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<h1>
Day 9 - Multiple Linear Regression
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<time class="published" datetime="2018-11-16T20:31:00+01:00">
16 novembre 2018
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<h2>Problem</h2>
<p>Here is a simple equation:
</p>
<div class="math">$$Y=a+b_1\cdot f_1++b_2\cdot f_2+...++b_m\cdot f_m$$</div>
<div class="math">$$Y=a+\sum_{i=1}^m b_i\cdot f_i$$</div>
<p>for <span class="math">\((m+1)\)</span> read constants <span class="math">\((a,f_1, f_2, ..., f_m)\)</span>. We can say that the value of <span class="math">\(Y\)</span> depends on <span class="math">\(m\)</span> features. We study this equation for <span class="math">\(n\)</span> different feature sets <span class="math">\((f_1, f_2, ..., f_m)\)</span> and records each respective value of <span class="math">\(Y\)</span>. </p>
<p>If we have <span class="math">\(q\)</span> new feature sets, and without accounting for bias and variance trade-offs,what is the value of <span class="math">\(Y\)</span> for each of the sets?</p>
<h2>Python implementation</h2>
<div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">m</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">7</span>
<span class="n">x_1</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.18</span><span class="p">,</span> <span class="mf">0.89</span><span class="p">]</span>
<span class="n">y_1</span> <span class="o">=</span> <span class="mf">109.85</span>
<span class="n">x_2</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.26</span><span class="p">]</span>
<span class="n">y_2</span> <span class="o">=</span> <span class="mf">155.72</span>
<span class="n">x_3</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.92</span><span class="p">,</span> <span class="mf">0.11</span><span class="p">]</span>
<span class="n">y_3</span> <span class="o">=</span> <span class="mf">137.66</span>
<span class="n">x_4</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.07</span><span class="p">,</span> <span class="mf">0.37</span><span class="p">]</span>
<span class="n">y_4</span> <span class="o">=</span> <span class="mf">76.17</span>
<span class="n">x_5</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.85</span><span class="p">,</span> <span class="mf">0.16</span><span class="p">]</span>
<span class="n">y_5</span> <span class="o">=</span> <span class="mf">139.75</span>
<span class="n">x_6</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.99</span><span class="p">,</span> <span class="mf">0.41</span><span class="p">]</span>
<span class="n">y_6</span> <span class="o">=</span> <span class="mf">162.6</span>
<span class="n">x_7</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.87</span><span class="p">,</span> <span class="mf">0.47</span><span class="p">]</span>
<span class="n">y_7</span> <span class="o">=</span> <span class="mf">151.77</span>
<span class="n">q_1</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.49</span><span class="p">,</span> <span class="mf">0.18</span><span class="p">]</span>
<span class="n">q_2</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.57</span><span class="p">,</span> <span class="mf">0.83</span><span class="p">]</span>
<span class="n">q_3</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.56</span><span class="p">,</span> <span class="mf">0.64</span><span class="p">]</span>
<span class="n">q_4</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.76</span><span class="p">,</span> <span class="mf">0.18</span><span class="p">]</span>
</pre></div>
<p>With scikit learn</p>
<div class="highlight"><pre><span></span><span class="n">X</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">x_1</span><span class="p">,</span> <span class="n">x_2</span><span class="p">,</span> <span class="n">x_3</span><span class="p">,</span> <span class="n">x_4</span><span class="p">,</span> <span class="n">x_5</span><span class="p">,</span> <span class="n">x_6</span><span class="p">,</span> <span class="n">x_7</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">array</span><span class="p">([</span><span class="n">y_1</span><span class="p">,</span> <span class="n">y_2</span><span class="p">,</span> <span class="n">y_3</span><span class="p">,</span> <span class="n">y_4</span><span class="p">,</span> <span class="n">y_5</span><span class="p">,</span> <span class="n">y_6</span><span class="p">,</span> <span class="n">y_7</span><span class="p">])</span>
<span class="n">X_q</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">q_1</span><span class="p">,</span> <span class="n">q_2</span><span class="p">,</span> <span class="n">q_3</span><span class="p">,</span> <span class="n">q_4</span><span class="p">])</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
<span class="n">lm</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">lm</span><span class="o">.</span><span class="n">fit</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">lm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_q</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="err">array([105.21455835, 142.67095131, 132.93605469, 129.70175405])</span>
</pre></div>
<p>without scikit learn (but with numpy)</p>
<div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">numpy.linalg</span> <span class="kn">import</span> <span class="n">inv</span>
<span class="c1">#center</span>
<span class="n">X_R</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">mean</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</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">mean</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
<span class="n">Y_R</span> <span class="o">=</span> <span class="n">Y</span><span class="o">-</span><span class="n">a</span>
<span class="c1">#calculate b</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">inv</span><span class="p">(</span><span class="n">X_R</span><span class="o">.</span><span class="n">T</span><span class="nd">@X_R</span><span class="p">)</span><span class="nd">@X_R</span><span class="o">.</span><span class="n">T</span><span class="nd">@Y_R</span>
<span class="c1">#predict</span>
<span class="n">X_new_R</span> <span class="o">=</span> <span class="n">X_q</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">Y_new_R</span> <span class="o">=</span> <span class="n">X_new_R</span><span class="nd">@B</span>
<span class="n">Y_new</span> <span class="o">=</span> <span class="n">Y_new_R</span> <span class="o">+</span> <span class="n">a</span>
<span class="n">Y_new</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="err">array([105.21455835, 142.67095131, 132.93605469, 129.70175405])</span>
</pre></div>
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