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<h1>
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Multiple Linear Regression
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</h1>
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<ol class="breadcrumb">
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<li>
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<time class="published" datetime="2018-11-16T20:25:00+01:00">
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16 novembre 2018
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</time>
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</li>
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<li>Machine Learning</li>
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<li>linear</li>
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</ol>
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</header>
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<div class='article_content'>
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<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>
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<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>
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<div class="math">$$\hat{Y}=a+\sum_{i=1}^{m}b_iX_i$$</div>
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<h2>Matrix Form of the Regression Equation</h2>
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<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>
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<div class="math">$$\begin{bmatrix}
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1 & X_1 & X_2\\
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\end{bmatrix}\cdot\begin{bmatrix}
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a \\
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b_1\\
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b_2\\
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\end{bmatrix}=a+b_1X_1+b_2X_2$$</div>
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<p>We define two matrices, <span class="math">\(X\)</span> and <span class="math">\(B\)</span> as:</p>
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<div class="math">$$X=\begin{bmatrix}1 & X_1 & X_2\\\end{bmatrix}$$</div>
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<div class="math">$$B=\begin{bmatrix}a \\b_1\\b_2\\\end{bmatrix}$$</div>
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<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>
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<h2>Generalized Matrix Form</h2>
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<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>
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<ul>
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<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>
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<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>
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<li><span class="math">\(...\)</span></li>
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<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>
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</ul>
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<p>Now, we can find the matrices:</p>
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<div class="math">$$X=\begin{bmatrix}1 & x_{1,1} & x_{2,1} & x_{3,1} & ... & x_{m,1} \\1 & x_{1,2} & x_{2,2} & x_{3,2} & ... & x_{m,2} \\1 & x_{1,3} & x_{2,3} & x_{3,3} & ... & x_{m,3} \\... & ... & ... & ... & ... & ... \\1 & x_{1,n} & x_{2,n} & x_{3,n} & ... & x_{m,n} \\\end{bmatrix}$$</div>
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<div class="math">$$Y=\begin{bmatrix}y_1 \\y_2\\y_3\\...\\y_n\\\end{bmatrix}$$</div>
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<h3>Finding the Matrix B</h3>
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<p>We know that <span class="math">\(Y=X\cdot B\)</span>
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</p>
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<div class="math">$$\Rightarrow X^T\cdot Y=X^T\cdot X \cdot B$$</div>
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<div class="math">$$\Rightarrow (X^T\cdot X)^{-1}\cdot X^T \cdot Y=I\cdot B$$</div>
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<div class="math">$$\Rightarrow B= (X^T\cdot X)^{-1}\cdot X^T \cdot Y$$</div>
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<h3>Finding the Value of Y</h3>
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<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>
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<div class="math">$$Y=\begin{bmatrix}
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1 & x_1 & x_2 & ... & x_m\\
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\end{bmatrix}\cdot B$$</div>
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<h2>Multiple Regression in Python</h2>
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<p>We can use the fit function in the sklearn.linear_model.LinearRegression class.</p>
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<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>
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<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>
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<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>
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<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>
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<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>
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<span class="n">a</span> <span class="o">=</span> <span class="n">lm</span><span class="o">.</span><span class="n">intercept_</span>
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<span class="n">b</span> <span class="o">=</span> <span class="n">lm</span><span class="o">.</span><span class="n">coef_</span>
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<span class="k">print</span><span class="p">(</span><span class="n">f</span><span class="s2">"Linear regression coefficients between Y and X : a={a}, b_0={b[0]}, b_1={b[1]}"</span><span class="p">)</span>
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</pre></div>
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<div class="highlight"><pre><span></span>Linear regression coefficients between Y and X : a=51.953488372092984, b_0=6.65116279069768, b_1=-11.162790697674419
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</pre></div>
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<p>Everything on this site is avaliable on GitHub. Head on over and <a href='https://github.com/redoules/redoules.github.io/issues/new'>submit an issue.</a> You can also message me directly by <a href='mailto:guillaume.redoules@gadz.org'>email</a>.</p>
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Copyright © Guillaume Redoulès,
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