filer/perf/simple-statistics/docs/simple_statistics.html

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<h1>simple_statistics.js</h1>
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<div class="content"><div class='highlight'><pre><span class="hljs-comment">/* global module */</span></pre></div></div>
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<h1 id="simple-statistics">simple-statistics</h1>
<p>A simple, literate statistics system. The code below uses the
<a href="http://www.adequatelygood.com/2010/3/JavaScript-Module-Pattern-In-Depth">Javascript module pattern</a>,
eventually assigning <code>simple-statistics</code> to <code>ss</code> in browsers or the
<code>exports</code> object for node.js</p>
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<div class="content"><div class='highlight'><pre>(<span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">()</span> {</span>
<span class="hljs-keyword">var</span> ss = {};
<span class="hljs-keyword">if</span> (<span class="hljs-keyword">typeof</span> module !== <span class="hljs-string">'undefined'</span>) {</pre></div></div>
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<p>Assign the <code>ss</code> object to exports, so that you can require
it in <a href="http://nodejs.org/">node.js</a></p>
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<div class="content"><div class='highlight'><pre> module.exports = ss;
} <span class="hljs-keyword">else</span> {</pre></div></div>
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<p>Otherwise, in a browser, we assign <code>ss</code> to the window object,
so you can simply refer to it as <code>ss</code>.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">this</span>.ss = ss;
}</pre></div></div>
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<h1 id="-linear-regression-http-en-wikipedia-org-wiki-linear_regression-"><a href="http://en.wikipedia.org/wiki/Linear_regression">Linear Regression</a></h1>
<p><a href="http://en.wikipedia.org/wiki/Simple_linear_regression">Simple linear regression</a>
is a simple way to find a fitted line
between a set of coordinates.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">linear_regression</span><span class="hljs-params">()</span> {</span>
<span class="hljs-keyword">var</span> linreg = {},
data = [];</pre></div></div>
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<p>Assign data to the model. Data is assumed to be an array.</p>
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<div class="content"><div class='highlight'><pre> linreg.data = <span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">(x)</span> {</span>
<span class="hljs-keyword">if</span> (!<span class="hljs-built_in">arguments</span>.length) <span class="hljs-keyword">return</span> data;
data = x.slice();
<span class="hljs-keyword">return</span> linreg;
};</pre></div></div>
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<p>Calculate the slope and y-intercept of the regression line
by calculating the least sum of squares</p>
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<div class="content"><div class='highlight'><pre> linreg.mb = <span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">()</span> {</span>
<span class="hljs-keyword">var</span> m, b;</pre></div></div>
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<p>Store data length in a local variable to reduce
repeated object property lookups</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> data_length = data.length;</pre></div></div>
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<p>if theres only one point, arbitrarily choose a slope of 0
and a y-intercept of whatever the y of the initial point is</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (data_length === <span class="hljs-number">1</span>) {
m = <span class="hljs-number">0</span>;
b = data[<span class="hljs-number">0</span>][<span class="hljs-number">1</span>];
} <span class="hljs-keyword">else</span> {</pre></div></div>
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<p>Initialize our sums and scope the <code>m</code> and <code>b</code>
variables that define the line.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> sum_x = <span class="hljs-number">0</span>, sum_y = <span class="hljs-number">0</span>,
sum_xx = <span class="hljs-number">0</span>, sum_xy = <span class="hljs-number">0</span>;</pre></div></div>
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<p>Use local variables to grab point values
with minimal object property lookups</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> point, x, y;</pre></div></div>
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<p>Gather the sum of all x values, the sum of all
y values, and the sum of x^2 and (x*y) for each
value.</p>
<p>In math notation, these would be SS_x, SS_y, SS_xx, and SS_xy</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; data_length; i++) {
point = data[i];
x = point[<span class="hljs-number">0</span>];
y = point[<span class="hljs-number">1</span>];
sum_x += x;
sum_y += y;
sum_xx += x * x;
sum_xy += x * y;
}</pre></div></div>
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<p><code>m</code> is the slope of the regression line</p>
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<div class="content"><div class='highlight'><pre> m = ((data_length * sum_xy) - (sum_x * sum_y)) /
((data_length * sum_xx) - (sum_x * sum_x));</pre></div></div>
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<p><code>b</code> is the y-intercept of the line.</p>
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<div class="content"><div class='highlight'><pre> b = (sum_y / data_length) - ((m * sum_x) / data_length);
}</pre></div></div>
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<p>Return both values as an object.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> { m: m, b: b };
};</pre></div></div>
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<p>a shortcut for simply getting the slope of the regression line</p>
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<div class="content"><div class='highlight'><pre> linreg.m = <span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">()</span> {</span>
<span class="hljs-keyword">return</span> linreg.mb().m;
};</pre></div></div>
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<p>a shortcut for simply getting the y-intercept of the regression
line.</p>
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<div class="content"><div class='highlight'><pre> linreg.b = <span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">()</span> {</span>
<span class="hljs-keyword">return</span> linreg.mb().b;
};</pre></div></div>
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<h2 id="fitting-the-regression-line">Fitting The Regression Line</h2>
<p>This is called after <code>.data()</code> and returns the
equation <code>y = f(x)</code> which gives the position
of the regression line at each point in <code>x</code>.</p>
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<div class="content"><div class='highlight'><pre> linreg.line = <span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">()</span> {</span></pre></div></div>
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<p>Get the slope, <code>m</code>, and y-intercept, <code>b</code>, of the line.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> mb = linreg.mb(),
m = mb.m,
b = mb.b;</pre></div></div>
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<p>Return a function that computes a <code>y</code> value for each
x value it is given, based on the values of <code>b</code> and <code>a</code>
that we just computed.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> <span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">(x)</span> {</span>
<span class="hljs-keyword">return</span> b + (m * x);
};
};
<span class="hljs-keyword">return</span> linreg;
}</pre></div></div>
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<h1 id="-r-squared-http-en-wikipedia-org-wiki-coefficient_of_determination-"><a href="http://en.wikipedia.org/wiki/Coefficient_of_determination">R Squared</a></h1>
<p>The r-squared value of data compared with a function <code>f</code>
is the sum of the squared differences between the prediction
and the actual value.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">r_squared</span><span class="hljs-params">(data, f)</span> {</span>
<span class="hljs-keyword">if</span> (data.length &lt; <span class="hljs-number">2</span>) <span class="hljs-keyword">return</span> <span class="hljs-number">1</span>;</pre></div></div>
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<p>Compute the average y value for the actual
