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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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