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added articles

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stat challenge day 8
corrected spearman
added least square regression
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@ -185,7 +185,7 @@ Mean : 43117.75, Median : 44627.5, Mode : 4978
<footer class="footer">
<div class="container">
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<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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>.

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blog/Statistics_10days-day1.html
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@ -219,7 +219,7 @@ Q1 : 6.0, Q2 : 12, Q3 : 16.0
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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
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@ -429,7 +429,7 @@
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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
blog/Statistics_10days-day3.html
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@ -256,7 +256,7 @@ we call B the event "a blue ball is drawn" and R the event "a red ball is drawn"
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<p class="text-muted">
<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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>.

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@ -311,7 +311,7 @@
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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
blog/Statistics_10days-day5.html
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@ -331,7 +331,7 @@ Between 20 and 22 hours?</p>
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<div class="container">
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<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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>.

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@ -270,7 +270,7 @@ B = 515.68
<footer class="footer">
<div class="container">
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<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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>.

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@ -146,12 +146,12 @@
<span class="k">def</span> <span class="nf">stdv</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">mu_x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="k">return</span> <span class="p">(</span><span class="nb">sum</span><span class="p">([(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mu_x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">X</span><span class="p">])</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span>
<span class="k">def</span> <span class="nf">spearman_1</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">n</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">pearson_1</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">n</span><span class="p">):</span>
<span class="n">std_x</span> <span class="o">=</span> <span class="n">stdv</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
<span class="n">std_y</span> <span class="o">=</span> <span class="n">stdv</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">Y</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cov</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">n</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">std_x</span><span class="o">*</span><span class="n">std_y</span><span class="p">)</span>
<span class="n">spearman_1</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">n</span><span class="p">)</span>
<span class="n">pearson_1</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">n</span><span class="p">)</span>
</pre></div>
@ -162,7 +162,7 @@
<h3>Python implementation</h3>
<p>Using the formula</p>
<div class="math">$$\rho_{X,Y}=\frac{\sum_{i}(x_i-\bar{x})(y_i-\bar{y})}{n\sigma_X\sigma_Y}$$</div>
<div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">spearman_2</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">n</span><span class="p">):</span>
<div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">pearson_2</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">n</span><span class="p">):</span>
<span class="n">std_x</span> <span class="o">=</span> <span class="n">stdv</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
<span class="n">std_y</span> <span class="o">=</span> <span class="n">stdv</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">Y</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
<span class="n">x_mean</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
@ -170,7 +170,7 @@
<span class="k">return</span> <span class="nb">sum</span><span class="p">([(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="n">x_mean</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">Y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="n">y_mean</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="n">n</span><span class="p">)])</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">std_x</span><span class="o">*</span><span class="n">std_y</span><span class="p">)</span>
<span class="n">spearman_2</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">n</span><span class="p">)</span>
<span class="n">pearson_2</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">n</span><span class="p">)</span>
</pre></div>
@ -272,7 +272,7 @@
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<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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>.