data set in order to compute the
<em>total sum of squares</em></p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> sum = <span class="hljs-number">0</span>, average;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; data.length; i++) {
sum += data[i][<span class="hljs-number">1</span>];
}
average = sum / data.length;</pre></div></div>
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<p>Compute the total sum of squares - the
squared difference between each point
and the average of all points.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> sum_of_squares = <span class="hljs-number">0</span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> j = <span class="hljs-number">0</span>; j &lt; data.length; j++) {
sum_of_squares += <span class="hljs-built_in">Math</span>.pow(average - data[j][<span class="hljs-number">1</span>], <span class="hljs-number">2</span>);
}</pre></div></div>
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<p>Finally estimate the error: the squared
difference between the estimate and the actual data
value at each point.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> err = <span class="hljs-number">0</span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> k = <span class="hljs-number">0</span>; k &lt; data.length; k++) {
err += <span class="hljs-built_in">Math</span>.pow(data[k][<span class="hljs-number">1</span>] - f(data[k][<span class="hljs-number">0</span>]), <span class="hljs-number">2</span>);
}</pre></div></div>
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<p>As the error grows larger, its ratio to the
sum of squares increases and the r squared
value grows lower.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> <span class="hljs-number">1</span> - (err / sum_of_squares);
}</pre></div></div>
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<h1 id="-bayesian-classifier-http-en-wikipedia-org-wiki-naive_bayes_classifier-"><a href="http://en.wikipedia.org/wiki/Naive_Bayes_classifier">Bayesian Classifier</a></h1>
<p>This is a naïve bayesian classifier that takes
singly-nested objects.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">bayesian</span><span class="hljs-params">()</span> {</span></pre></div></div>
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<p>The <code>bayes_model</code> object is what will be exposed
by this closure, with all of its extended methods, and will
have access to all scope variables, like <code>total_count</code>.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> bayes_model = {},</pre></div></div>
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<p>The number of items that are currently
classified in the model</p>
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<div class="content"><div class='highlight'><pre> total_count = <span class="hljs-number">0</span>,</pre></div></div>
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<p>Every item classified in the model</p>
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<div class="content"><div class='highlight'><pre> data = {};</pre></div></div>
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<h2 id="train">Train</h2>
<p>Train the classifier with a new item, which has a single
dimension of Javascript literal keys and values.</p>
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<div class="content"><div class='highlight'><pre> bayes_model.train = <span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">(item, category)</span> {</span></pre></div></div>
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<p>If the data object doesnt have any values
for this category, create a new object for it.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (!data[category]) data[category] = {};</pre></div></div>
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<p>Iterate through each key in the item.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> k <span class="hljs-keyword">in</span> item) {
<span class="hljs-keyword">var</span> v = item[k];</pre></div></div>
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<p>Initialize the nested object <code>data[category][k][item[k]]</code>
with an object of keys that equal 0.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (data[category][k] === <span class="hljs-literal">undefined</span>) data[category][k] = {};
<span class="hljs-keyword">if</span> (data[category][k][v] === <span class="hljs-literal">undefined</span>) data[category][k][v] = <span class="hljs-number">0</span>;</pre></div></div>
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<p>And increment the key for this key/value combination.</p>
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<div class="content"><div class='highlight'><pre> data[category][k][item[k]]++;
}</pre></div></div>
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<p>Increment the number of items classified</p>
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<div class="content"><div class='highlight'><pre> total_count++;
};</pre></div></div>
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<h2 id="score">Score</h2>
<p>Generate a score of how well this item matches all
possible categories based on its attributes</p>
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<div class="content"><div class='highlight'><pre> bayes_model.score = <span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">(item)</span> {</span></pre></div></div>
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<p>Initialize an empty array of odds per category.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> odds = {}, category;</pre></div></div>
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<p>Iterate through each key in the item,
then iterate through each category that has been used
in previous calls to <code>.train()</code></p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> k <span class="hljs-keyword">in</span> item) {
<span class="hljs-keyword">var</span> v = item[k];
<span class="hljs-keyword">for</span> (category <span class="hljs-keyword">in</span> data) {</pre></div></div>
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<p>Create an empty object for storing key - value combinations
for this category.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (odds[category] === <span class="hljs-literal">undefined</span>) odds[category] = {};</pre></div></div>
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<p>If this item doesnt even have a property, it counts for nothing,
but if it does have the property that were looking for from
the item to categorize, it counts based on how popular it is
versus the whole population.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (data[category][k]) {
odds[category][k + <span class="hljs-string">'_'</span> + v] = (data[category][k][v] || <span class="hljs-number">0</span>) / total_count;
} <span class="hljs-keyword">else</span> {
odds[category][k + <span class="hljs-string">'_'</span> + v] = <span class="hljs-number">0</span>;
}
}
}</pre></div></div>
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<p>Set up a new object that will contain sums of these odds by category</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> odds_sums = {};
<span class="hljs-keyword">for</span> (category <span class="hljs-keyword">in</span> odds) {</pre></div></div>
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<p>Tally all of the odds for each category-combination pair -
the non-existence of a category does not add anything to the
score.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> combination <span class="hljs-keyword">in</span> odds[category]) {
<span class="hljs-keyword">if</span> (odds_sums[category] === <span class="hljs-literal">undefined</span>) odds_sums[category] = <span class="hljs-number">0</span>;
odds_sums[category] += odds[category][combination];
}
}
<span class="hljs-keyword">return</span> odds_sums;
};</pre></div></div>
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<p>Return the completed model.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> bayes_model;
}</pre></div></div>
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<h1 id="sum">sum</h1>
<p>is simply the result of adding all numbers
together, starting from zero.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">sum</span><span class="hljs-params">(x)</span> {</span>
<span class="hljs-keyword">var</span> value = <span class="hljs-number">0</span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; x.length; i++) {
value += x[i];
}
<span class="hljs-keyword">return</span> value;
}</pre></div></div>
