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stat challenge day 7
pearson coeff
spearman rank coeff
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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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<h1>
Day 7 - Pearson and spearman correlations
</h1>
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13 novembre 2018
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<h2>Pearson correlation coefficient</h2>
<h3>Problem</h3>
<p>Given two n-element data sets, <span class="math">\(X\)</span> and <span class="math">\(Y\)</span>, calculate the value of the Pearson correlation coefficient.</p>
<h3>Python implementation</h3>
<p>Using the formula</p>
<div class="math">$$\rho_{X,Y}=\frac{cov(X,Y)}{\sigma_X\sigma_Y}$$</div>
<p>where
</p>
<div class="math">$$cov(X,Y)=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})\cdot(y_i-\bar{y})$$</div>
<div class="highlight"><pre><span></span><span class="n">n</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">X</span> <span class="o">=</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mf">9.8</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mf">7.8</span><span class="p">,</span> <span class="mf">7.7</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">200</span><span class="p">,</span> <span class="mi">44</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="mi">24</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">4</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">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">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>
</pre></div>
<div class="highlight"><pre><span></span>0.6124721937208479
</pre></div>
<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>
<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>
<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="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>
</pre></div>
<div class="highlight"><pre><span></span>0.6124721937208479
</pre></div>
<h2>Spearman's rank correlation coefficient</h2>
<h3>Problem</h3>
<p>Given two <span class="math">\(n\)</span>-element data sets, <span class="math">\(X\)</span> and <span class="math">\(Y\)</span>, calculate the value of Spearman's rank correlation coefficient.</p>
<h3>Python implementation</h3>
<p>We knwo that in this case, the values in each dataset are unique. Hence we can use the formula :
</p>
<div class="math">$$r_s=1-\frac{6\sum d_i^2}{n(n^2-1)}$$</div>
<div class="highlight"><pre><span></span><span class="n">n</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">X</span> <span class="o">=</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mf">9.8</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mf">7.8</span><span class="p">,</span> <span class="mf">7.7</span><span class="p">,</span> <span class="mf">1.7</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mf">1.4</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">200</span><span class="p">,</span> <span class="mi">44</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="mi">24</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">spearman_rank</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">rank_X</span> <span class="o">=</span> <span class="p">[</span><span class="nb">sorted</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">X</span><span class="p">]</span>
<span class="n">rank_Y</span> <span class="o">=</span> <span class="p">[</span><span class="nb">sorted</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">]</span>
<span class="n">d</span> <span class="o">=</span> <span class="p">[(</span><span class="n">rank_X</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="n">rank_Y</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">**</span><span class="mi">2</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">return</span> <span class="mi">1</span><span class="o">-</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">d</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
<span class="n">spearman_rank</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>
<div class="highlight"><pre><span></span>0.9030303030303031
</pre></div>
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<h1>
Pearson correlation coefficient
</h1>
<ol class="breadcrumb">
<li>
<time class="published" datetime="2018-11-14T21:21:00+01:00">
14 novembre 2018
</time>
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<li>Mathematics</li>
<li>Statistics</li>
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<h2>Covariance</h2>
<p>This is a measure of how two random variables change together, or the strength of their correlation.</p>
