw that the predicted $Y$ value corresponds to the orthogonal projection $\hat{Y}=(Y\cdot e)e+(Y\cdot x)x$. The points $((Y\cdot e),(Y\cdot x))$ and
$(Y\cdot x)$ are distributed around $((X+1)\cdot e),((X+1)\cdot x))$ -- basically we are looking at the random vectors $\epsilon$
projected orthogonally onto the $e,x$ plane. Since the $\epsilon$ fall in a sphere around zero, the projection is a circle.</p>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">P</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Y</span><span class="p">):</span>
<span class="n">N</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">Xbar</span><span class="o">=</span><span class="nb">sum</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">/</span><span class="n">N</span>
<span class="n">Sx</span><span class="o">=</span><span class="nb">sum</span><span class="p">((</span><span class="n">X</span><span class="o">-</span><span class="n">Xbar</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">N</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">A</span><span class="o">=</span><span class="nb">sum</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">Z</span><span class="o">=</span><span class="p">(</span><span class="n">X</span><span class="o">-</span><span class="n">Xbar</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">N</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">Sx</span><span class="p">)</span>
<span class="n">B</span><span class="o">=</span><span class="nb">sum</span><span class="p">([</span><span class="n">Z</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">Y</span><span class="p">[</span><span class="n">i</span><span class="p">]</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">N</span><span class="p">)])</span>
<span class="k">return</span> <span class="n">A</span><span class="p">,</span><span class="n">B</span>
<span class="n">A</span><span class="o">=</span><span class="p">[]</span>
<span class="n">B</span><span class="o">=</span><span class="p">[]</span>
<span class="n">a0</span><span class="p">,</span><span class="n">b0</span><span class="o">=</span><span class="n">P</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">X</span><span class="o">+</span><span class="mi">1</span><span class="p">)</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="mi">5000</span><span class="p">):</span>
<span class="n">Yn</span><span class="o">=</span><span class="n">Y</span><span class="o">+</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="o">=</span><span class="n">P</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">Yn</span><span class="p">)</span>
<span class="n">A</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="n">a0</span><span class="p">)</span>
<span class="n">B</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">b0</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span><span class="n">axes</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&quot;Orthogonal</span><span class="se">\n</span><span class="s"> Projection of Residuals&quot;</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s">&#39;equal&#39;</span><span class="p">)</span>
<span class="c">#plt.gca().set_aspect(&#39;equal&#39;)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">s</span><span class="o">=.</span><span class="mi">3</span><span class="p">,</span><span class="n">alpha</span><span class="o">=.</span><span class="mi">3</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;blue&#39;</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">normed</span><span class="o">=</span><span class="s">&#39;True&#39;</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">B</span><span class="p">,</span><span class="n">normed</span><span class="o">=</span><span class="s">&#39;True&#39;</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</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="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="o">.</span><span class="mi">1</span><span class="p">),</span><span class="n">norm</span><span class="o">.</span><span class="n">pdf</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="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="o">.</span><span class="mi">1</span><span class="p">),</span><span class="n">scale</span><span class="o">=</span><span class="mi">2</span><span class="p">),</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;black&#39;</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&quot;Distribution of Residuals&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
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<p>We can make a change of variables from the $x$,$e$ coordinate system to the $X$, $E$ coordinates; the coefficients of the $\hat{Y}$ vector in these coordinates are $m$ and $b$. The circular cloud of points becomes this ellipse. The ellipses correspond to the circles where about 68\% of the points lie, 95\%; and 99.7\%.</p>
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<div class=" highlight hl-ipython3"><pre><span class="c">#plt.xlabel(&quot;Slope&quot;)</span>
<span class="c">#plt.ylabel(&quot;Intercept&quot;)</span>
<span class="c">#axes[0].set_xticks(np.arange(-.5,.5,.1))</span>
<span class="n">fig</span><span class="o">=</span><span class="n">figure</span><span class="p">(</span><span class="n">num</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">axes</span><span class="o">=</span><span class="n">fig</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span>
<span class="n">A</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="n">B</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
<span class="n">axes</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s">&#39;Projection of Residuals in Slope,Intercept&#39;</span><span class="p">)</span>
<span class="n">axes</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="o">-.</span><span class="mi">3</span><span class="p">,</span><span class="o">.</span><span class="mi">3</span><span class="p">])</span>
<span class="n">axes</span><span class="o">.</span><span class="n">set_xticks</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="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
<span class="n">N</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">Xbar</span><span class="o">=</span><span class="nb">sum</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">/</span><span class="n">N</span>
<span class="n">Sx</span><span class="o">=</span><span class="nb">sum</span><span class="p">((</span><span class="n">X</span><span class="o">-</span><span class="n">Xbar</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">N</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">R</span><span class="o">=</span><span class="n">Sx</span><span class="o">+</span><span class="n">N</span><span class="o">*</span><span class="n">Xbar</span><span class="o">**</span><span class="mi">2</span>
<span class="n">S</span><span class="o">=</span><span class="mi">2</span><span class="o">*</span><span class="n">N</span><span class="o">*</span><span class="n">Xbar</span>
<span class="n">T</span><span class="o">=</span><span class="n">N</span>
<span class="n">u</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="o">.</span><span class="mi">01</span><span class="p">)</span>
<span class="n">v</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="o">.</span><span class="mi">01</span><span class="p">)</span>
<span class="n">U</span><span class="p">,</span><span class="n">V</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)</span>
<span class="n">axes</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">U</span><span class="p">,</span><span class="n">V</span><span class="p">,</span><span class="n">T</span><span class="o">*</span><span class="n">U</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">S</span><span class="o">*</span><span class="n">U</span><span class="o">*</span><span class="n">V</span><span class="o">+</span><span class="n">R</span><span class="o">*</span><span class="n">V</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="n">levels</span><span class="o">=</span><span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">16</span><span class="p">,</span><span class="mi">36</span><span class="p">],</span><span class="n">colors</span><span class="o">=</span><span class="p">[</span><span class="s">&#39;black&#39;</span><span class="p">,</span><span class="s">&#39;black&#39;</span><span class="p">,</span><span class="s">&#39;black&#39;</span><span class="p">])</span>
<span class="c">#axes.set_aspect(&#39;equal&#39;,adjustable=&#39;box&#39;)</span>
<span class="n">axes</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">*</span><span class="n">A</span><span class="o">-</span><span class="n">Xbar</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">Sx</span><span class="p">)</span><span class="o">*</span><span class="n">B</span><span class="p">,</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">Sx</span><span class="p">)</span><span class="o">*</span><span class="n">B</span><span class="p">,</span><span class="n">s</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span><span class="n">alpha</span><span class="o">=.</span><span class="mi">3</span><span class="p">,</span><span class="n">color</span><span class="o">=</span><span class="s">&#39;red&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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