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<h1 id="mean">mean</h1>
<p>is the sum over the number of values</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">mean</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The mean of no numbers is null</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
<span class="hljs-keyword">return</span> sum(x) / x.length;
}</pre></div></div>
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<h1 id="geometric-mean">geometric mean</h1>
<p>a mean function that is more useful for numbers in different
ranges.</p>
<p>this is the nth root of the input numbers multiplied by each other</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">geometric_mean</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The mean of no numbers is null</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;</pre></div></div>
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<p>the starting value.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> value = <span class="hljs-number">1</span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; x.length; i++) {</pre></div></div>
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<p>the geometric mean is only valid for positive numbers</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x[i] &lt;= <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;</pre></div></div>
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<p>repeatedly multiply the value by each number</p>
</div>
<div class="content"><div class='highlight'><pre> value *= x[i];
}
<span class="hljs-keyword">return</span> <span class="hljs-built_in">Math</span>.pow(value, <span class="hljs-number">1</span> / x.length);
}</pre></div></div>
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<h1 id="harmonic-mean">harmonic mean</h1>
<p>a mean function typically used to find the average of rates</p>
<p>this is the reciprocal of the arithmetic mean of the reciprocals
of the input numbers</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">harmonic_mean</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The mean of no numbers is null</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
<span class="hljs-keyword">var</span> reciprocal_sum = <span class="hljs-number">0</span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; x.length; i++) {</pre></div></div>
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<p>the harmonic mean is only valid for positive numbers</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x[i] &lt;= <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
reciprocal_sum += <span class="hljs-number">1</span> / x[i];
}</pre></div></div>
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<p>divide n by the the reciprocal sum</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> x.length / reciprocal_sum;
}</pre></div></div>
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<h1 id="min">min</h1>
<p>This is simply the minimum number in the set.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">min</span><span class="hljs-params">(x)</span> {</span>
<span class="hljs-keyword">var</span> value;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; x.length; i++) {</pre></div></div>
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<p>On the first iteration of this loop, min is
undefined and is thus made the minimum element in the array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x[i] &lt; value || value === <span class="hljs-literal">undefined</span>) value = x[i];
}
<span class="hljs-keyword">return</span> value;
}</pre></div></div>
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<h1 id="max">max</h1>
<p>This is simply the maximum number in the set.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">max</span><span class="hljs-params">(x)</span> {</span>
<span class="hljs-keyword">var</span> value;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; x.length; i++) {</pre></div></div>
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<p>On the first iteration of this loop, max is
undefined and is thus made the maximum element in the array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x[i] &gt; value || value === <span class="hljs-literal">undefined</span>) value = x[i];
}
<span class="hljs-keyword">return</span> value;
}</pre></div></div>
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<h1 id="-variance-http-en-wikipedia-org-wiki-variance-"><a href="http://en.wikipedia.org/wiki/Variance">variance</a></h1>
<p>is the sum of squared deviations from the mean</p>
<p>depends on <code>mean()</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">variance</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The variance of no numbers is null</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
<span class="hljs-keyword">var</span> mean_value = mean(x),
deviations = [];</pre></div></div>
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<p>Make a list of squared deviations from the mean.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; x.length; i++) {
deviations.push(<span class="hljs-built_in">Math</span>.pow(x[i] - mean_value, <span class="hljs-number">2</span>));
}</pre></div></div>
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<p>Find the mean value of that list</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> mean(deviations);
}</pre></div></div>
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<h1 id="-standard-deviation-http-en-wikipedia-org-wiki-standard_deviation-"><a href="http://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a></h1>
<p>is just the square root of the variance.</p>
<p>depends on <code>variance()</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">standard_deviation</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The standard deviation of no numbers is null</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
<span class="hljs-keyword">return</span> <span class="hljs-built_in">Math</span>.sqrt(variance(x));
}</pre></div></div>
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<p>The sum of deviations to the Nth power.
When n=2 its the sum of squared deviations.
When n=3 its the sum of cubed deviations.</p>
<p>depends on <code>mean()</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">sum_nth_power_deviations</span><span class="hljs-params">(x, n)</span> {</span>
<span class="hljs-keyword">var</span> mean_value = mean(x),
sum = <span class="hljs-number">0</span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; x.length; i++) {
sum += <span class="hljs-built_in">Math</span>.pow(x[i] - mean_value, n);
}
<span class="hljs-keyword">return</span> sum;
}</pre></div></div>
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<h1 id="-variance-http-en-wikipedia-org-wiki-variance-"><a href="http://en.wikipedia.org/wiki/Variance">variance</a></h1>
<p>is the sum of squared deviations from the mean</p>
<p>depends on <code>sum_nth_power_deviations</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">sample_variance</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The variance of no numbers is null</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length &lt;= <span class="hljs-number">1</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
<span class="hljs-keyword">var</span> sum_squared_deviations_value = sum_nth_power_deviations(x, <span class="hljs-number">2</span>);</pre></div></div>
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<p>Find the mean value of that list</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> sum_squared_deviations_value / (x.length - <span class="hljs-number">1</span>);
}</pre></div></div>
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<h1 id="-standard-deviation-http-en-wikipedia-org-wiki-standard_deviation-"><a href="http://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a></h1>
<p>is just the square root of the variance.</p>
<p>depends on <code>sample_variance()</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">sample_standard_deviation</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The standard deviation of no numbers is null</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length &lt;= <span class="hljs-number">1</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
<span class="hljs-keyword">return</span> <span class="hljs-built_in">Math</span>.sqrt(sample_variance(x));
}</pre></div></div>
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<h1 id="-covariance-http-en-wikipedia-org-wiki-covariance-"><a href="http://en.wikipedia.org/wiki/Covariance">covariance</a></h1>
<p>sample covariance of two datasets:
how much do the two datasets move together?