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<h1>
Day 8 - Least Square Regression Line
</h1>
<ol class="breadcrumb">
<li>
<time class="published" datetime="2018-11-15T20:22:00+01:00">
15 novembre 2018
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<li>Blog</li>
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<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>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>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">cov</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">n</span><span class="p">):</span>
<span class="n">x_mean</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">y_mean</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">([(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="n">x_mean</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">Y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="n">y_mean</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="n">n</span><span class="p">)])</span>
<span class="k">def</span> <span class="nf">stdv</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">mu_x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="k">return</span> <span class="p">(</span><span class="nb">sum</span><span class="p">([(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mu_x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">X</span><span class="p">])</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span>
<span class="k">def</span> <span class="nf">pearson_1</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">n</span><span class="p">):</span>
<span class="n">std_x</span> <span class="o">=</span> <span class="n">stdv</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
<span class="n">std_y</span> <span class="o">=</span> <span class="n">stdv</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">Y</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cov</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">n</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">std_x</span><span class="o">*</span><span class="n">std_y</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">pearson_1</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">n</span><span class="p">)</span><span class="o">*</span><span class="n">stdv</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="nb">sum</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span><span class="o">/</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">stdv</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="nb">sum</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">/</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span><span class="o">/</span><span class="n">n</span> <span class="o">-</span> <span class="n">b</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
<span class="k">print</span><span class="p">(</span><span class="n">f</span><span class="s2">&quot;If a student scored 80 on the math test, he would most likely score a {round(a+80*b,3)} in statistics&quot;</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span>If a student scored 80 on the math test, he would most likely score a 78.288 in statistics
</pre></div>
<h2>Pearson correlation coefficient</h2>
<h3>Problem</h3>
<p>The regression line of <span class="math">\(y\)</span> on <span class="math">\(x\)</span> is <span class="math">\(3x+4y+8=0\)</span>, and the regression line of <span class="math">\(x\)</span> on <span class="math">\(y\)</span> is <span class="math">\(4x+3y+7=0\)</span>. What is the value of the Pearson correlation coefficient?</p>
<h3>Mathematical explanation</h3>
<p>The initial equation system is :
</p>
<div class="math">$$
\left\{\begin{array}{ r @{{}={}} r &gt;{{}}c&lt;{{}} r &gt;{{}}c&lt;{{}} r }
3x+4y+8=0 &amp; (1)\\
4x+3y+7=0 &amp; (2)\\
\end{array}
\right.
$$</div>
<p>So we can rewrite the 2 lines this way :
</p>
<div class="math">$$
\left\{\begin{array}{ r @{{}={}} r &gt;{{}}c&lt;{{}} r &gt;{{}}c&lt;{{}} r }
y=-2+(\frac{-3}{4})x &amp; (1)\\
x=-\frac{7}{4}+(-\frac{3}{4})y &amp; (2)\\
\end{array}
\right.
$$</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>We hence have </p>
<div class="math">$$
\left\{\begin{array}{ r @{{}={}} r &gt;{{}}c&lt;{{}} r &gt;{{}}c&lt;{{}} r }
p=b_1\left(\frac{\sigma_X}{\sigma_Y}\right) &amp; (1)\\
p=b_2\left(\frac{\sigma_Y}{\sigma_X}\right) &amp; (2)\\
\end{array}
\right.
$$</div>
<p>by multiplying theses 2 equations together we get
</p>
<div class="math">$$p^2=b_1\cdot b_2$$</div>
<div class="math">$$p^2=\left(-\frac{3}{4}\right)\left(-\frac{3}{4}\right)$$</div>
<div class="math">$$p^2=\left(-\frac{9}{16}\right)$$</div>
<p>finally we get <span class="math">\(p=\left(-\frac{3}{4}\right)\)</span> or <span class="math">\(p=\left(\frac{3}{4}\right)\)</span></p>
<p>Since <span class="math">\(X\)</span> and <span class="math">\(Y\)</span> are negatively correlated we have <span class="math">\(p=\left(-\frac{3}{4}\right)\)</span></p>
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index.html
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@ -120,6 +120,7 @@
<div class="panel panel-default">
<div class="panel-body">
<ul>
<li><a href="./blog/Statistics_10days-day8.html">11/18 Day 8 - Least Square Regression Line</a></li>
<li><a href="./blog/Statistics_10days-day7.html">11/18 Day 7 - Pearson and spearman correlations</a></li>
<li><a href="./blog/Statistics_10days-day6.html">11/18 Day 6 - The Central Limit Theorem</a></li>
<li><a href="./blog/Statistics_10days-day5.html">11/18 Day 5 - Poisson and Normal distributions</a></li>
@ -231,7 +232,7 @@
</div>
<div class="panel-body">
<ul>
<li>Coming soon.</li>
<li><a href="./machine-learning/LeastSquareRegressionLine.html">Least Square Regression Line</a></li>
</ul>
</div>
</div>
@ -711,7 +712,7 @@
<footer class="footer">
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<p class="text-muted">
<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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>.

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<footer class="footer">
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<p class="text-muted">
<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 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>.

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<footer class="footer">
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<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
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linux/Reloading_.bashrc.html
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@ -144,7 +144,7 @@ However, you can use the following command to do it </p>
<footer class="footer">
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<p class="text-muted">
<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 65 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
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<center>This project contains 63 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
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<h1>
Least Square Regression Line
</h1>
<ol class="breadcrumb">
<li>
<time class="published" datetime="2018-11-15T20:25:00+01:00">
15 novembre 2018
</time>
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<li>Machine Learning</li>
<li>linear</li>
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<div class='article_content'>
<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>
<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>
<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>
<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="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="n">xl</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</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">asarray</span><span class="p">(</span><span class="n">xl</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</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="k">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;a = {lm.intercept_}&#39;</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;b = {lm.coef_[0]}&#39;</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s2">&quot;Where Y=a+b*X&quot;</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span>a = 0.5999999999999996
b = 0.8000000000000002
Where Y=a+b*X
</pre></div>
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<em> <span class="math">\(X=\{x_i, x_2, x_3, ..., x_n\}\)</span>
</em> <span class="math">\(Y=\{y_i, y_2, y_3, ..., y_n\}\)</span></p>
<p>if <span class="math">\(Rank_X\)</span> and <span class="math">\(Rank_Y\)</span> denote the respective ranks of each data point, then the Spearman's rank correlation coefficient, <span class="math">\(r_s\)</span>, is the Pearson correlation coefficient of <span class="math">\(Rank_X\)</span> and <span class="math">\(Rank_Y\)</span>.</p>
<h2>What does it means?</h2>
<p>The Spearman's rank correlation coefficientis is a nonparametric measure of rank correlation (statistical dependence between the rankings of two variables). It assesses how well the relationship between two variables can be described using a monotonic function.
The Spearman correlation between two variables is equal to the Pearson correlation between the rank values of those two variables; while Pearson's correlation assesses linear relationships, Spearman's correlation assesses monotonic relationships (whether linear or not). If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other. </p>
<p><img alt="Spearman" src="../images/spearman/Spearman_fig1.svg"></p>
<p>A Spearman correlation of 1 results when the two variables being compared are monotonically related, even if their relationship is not linear. This means that all data-points with greater x-values than that of a given data-point will have greater y-values as well. In contrast, this does not give a perfect Pearson correlation.</p>
<h2>Example</h2>
<ul>
<li><span class="math">\(X=\{0.2, 1.3, 0.2, 1.1, 1.4, 1.5\}\)</span></li>
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