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blog/Statistics_10days-day8.html
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@ -127,12 +127,14 @@
<div class='article_content'>
<h2>Least Square Regression Line</h2>
<h2>Problem</h2>
<p>A group of five students enrolls in Statistics immediately after taking a Math aptitude test. Each student's Math aptitude test score, <span class="math">\(x\)</span>, and Statistics course grade, <span class="math">\(y\)</span>, can be expressed as the following list <span class="math">\((x,y)\)</span> of points:
<em> <span class="math">\((95, 85)\)</span>
</em> <span class="math">\((85, 95)\)</span>
<em> <span class="math">\((80, 70)\)</span>
</em> <span class="math">\((70, 65)\)</span>
* <span class="math">\((60, 70)\)</span></p>
<p>A group of five students enrolls in Statistics immediately after taking a Math aptitude test. Each student's Math aptitude test score, <span class="math">\(x\)</span>, and Statistics course grade, <span class="math">\(y\)</span>, can be expressed as the following list <span class="math">\((x,y)\)</span> of points: </p>
<ul>
<li><span class="math">\((95, 85)\)</span></li>
<li><span class="math">\((85, 95)\)</span></li>
<li><span class="math">\((80, 70)\)</span></li>
<li><span class="math">\((70, 65)\)</span></li>
<li><span class="math">\((60, 70)\)</span></li>
</ul>
<p>If a student scored an 80 on the Math aptitude test, what grade would we expect them to achieve in Statistics? Determine the equation of the best-fit line using the least squares method, then compute and print the value of <span class="math">\(y\)</span> when <span class="math">\(x=80\)</span>.</p>
<div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="p">[</span><span class="mi">95</span><span class="p">,</span> <span class="mi">85</span><span class="p">,</span> <span class="mi">80</span><span class="p">,</span> <span class="mi">70</span><span class="p">,</span> <span class="mi">60</span><span class="p">]</span>
<span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">85</span><span class="p">,</span> <span class="mi">95</span><span class="p">,</span> <span class="mi">70</span><span class="p">,</span> <span class="mi">65</span><span class="p">,</span> <span class="mi">70</span><span class="p">]</span>
@ -186,10 +188,12 @@ $$</div>
\right.
$$</div>
<p>so <span class="math">\(b_1=-\frac{3}{4}\)</span> and <span class="math">\(b_2=-\frac{3}{4}\)</span></p>
<p>When we apply the Pearson's coefficient formula :
<em> let <span class="math">\(p\)</span> be the pearson coefficient
</em> let <span class="math">\(\sigma_X\)</span> be the standard deviation of <span class="math">\(x\)</span>
* let <span class="math">\(\sigma_Y\)</span> be the standard deviation of <span class="math">\(y\)</span></p>
<p>When we apply the Pearson's coefficient formula : </p>
<ul>
<li>let <span class="math">\(p\)</span> be the pearson coefficient</li>
<li>let <span class="math">\(\sigma_X\)</span> be the standard deviation of <span class="math">\(x\)</span></li>
<li>let <span class="math">\(\sigma_Y\)</span> be the standard deviation of <span class="math">\(y\)</span></li>
</ul>
<p>We hence have </p>
<div class="math">$$
\left\{\begin{array}{ r @{{}={}} r &gt;{{}}c&lt;{{}} r &gt;{{}}c&lt;{{}} r }

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machine-learning/LeastSquareRegressionLine.html
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@ -128,18 +128,22 @@
<h2>Linear Regression</h2>
<p>If our data shows a linear relationship between <span class="math">\(X\)</span> and <span class="math">\(Y\)</span>, then the straight line which best describes the relationship is the regression line. The regression line is given by <span class="math">\(\hat{Y}\)</span>=a+bX$. </p>
<h3>Finding the value of b</h3>
<p>The value of <span class="math">\(b\)</span> can be calculated using either of the following formulae:
<em> <span class="math">\(b=\frac{n\sum(x_iy_i)-(\sum x_i)(\sum y_i)}{n\sum(x_i^2)-(\sum x_i)^2}\)</span>
</em> <span class="math">\(b=\rho\frac{\sigma_Y}{\sigma_X}\)</span>, where <span class="math">\(\rho\)</span> is the Pearson correlation coefficient, <span class="math">\(\sigma_X\)</span></p>
<p>The value of <span class="math">\(b\)</span> can be calculated using either of the following formulae:</p>
<ul>
<li><span class="math">\(b=\frac{n\sum(x_iy_i)-(\sum x_i)(\sum y_i)}{n\sum(x_i^2)-(\sum x_i)^2}\)</span></li>
<li><span class="math">\(b=\rho\frac{\sigma_Y}{\sigma_X}\)</span>, where <span class="math">\(\rho\)</span> is the Pearson correlation coefficient, <span class="math">\(\sigma_X\)</span></li>
</ul>
<h3>Finding the value of a</h3>
<p><span class="math">\(a=\bar{y}-b\cdot\bar{x}\)</span>, where <span class="math">\(\bar{x}\)</span> is the mean of <span class="math">\(X\)</span> and <span class="math">\(\bar{y}\)</span> is the mean of <span class="math">\(Y\)</span>.</p>
<h2>Coefficient of determination (<span class="math">\(R^2\)</span>)</h2>
<p>The coefficient of determination can be computer with :
<span class="math">\(R^2 = \frac{SSR}{SST}=1-\frac{SSE}{SST}\)</span>
Where :
<em> <span class="math">\(SST\)</span> is the total Sum of Squares : <span class="math">\(SST=\sum (y_i-\bar{y})^2\)</span>
</em> <span class="math">\(SSR\)</span> is the regression Sum of Squares : <span class="math">\(SSR=\sum (\hat{y_i}-\bar{y})^2\)</span>
* <span class="math">\(SSE\)</span> is the error Sum of Squares : <span class="math">\(SSE=\sum (\hat{y_i}-y)^2\)</span></p>
Where :</p>
<ul>
<li><span class="math">\(SST\)</span> is the total Sum of Squares : <span class="math">\(SST=\sum (y_i-\bar{y})^2\)</span></li>
<li><span class="math">\(SSR\)</span> is the regression Sum of Squares : <span class="math">\(SSR=\sum (\hat{y_i}-\bar{y})^2\)</span></li>
<li><span class="math">\(SSE\)</span> is the error Sum of Squares : <span class="math">\(SSE=\sum (\hat{y_i}-y)^2\)</span></li>
</ul>
<p>If <span class="math">\(SSE\)</span> is small, we can assume that our fit is good. </p>
<h2>Linear Regression in Python</h2>
<p>We can use the fit function in the sklearn.linear_model.LinearRegression class.</p>

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