x and y are two datasets, represented as arrays of numbers.</p>
<p>depends on <code>mean()</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">sample_covariance</span><span class="hljs-params">(x, y)</span> {</span></pre></div></div>
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<p>The two datasets must have the same length which must be more than 1</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length &lt;= <span class="hljs-number">1</span> || x.length != y.length){
<span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
}</pre></div></div>
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<p>determine the mean of each dataset so that we can judge each
value of the dataset fairly as the difference from the mean. this
way, if one dataset is [1, 2, 3] and [2, 3, 4], their covariance
does not suffer because of the difference in absolute values</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> xmean = mean(x),
ymean = mean(y),
sum = <span class="hljs-number">0</span>;</pre></div></div>
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<p>for each pair of values, the covariance increases when their
difference from the mean is associated - if both are well above
or if both are well below
the mean, the covariance increases significantly.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; x.length; i++){
sum += (x[i] - xmean) * (y[i] - ymean);
}</pre></div></div>
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<p>the covariance is weighted by the length of the datasets.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> sum / (x.length - <span class="hljs-number">1</span>);
}</pre></div></div>
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<h1 id="-correlation-http-en-wikipedia-org-wiki-correlation_and_dependence-"><a href="http://en.wikipedia.org/wiki/Correlation_and_dependence">correlation</a></h1>
<p>Gets a measure of how correlated two datasets are, between -1 and 1</p>
<p>depends on <code>sample_standard_deviation()</code> and <code>sample_covariance()</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">sample_correlation</span><span class="hljs-params">(x, y)</span> {</span>
<span class="hljs-keyword">var</span> cov = sample_covariance(x, y),
xstd = sample_standard_deviation(x),
ystd = sample_standard_deviation(y);
<span class="hljs-keyword">if</span> (cov === <span class="hljs-literal">null</span> || xstd === <span class="hljs-literal">null</span> || ystd === <span class="hljs-literal">null</span>) {
<span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
}
<span class="hljs-keyword">return</span> cov / xstd / ystd;
}</pre></div></div>
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<h1 id="-median-http-en-wikipedia-org-wiki-median-"><a href="http://en.wikipedia.org/wiki/Median">median</a></h1>
<p>The middle number of a list. This is often a good indicator of the middle
when there are outliers that skew the <code>mean()</code> value.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">median</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The median of an empty list is null</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;</pre></div></div>
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<p>Sorting the array makes it easy to find the center, but
use <code>.slice()</code> to ensure the original array <code>x</code> is not modified</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> sorted = x.slice().sort(<span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-params">(a, b)</span> {</span> <span class="hljs-keyword">return</span> a - b; });</pre></div></div>
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<p>If the length of the list is odd, its the central number</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (sorted.length % <span class="hljs-number">2</span> === <span class="hljs-number">1</span>) {
<span class="hljs-keyword">return</span> sorted[(sorted.length - <span class="hljs-number">1</span>) / <span class="hljs-number">2</span>];</pre></div></div>
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<p>Otherwise, the median is the average of the two numbers
at the center of the list</p>
</div>
<div class="content"><div class='highlight'><pre> } <span class="hljs-keyword">else</span> {
<span class="hljs-keyword">var</span> a = sorted[(sorted.length / <span class="hljs-number">2</span>) - <span class="hljs-number">1</span>];
<span class="hljs-keyword">var</span> b = sorted[(sorted.length / <span class="hljs-number">2</span>)];
<span class="hljs-keyword">return</span> (a + b) / <span class="hljs-number">2</span>;
}
}</pre></div></div>
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<h1 id="-mode-http-bit-ly-w5k4yt-"><a href="http://bit.ly/W5K4Yt">mode</a></h1>
<p>The mode is the number that appears in a list the highest number of times.
There can be multiple modes in a list: in the event of a tie, this
algorithm will return the most recently seen mode.</p>
<p>This implementation is inspired by <a href="https://github.com/jasondavies/science.js/blob/master/src/stats/mode.js">science.js</a></p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">mode</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>Handle edge cases:
The median of an empty list is null</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
<span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (x.length === <span class="hljs-number">1</span>) <span class="hljs-keyword">return</span> x[<span class="hljs-number">0</span>];</pre></div></div>
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<p>Sorting the array lets us iterate through it below and be sure
that every time we see a new number its new and well never
see the same number twice</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> sorted = x.slice().sort(<span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-params">(a, b)</span> {</span> <span class="hljs-keyword">return</span> a - b; });</pre></div></div>
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<p>This assumes it is dealing with an array of size &gt; 1, since size
0 and 1 are handled immediately. Hence it starts at index 1 in the
array.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> last = sorted[<span class="hljs-number">0</span>],</pre></div></div>
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<p>store the mode as we find new modes</p>
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<div class="content"><div class='highlight'><pre> value,</pre></div></div>
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<p>store how many times weve seen the mode</p>
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<div class="content"><div class='highlight'><pre> max_seen = <span class="hljs-number">0</span>,</pre></div></div>
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<p>how many times the current candidate for the mode
has been seen</p>
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<div class="content"><div class='highlight'><pre> seen_this = <span class="hljs-number">1</span>;</pre></div></div>
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<p>end at sorted.length + 1 to fix the case in which the mode is
the highest number that occurs in the sequence. the last iteration
compares sorted[i], which is undefined, to the highest number
in the series</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">1</span>; i &lt; sorted.length + <span class="hljs-number">1</span>; i++) {</pre></div></div>
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<p>were seeing a new number pass by</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (sorted[i] !== last) {</pre></div></div>
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<p>the last number is the new mode since we saw it more
often than the old one</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (seen_this &gt; max_seen) {
max_seen = seen_this;
value = last;
}
seen_this = <span class="hljs-number">1</span>;
last = sorted[i];</pre></div></div>
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<p>if this isnt a new number, its one more occurrence of
the potential mode</p>
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<div class="content"><div class='highlight'><pre> } <span class="hljs-keyword">else</span> { seen_this++; }
}
<span class="hljs-keyword">return</span> value;
}</pre></div></div>
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<h1 id="-t-test-http-en-wikipedia-org-wiki-student-s_t-test-"><a href="http://en.wikipedia.org/wiki/Student&#39;s_t-test">t-test</a></h1>
<p>This is to compute a one-sample t-test, comparing the mean
of a sample to a known value, x.</p>
<p>in this case, were trying to determine whether the
population mean is equal to the value that we know, which is <code>x</code>
here. usually the results here are used to look up a
<a href="http://en.wikipedia.org/wiki/P-value">p-value</a>, which, for
a certain level of significance, will let you determine that the
null hypothesis can or cannot be rejected.</p>
<p>Depends on <code>standard_deviation()</code> and <code>mean()</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">t_test</span><span class="hljs-params">(sample, x)</span> {</span></pre></div></div>
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<p>The mean of the sample</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> sample_mean = mean(sample);</pre></div></div>
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<p>The standard deviation of the sample</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> sd = standard_deviation(sample);</pre></div></div>
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<p>Square root the length of the sample</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> rootN = <span class="hljs-built_in">Math</span>.sqrt(sample.length);</pre></div></div>
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<p>Compute the known value against the sample,
returning the t value</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> (sample_mean - x) / (sd / rootN);
}</pre></div></div>
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<h1 id="-2-sample-t-test-http-en-wikipedia-org-wiki-student-s_t-test-"><a href="http://en.wikipedia.org/wiki/Student&#39;s_t-test">2-sample t-test</a></h1>
<p>This is to compute two sample t-test.