<p>Consider two random variables, <span class="math">\(X\)</span> and <span class="math">\(Y\)</span>, each with <span class="math">\(n\)</span> values (i.e., <span class="math">\(x_1\)</span>, <span class="math">\(x_2\)</span>, <span class="math">\(...\)</span>, <span class="math">\(x_n\)</span> and <span class="math">\(y_1\)</span>, <span class="math">\(y_2\)</span>, <span class="math">\(...\)</span>, <span class="math">\(y_n\)</span>). The covariance of <span class="math">\(X\)</span> and <span class="math">\(Y\)</span> can be found using either of the following equivalent formulas: </p>
<div class="math">$$cov(X,Y)=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})\cdot(y_i-\bar{y})$$</div>
<p>or </p>
<div class="math">$$cov(X,Y)=\frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}(x_i-x_j)\cdot(y_i-y_j))$$</div>
<div class="math">$$cov(X,Y)=\frac{1}{n^2}\sum_{i}\sum_{j\gt i}^{n}(x_i-x_j)\cdot(y_i-y_j)$$</div>
<p>where, <span class="math">\(\bar{x}\)</span> is the mean of <span class="math">\(X\)</span> (or <span class="math">\(\mu_X\)</span>) and <span class="math">\(\bar{y}\)</span> is the mean of <span class="math">\(Y\)</span> (or <span class="math">\(\mu_Y\)</span>)</p>
<h2>Pearson correlation coefficient</h2>
<p>The pearson correlation coefficient, <span class="math">\(\rho_{X,Y}\)</span>, is given by : </p>
<div class="math">$$\rho_{X,Y}=\frac{cov(X,Y)}{\sigma_X\sigma_Y}=\frac{\sum_{i}(x_i-\bar{x})(y_i-\bar{y})}{n\sigma_X\sigma_Y}$$</div>
<p>Here, <span class="math">\(\sigma_X\)</span> is the standard deviation of <span class="math">\(X\)</span> and <span class="math">\(\sigma_Y\)</span> is the standard deviation of <span class="math">\(Y\)</span>. You may also see <span class="math">\(\rho_{X,Y}\)</span> written as <span class="math">\(r_{X,Y}\)</span>.</p>
<p>The pearson correlation coefficient is a measure of the linear correlation between two variables X and Y.</p>
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<h1>
Spearman's Rank Correlation Coefficient
</h1>
<ol class="breadcrumb">
<li>
<time class="published" datetime="2018-11-14T22:09:00+01:00">
14 novembre 2018
</time>
</li>
<li>Mathematics</li>
<li>Statistics</li>
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<div class='article_content'>
<p>A rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them. </p>
<p>We have two random variables <span class="math">\(X\)</span> and <span class="math">\(Y\)</span>:
<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>Example</h2>
<ul>
<li><span class="math">\(X=\{0.2, 1.3, 0.2, 1.1, 1.4, 1.5\}\)</span></li>
<li><span class="math">\(Y=\{1.9, 2.2, 3.1, 1.2, 2.2, 2.2\}\)</span></li>
</ul>
<div class="math">$$
Rank_X
\quad
\begin{bmatrix}
X: &amp; 0.2 &amp; 1.3 &amp; 0.2 &amp; 1.1 &amp; 1.4 &amp; 1.5 \\
Rank: &amp; 1 &amp; 3 &amp; 1 &amp; 2 &amp; 4 &amp; 5
\end{bmatrix}
\quad
$$</div>
<p>so, <span class="math">\(Rank_X = \{1, 3, 1, 2, 4, 5\}\)</span></p>
<p>similarly, <span class="math">\(Rank_Y=\{2,3,4,1,3,3\}\)</span></p>
<p><span class="math">\(r_s\)</span> equals the Pearson correlation coefficient of <span class="math">\(Rank_X\)</span> and <span class="math">\(Rank_Y\)</span>, meaning that <span class="math">\(r=0.158114\)</span></p>
<h2>Special case : <span class="math">\(X\)</span> and <span class="math">\(Y\)</span> don't contain duplicates</h2>
<div class="math">$$r_s=1-\frac{6\sum d_i^2}{n(n^2-1)}$$</div>
<p>Where, <span class="math">\(d_i\)</span> is the difference between the respective values of <span class="math">\(Rank_X\)</span> and <span class="math">\(Rank_Y\)</span>.</p>
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python/Optimized_numpy_random_intel.html
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python/Reading_data_from_a_sql_database_with_pandas.html
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python/Saving_a_matplotlib_figure_with_a_high_resolution.html
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python/Setting_up_the_notebook_for_plotting_with_matplotlib.html
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python/Writing_data_to_a_sql_database_with_pandas.html
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python/counting.html
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python/dask_distributed_parallelism.html
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python/design_own_libs.html
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python/sorting.html
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@ -192,7 +192,7 @@ Sorted array : [0.13876208 0.19939047 0.40021786 0.43734158 0.46015169]
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python/unique.html
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@ -5,7 +5,7 @@ xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
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sql/Sorting_results.html
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tipuesearch_content.json
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