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stat challenge day 9
multiple linear regression
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<h1>
Day 9 - Multiple Linear Regression
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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="kn">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>array([105.21455835, 142.67095131, 132.93605469, 129.70175405])
</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.T@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>array([105.21455835, 142.67095131, 132.93605469, 129.70175405])
</pre></div>
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<h1>
Multiple Linear Regression
</h1>
<ol class="breadcrumb">
<li>
<time class="published" datetime="2018-11-16T20:25:00+01:00">
16 novembre 2018
</time>
</li>
<li>Machine Learning</li>
<li>linear</li>
</ol>
</header>
<div class='article_content'>
<p>If <span class="math">\(Y\)</span> is linearly dependent only on <span class="math">\(X\)</span>, then we can use the ordinary least square regression line, <span class="math">\(\hat{Y}=a+bX\)</span>. </p>
<p>However, if <span class="math">\(Y\)</span> shows linear dependency on <span class="math">\(m\)</span> variables <span class="math">\(X_1\)</span>, <span class="math">\(X_2\)</span>, ..., <span class="math">\(X_m\)</span>, then we need to find the values of <span class="math">\(a\)</span> and <span class="math">\(m\)</span> other constants (<span class="math">\(b_1\)</span>, <span class="math">\(b_2\)</span>, ..., <span class="math">\(b_m\)</span>). We can then write the regression equation as: </p>
<div class="math">$$\hat{Y}=a+\sum_{i=1}^{m}b_iX_i$$</div>
<h2>Matrix Form of the Regression Equation</h2>
<p>Let's consider that <span class="math">\(Y\)</span> depends on two variables, <span class="math">\(X_1\)</span> and <span class="math">\(X_2\)</span>. We write the regression relation as <span class="math">\(\hat{Y}=a+b_1X_1+b_2X_2\)</span>. Consider the following matrix operation: </p>
<div class="math">$$\begin{bmatrix}
1 &amp; X_1 &amp; X_2\\
\end{bmatrix}\cdot\begin{bmatrix}
a \\
b_1\\
b_2\\
\end{bmatrix}=a+b_1X_1+b_2X_2$$</div>
<p>We define two matrices, <span class="math">\(X\)</span> and <span class="math">\(B\)</span> as:</p>
<ul>
<li><span class="math">\(X=<div class="math">\begin{bmatrix}1 &amp; X_1 &amp; X_2\\\end{bmatrix}</div>\)</span></li>
<li><span class="math">\(B=<div class="math">\begin{bmatrix}a \\b_1\\b_2\\\end{bmatrix}</div>\)</span></li>
</ul>
<p>Now, we rewrite the regression relation as <span class="math">\(\hat{Y}=X\cdot B\)</span>. This transforms the regression relation into matrix form.</p>
<h2>Generalized Matrix Form</h2>
<p>We will consider that <span class="math">\(Y\)</span> shows a linear relationship with <span class="math">\(m\)</span> variables, <span class="math">\(X_1\)</span>, <span class="math">\(X_2\)</span>, ..., <span class="math">\(X_m\)</span>. Let's say that we made <span class="math">\(n\)</span> observations on different tuples <span class="math">\((x_1, x_2, ..., x_m)\)</span>:</p>
<ul>
<li><span class="math">\(y_1=a+b_1\cdot x_{1,1} + b_2\cdot x_{2,1} + ... + b_m\cdot x_{m,1}\)</span></li>
<li><span class="math">\(y_2=a+b_2\cdot x_{1,2} + b_2\cdot x_{2,2} + ... + b_m\cdot x_{m,2}\)</span></li>
<li><span class="math">\(...\)</span></li>
<li><span class="math">\(y_n=a+b_n\cdot x_{1,n} + b_2\cdot x_{2,n} + ... + b_m\cdot x_{m,n}\)</span></li>
</ul>
<p>Now, we can find the matrices:</p>
<div class="math">$$X=\begin{bmatrix}1 &amp; x_{1,1} &amp; x_{2,1} &amp; x_{3,1} &amp; ... &amp; x_{m,1} \\1 &amp; x_{1,2} &amp; x_{2,2} &amp; x_{3,2} &amp; ... &amp; x_{m,2} \\1 &amp; x_{1,3} &amp; x_{2,3} &amp; x_{3,3} &amp; ... &amp; x_{m,3} \\... &amp; ... &amp; ... &amp; ... &amp; ... &amp; ... \\1 &amp; x_{1,n} &amp; x_{2,n} &amp; x_{3,n} &amp; ... &amp; x_{m,n} \\\end{bmatrix}$$</div>
<div class="math">$$Y=\begin{bmatrix}y_1 \\y_2\\y_3\\...\\y_n\\\end{bmatrix}$$</div>
<h3>Finding the Matrix B</h3>
<p>We know that <span class="math">\(Y=X\cdot B\)</span>
</p>
<div class="math">$$\Rightarrow X^T\cdot Y=X^T\cdot X \cdot B$$</div>
<div class="math">$$\Rightarrow (X^T\cdot X)^{-1}\cdot X^T \cdot Y=I\cdot B$$</div>
<div class="math">$$\Rightarrow B= (X^T\cdot X)^{-1}\cdot X^T \cdot Y$$</div>
<h3>Finding the Value of Y</h3>
<p>Suppose we want to find the value of for some tuple <span class="math">\(Y\)</span>, then <span class="math">\((x_1, x_2, ..., x_m)\)</span>,</p>
<div class="math">$$Y=\begin{bmatrix}
1 &amp; x_1 &amp; x_2 &amp; ... &amp; x_m\\
\end{bmatrix}\cdot B$$</div>
<h2>Multiple Regression in Python</h2>
<p>We can use the fit function in the sklearn.linear_model.LinearRegression class.</p>
<div class="highlight"><pre><span></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">x</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</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="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">6</span><span class="p">]]</span>
<span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="mi">40</span><span class="p">,</span> <span class="mi">50</span><span class="p">]</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">a</span> <span class="o">=</span> <span class="n">lm</span><span class="o">.</span><span class="n">intercept_</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">lm</span><span class="o">.</span><span class="n">coef_</span>
<span class="k">print</span><span class="p">(</span><span class="n">f</span><span class="s2">&quot;Linear regression coefficients between Y and X : a={a}, b_0={b[0]}, b_1={b[1]}&quot;</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span>Linear regression coefficients between Y and X : a=51.953488372092984, b_0=6.65116279069768, b_1=-11.162790697674419
</pre></div>
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2
sql/Sorting_results.html
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@ -344,7 +344,7 @@ Done.
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 65 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 67 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
sql/WHERE_SQL_keywords.html
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@ -631,7 +631,7 @@ Done.
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 65 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 67 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
sql/display_table.html
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@ -216,7 +216,7 @@ Done.
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 65 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 67 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
sql/display_table_filter.html
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@ -346,7 +346,7 @@ Done.
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 65 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 67 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
tipuesearch_content.json
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