Tests whether “mean(X)-mean(Y) = difference”, (
in the most common case, we often have <code>difference == 0</code> to test if two samples
are likely to be taken from populations with the same mean value) with
no prior knowledge on standard deviations of both samples
other than the fact that they have the same standard deviation.</p>
<p>Usually the results here are used to look up a
<a href="http://en.wikipedia.org/wiki/P-value">p-value</a>, which, for
a certain level of significance, will let you determine that the
null hypothesis can or cannot be rejected.</p>
<p><code>diff</code> can be omitted if it equals 0.</p>
<p><a href="http://www.monarchlab.org/Lab/Research/Stats/2SampleT.aspx">This is used to confirm or deny</a>
a null hypothesis that the two populations that have been sampled into
<code>sample_x</code> and <code>sample_y</code> are equal to each other.</p>
<p>Depends on <code>sample_variance()</code> and <code>mean()</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">t_test_two_sample</span><span class="hljs-params">(sample_x, sample_y, difference)</span> {</span>
<span class="hljs-keyword">var</span> n = sample_x.length,
m = sample_y.length;</pre></div></div>
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<p>If either sample doesnt actually have any values, we cant
compute this at all, so we return <code>null</code>.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (!n || !m) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span> ;</pre></div></div>
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<p>default difference (mu) is zero</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (!difference) difference = <span class="hljs-number">0</span>;
<span class="hljs-keyword">var</span> meanX = mean(sample_x),
meanY = mean(sample_y);
<span class="hljs-keyword">var</span> weightedVariance = ((n - <span class="hljs-number">1</span>) * sample_variance(sample_x) +
(m - <span class="hljs-number">1</span>) * sample_variance(sample_y)) / (n + m - <span class="hljs-number">2</span>);
<span class="hljs-keyword">return</span> (meanX - meanY - difference) /
<span class="hljs-built_in">Math</span>.sqrt(weightedVariance * (<span class="hljs-number">1</span> / n + <span class="hljs-number">1</span> / m));
}</pre></div></div>
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<h1 id="chunk">chunk</h1>
<p>Split an array into chunks of a specified size. This function
has the same behavior as <a href="http://php.net/manual/en/function.array-chunk.php">PHPs array_chunk</a>
function, and thus will insert smaller-sized chunks at the end if
the input size is not divisible by the chunk size.</p>
<p><code>sample</code> is expected to be an array, and <code>chunkSize</code> a number.
The <code>sample</code> array can contain any kind of data.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">chunk</span><span class="hljs-params">(sample, chunkSize)</span> {</span></pre></div></div>
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<p>a list of result chunks, as arrays in an array</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> output = [];</pre></div></div>
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<p><code>chunkSize</code> must be zero or higher - otherwise the loop below,
in which we call <code>start += chunkSize</code>, will loop infinitely.
So, well detect and return null in that case to indicate
invalid input.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (chunkSize &lt;= <span class="hljs-number">0</span>) {
<span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
}</pre></div></div>
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<p><code>start</code> is the index at which <code>.slice</code> will start selecting
new array elements</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> start = <span class="hljs-number">0</span>; start &lt; sample.length; start += chunkSize) {</pre></div></div>
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<p>for each chunk, slice that part of the array and add it
to the output. The <code>.slice</code> function does not change
the original array.</p>
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<div class="content"><div class='highlight'><pre> output.push(sample.slice(start, start + chunkSize));
}
<span class="hljs-keyword">return</span> output;
}</pre></div></div>
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<h1 id="shuffle_in_place">shuffle_in_place</h1>
<p>A <a href="http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle">Fisher-Yates shuffle</a>
in-place - which means that it will change the order of the original
array by reference.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">shuffle_in_place</span><span class="hljs-params">(sample, randomSource)</span> {</span></pre></div></div>
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<p>a custom random number source can be provided if you want to use
a fixed seed or another random number generator, like
<a href="https://www.npmjs.org/package/random-js">random-js</a></p>
</div>
<div class="content"><div class='highlight'><pre> randomSource = randomSource || <span class="hljs-built_in">Math</span>.random;</pre></div></div>
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<p>store the current length of the sample to determine
when no elements remain to shuffle.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> length = sample.length;</pre></div></div>
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<p>temporary is used to hold an item when it is being
swapped between indices.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> temporary;</pre></div></div>
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<p>The index to swap at each stage.</p>
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<p>While there are still items to shuffle</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">while</span> (length &gt; <span class="hljs-number">0</span>) {</pre></div></div>
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<p>chose a random index within the subset of the array
that is not yet shuffled</p>
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<div class="content"><div class='highlight'><pre> index = <span class="hljs-built_in">Math</span>.floor(randomSource() * length--);</pre></div></div>
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<p>store the value that well move temporarily</p>
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<p>swap the value at <code>sample[length]</code> with <code>sample[index]</code></p>
</div>
<div class="content"><div class='highlight'><pre> sample[length] = sample[index];
sample[index] = temporary;
}
<span class="hljs-keyword">return</span> sample;
}</pre></div></div>
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<h1 id="shuffle">shuffle</h1>
<p>A <a href="http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle">Fisher-Yates shuffle</a>
is a fast way to create a random permutation of a finite set.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">shuffle</span><span class="hljs-params">(sample, randomSource)</span> {</span></pre></div></div>
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<p>slice the original array so that it is not modified</p>
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<div class="content"><div class='highlight'><pre> sample = sample.slice();</pre></div></div>
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<p>and then shuffle that shallow-copied array, in place</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> shuffle_in_place(sample.slice(), randomSource);
}</pre></div></div>
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<h1 id="sample">sample</h1>
<p>Create a <a href="http://en.wikipedia.org/wiki/Simple_random_sample">simple random sample</a>
from a given array of <code>n</code> elements.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">sample</span><span class="hljs-params">(array, n, randomSource)</span> {</span></pre></div></div>
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<p>shuffle the original array using a fisher-yates shuffle</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> shuffled = shuffle(array, randomSource);</pre></div></div>
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<p>and then return a subset of it - the first <code>n</code> elements.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> shuffled.slice(<span class="hljs-number">0</span>, n);
}</pre></div></div>
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<h1 id="quantile">quantile</h1>
<p>This is a population quantile, since we assume to know the entire
dataset in this library. Thus Im trying to follow the
<a href="http://en.wikipedia.org/wiki/Quantile#Quantiles_of_a_population">Quantiles of a Population</a>
algorithm from wikipedia.</p>
<p>Sample is a one-dimensional array of numbers,
and p is either a decimal number from 0 to 1 or an array of decimal
numbers from 0 to 1.
In terms of a k/q quantile, p = k/q - its just dealing with fractions or dealing
with decimal values.
When p is an array, the result of the function is also an array containing the appropriate
quantiles in input order</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">quantile</span><span class="hljs-params">(sample, p)</span> {</span></pre></div></div>
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<p>We cant derive quantiles from an empty list</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (sample.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;</pre></div></div>
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<p>Sort a copy of the array. Well need a sorted array to index
the values in sorted order.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> sorted = sample.slice().sort(<span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-params">(a, b)</span> {</span> <span class="hljs-keyword">return</span> a - b; });
<span class="hljs-keyword">if</span> (p.length) {</pre></div></div>
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<p>Initialize the result array</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> results = [];</pre></div></div>
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<p>For each requested quantile</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; p.length; i++) {
results[i] = quantile_sorted(sorted, p[i]);
}
<span class="hljs-keyword">return</span> results;
} <span class="hljs-keyword">else</span> {
<span class="hljs-keyword">return</span> quantile_sorted(sorted, p);
}
}</pre></div></div>
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<h1 id="quantile">quantile</h1>
<p>This is the internal implementation of quantiles: when you know
that the order is sorted, you dont need to re-sort it, and the computations
are much faster.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">quantile_sorted</span><span class="hljs-params">(sample, p)</span> {</span>
<span class="hljs-keyword">var</span> idx = (sample.length) * p;
<span class="hljs-keyword">if</span> (p &lt; <span class="hljs-number">0</span> || p &gt; <span class="hljs-number">1</span>) {
<span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
} <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (p === <span class="hljs-number">1</span>) {</pre></div></div>
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<p>If p is 1, directly return the last element</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> sample[sample.length - <span class="hljs-number">1</span>];
} <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (p === <span class="hljs-number">0</span>) {</pre></div></div>
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<p>If p is 0, directly return the first element</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> sample[<span class="hljs-number">0</span>];
} <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (idx % <span class="hljs-number">1</span> !== <span class="hljs-number">0</span>) {</pre></div></div>
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<p>If p is not integer, return the next element in array</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> sample[<span class="hljs-built_in">Math</span>.ceil(idx) - <span class="hljs-number">1</span>];
} <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (sample.length % <span class="hljs-number">2</span> === <span class="hljs-number">0</span>) {</pre></div></div>
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<p>If the list has even-length, well take the average of this number
and the next value, if there is one</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> (sample[idx - <span class="hljs-number">1</span>] + sample[idx]) / <span class="hljs-number">2</span>;
} <span class="hljs-keyword">else</span> {</pre></div></div>
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<p>Finally, in the simple case of an integer value
with an odd-length list, return the sample value at the index.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> sample[idx];
}
}</pre></div></div>
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<h1 id="-interquartile-range-http-en-wikipedia-org-wiki-interquartile_range-"><a href="http://en.wikipedia.org/wiki/Interquartile_range">Interquartile range</a></h1>
<p>A measure of statistical dispersion, or how scattered, spread, or
concentrated a distribution is. Its computed as the difference between
the third quartile and first quartile.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">iqr</span><span class="hljs-params">(sample)</span> {</span></pre></div></div>
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<p>We cant derive quantiles from an empty list</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (sample.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;</pre></div></div>
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<p>Interquartile range is the span between the upper quartile,
at <code>0.75</code>, and lower quartile, <code>0.25</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> quantile(sample, <span class="hljs-number">0.75</span>) - quantile(sample, <span class="hljs-number">0.25</span>);
}</pre></div></div>
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<h1 id="-median-absolute-deviation-http-en-wikipedia-org-wiki-median_absolute_deviation-"><a href="http://en.wikipedia.org/wiki/Median_absolute_deviation">Median Absolute Deviation</a></h1>
<p>The Median Absolute Deviation (MAD) is a robust measure of statistical
dispersion. It is more resilient to outliers than the standard deviation.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">mad</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The mad of nothing is null</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (!x || x.length === <span class="hljs-number">0</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
<span class="hljs-keyword">var</span> median_value = median(x),
median_absolute_deviations = [];</pre></div></div>
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<p>Make a list of absolute deviations from the median</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; x.length; i++) {
median_absolute_deviations.push(<span class="hljs-built_in">Math</span>.abs(x[i] - median_value));
}</pre></div></div>
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<p>Find the median value of that list</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> median(median_absolute_deviations);
}</pre></div></div>
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<h2 id="compute-matrices-for-jenks">Compute Matrices for Jenks</h2>
<p>Compute the matrices required for Jenks breaks. These matrices
can be used for any classing of data with <code>classes &lt;= n_classes</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">jenksMatrices</span><span class="hljs-params">(data, n_classes)</span> {</span></pre></div></div>
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<p>in the original implementation, these matrices are referred to
as <code>LC</code> and <code>OP</code></p>
<ul>
<li>lower_class_limits (LC): optimal lower class limits</li>
<li>variance_combinations (OP): optimal variance combinations for all classes</li>
</ul>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> lower_class_limits = [],
variance_combinations = [],</pre></div></div>
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<p>loop counters</p>
</div>
<div class="content"><div class='highlight'><pre> i, j,</pre></div></div>
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<p>the variance, as computed at each step in the calculation</p>
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<div class="content"><div class='highlight'><pre> variance = <span class="hljs-number">0</span>;</pre></div></div>
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<p>Initialize and fill each matrix with zeroes</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i &lt; data.length + <span class="hljs-number">1</span>; i++) {
<span class="hljs-keyword">var</span> tmp1 = [], tmp2 = [];</pre></div></div>
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<p>despite these arrays having the same values, we need
to keep them separate so that changing one does not change
the other</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j &lt; n_classes + <span class="hljs-number">1</span>; j++) {
tmp1.push(<span class="hljs-number">0</span>);
tmp2.push(<span class="hljs-number">0</span>);
}
lower_class_limits.push(tmp1);
variance_combinations.push(tmp2);
}
<span class="hljs-keyword">for</span> (i = <span class="hljs-number">1</span>; i &lt; n_classes + <span class="hljs-number">1</span>; i++) {
lower_class_limits[<span class="hljs-number">1</span>][i] = <span class="hljs-number">1</span>;
variance_combinations[<span class="hljs-number">1</span>][i] = <span class="hljs-number">0</span>;</pre></div></div>
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<p>in the original implementation, 9999999 is used but
since Javascript has <code>Infinity</code>, we use that.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (j = <span class="hljs-number">2</span>; j &lt; data.length + <span class="hljs-number">1</span>; j++) {
variance_combinations[j][i] = <span class="hljs-literal">Infinity</span>;
}
}
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> l = <span class="hljs-number">2</span>; l &lt; data.length + <span class="hljs-number">1</span>; l++) {</pre></div></div>
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<p><code>SZ</code> originally. this is the sum of the values seen thus
far when calculating variance.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> sum = <span class="hljs-number">0</span>,</pre></div></div>
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<p><code>ZSQ</code> originally. the sum of squares of values seen
thus far</p>
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<div class="content"><div class='highlight'><pre> sum_squares = <span class="hljs-number">0</span>,</pre></div></div>
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<p><code>WT</code> originally. This is the number of</p>
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<div class="content"><div class='highlight'><pre> w = <span class="hljs-number">0</span>,</pre></div></div>
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<p><code>IV</code> originally</p>
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<div class="content"><div class='highlight'><pre> i4 = <span class="hljs-number">0</span>;</pre></div></div>
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<p>in several instances, you could say <code>Math.pow(x, 2)</code>
instead of <code>x * x</code>, but this is slower in some browsers
introduces an unnecessary concept.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> m = <span class="hljs-number">1</span>; m &lt; l + <span class="hljs-number">1</span>; m++) {</pre></div></div>
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<p><code>III</code> originally</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> lower_class_limit = l - m + <span class="hljs-number">1</span>,
val = data[lower_class_limit - <span class="hljs-number">1</span>];</pre></div></div>
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<p>here were estimating variance for each potential classing
of the data, for each potential number of classes. <code>w</code>
is the number of data points considered so far.</p>
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<div class="content"><div class='highlight'><pre> w++;</pre></div></div>
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<p>increase the current sum and sum-of-squares</p>
</div>
<div class="content"><div class='highlight'><pre> sum += val;
sum_squares += val * val;</pre></div></div>
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<p>the variance at this point in the sequence is the difference
between the sum of squares and the total x 2, over the number
of samples.</p>
</div>
<div class="content"><div class='highlight'><pre> variance = sum_squares - (sum * sum) / w;
i4 = lower_class_limit - <span class="hljs-number">1</span>;
<span class="hljs-keyword">if</span> (i4 !== <span class="hljs-number">0</span>) {
<span class="hljs-keyword">for</span> (j = <span class="hljs-number">2</span>; j &lt; n_classes + <span class="hljs-number">1</span>; j++) {</pre></div></div>
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<p>if adding this element to an existing class
will increase its variance beyond the limit, break
the class at this point, setting the <code>lower_class_limit</code>
at this point.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (variance_combinations[l][j] &gt;=
(variance + variance_combinations[i4][j - <span class="hljs-number">1</span>])) {
lower_class_limits[l][j] = lower_class_limit;
variance_combinations[l][j] = variance +
variance_combinations[i4][j - <span class="hljs-number">1</span>];
}
}
}
}
lower_class_limits[l][<span class="hljs-number">1</span>] = <span class="hljs-number">1</span>;
variance_combinations[l][<span class="hljs-number">1</span>] = variance;
}</pre></div></div>
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<p>return the two matrices. for just providing breaks, only
<code>lower_class_limits</code> is needed, but variances can be useful to
evaluate goodness of fit.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> {
lower_class_limits: lower_class_limits,
variance_combinations: variance_combinations
};
}</pre></div></div>
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<h2 id="pull-breaks-values-for-jenks">Pull Breaks Values for Jenks</h2>
<p>the second part of the jenks recipe: take the calculated matrices
and derive an array of n breaks.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">jenksBreaks</span><span class="hljs-params">(data, lower_class_limits, n_classes)</span> {</span>
<span class="hljs-keyword">var</span> k = data.length - <span class="hljs-number">1</span>,
kclass = [],
countNum = n_classes;</pre></div></div>
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<p>the calculation of classes will never include the upper and
lower bounds, so we need to explicitly set them</p>
</div>
<div class="content"><div class='highlight'><pre> kclass[n_classes] = data[data.length - <span class="hljs-number">1</span>];
kclass[<span class="hljs-number">0</span>] = data[<span class="hljs-number">0</span>];</pre></div></div>
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<p>the lower_class_limits matrix is used as indices into itself
here: the <code>k</code> variable is reused in each iteration.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">while</span> (countNum &gt; <span class="hljs-number">1</span>) {
kclass[countNum - <span class="hljs-number">1</span>] = data[lower_class_limits[k][countNum] - <span class="hljs-number">2</span>];
k = lower_class_limits[k][countNum] - <span class="hljs-number">1</span>;
countNum--;
}
<span class="hljs-keyword">return</span> kclass;
}</pre></div></div>
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<h1 id="-jenks-natural-breaks-optimization-http-en-wikipedia-org-wiki-jenks_natural_breaks_optimization-"><a href="http://en.wikipedia.org/wiki/Jenks_natural_breaks_optimization">Jenks natural breaks optimization</a></h1>
<p>Implementations: <a href="http://danieljlewis.org/files/2010/06/Jenks.pdf">1</a> (python),
<a href="https://github.com/vvoovv/djeo-jenks/blob/master/main.js">2</a> (buggy),
<a href="https://github.com/simogeo/geostats/blob/master/lib/geostats.js#L407">3</a> (works)</p>
<p>Depends on <code>jenksBreaks()</code> and <code>jenksMatrices()</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">jenks</span><span class="hljs-params">(data, n_classes)</span> {</span>
<span class="hljs-keyword">if</span> (n_classes &gt; data.length) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;</pre></div></div>
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<p>sort data in numerical order, since this is expected
by the matrices function</p>
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<div class="content"><div class='highlight'><pre> data = data.slice().sort(<span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-params">(a, b)</span> {</span> <span class="hljs-keyword">return</span> a - b; });</pre></div></div>
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<p>get our basic matrices</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> matrices = jenksMatrices(data, n_classes),</pre></div></div>
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<p>we only need lower class limits here</p>
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<div class="content"><div class='highlight'><pre> lower_class_limits = matrices.lower_class_limits;</pre></div></div>
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<p>extract n_classes out of the computed matrices</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> jenksBreaks(data, lower_class_limits, n_classes);
}</pre></div></div>
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<h1 id="-skewness-http-en-wikipedia-org-wiki-skewness-"><a href="http://en.wikipedia.org/wiki/Skewness">Skewness</a></h1>
<p>A measure of the extent to which a probability distribution of a
real-valued random variable “leans” to one side of the mean.
The skewness value can be positive or negative, or even undefined.</p>
<p>Implementation is based on the adjusted Fisher-Pearson standardized
moment coefficient, which is the version found in Excel and several
statistical packages including Minitab, SAS and SPSS.</p>
<p>Depends on <code>sum_nth_power_deviations()</code> and <code>sample_standard_deviation</code></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">sample_skewness</span><span class="hljs-params">(x)</span> {</span></pre></div></div>
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<p>The skewness of less than three arguments is null</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (x.length &lt; <span class="hljs-number">3</span>) <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
<span class="hljs-keyword">var</span> n = x.length,
cubed_s = <span class="hljs-built_in">Math</span>.pow(sample_standard_deviation(x), <span class="hljs-number">3</span>),
sum_cubed_deviations = sum_nth_power_deviations(x, <span class="hljs-number">3</span>);
<span class="hljs-keyword">return</span> n * sum_cubed_deviations / ((n - <span class="hljs-number">1</span>) * (n - <span class="hljs-number">2</span>) * cubed_s);
}</pre></div></div>
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<h1 id="standard-normal-table">Standard Normal Table</h1>
<p>A standard normal table, also called the unit normal table or Z table,
is a mathematical table for the values of Φ (phi), which are the values of
the cumulative distribution function of the normal distribution.
It is used to find the probability that a statistic is observed below,
above, or between values on the standard normal distribution, and by
extension, any normal distribution.</p>
<p>The probabilities are taken from <a href="http://en.wikipedia.org/wiki/Standard_normal_table">http://en.wikipedia.org/wiki/Standard_normal_table</a>
The table used is the cumulative, and not cumulative from 0 to mean
(even though the latter has 5 digits precision, instead of 4).</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> standard_normal_table = [
<span class="hljs-comment">/* z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 */</span>
<span class="hljs-comment">/* 0.0 */</span>
<span class="hljs-number">0.5000</span>, <span class="hljs-number">0.5040</span>, <span class="hljs-number">0.5080</span>, <span class="hljs-number">0.5120</span>, <span class="hljs-number">0.5160</span>, <span class="hljs-number">0.5199</span>, <span class="hljs-number">0.5239</span>, <span class="hljs-number">0.5279</span>, <span class="hljs-number">0.5319</span>, <span class="hljs-number">0.5359</span>,
<span class="hljs-comment">/* 0.1 */</span>
<span class="hljs-number">0.5398</span>, <span class="hljs-number">0.5438</span>, <span class="hljs-number">0.5478</span>, <span class="hljs-number">0.5517</span>, <span class="hljs-number">0.5557</span>, <span class="hljs-number">0.5596</span>, <span class="hljs-number">0.5636</span>, <span class="hljs-number">0.5675</span>, <span class="hljs-number">0.5714</span>, <span class="hljs-number">0.5753</span>,
<span class="hljs-comment">/* 0.2 */</span>
<span class="hljs-number">0.5793</span>, <span class="hljs-number">0.5832</span>, <span class="hljs-number">0.5871</span>, <span class="hljs-number">0.5910</span>, <span class="hljs-number">0.5948</span>, <span class="hljs-number">0.5987</span>, <span class="hljs-number">0.6026</span>, <span class="hljs-number">0.6064</span>, <span class="hljs-number">0.6103</span>, <span class="hljs-number">0.6141</span>,
<span class="hljs-comment">/* 0.3 */</span>
<span class="hljs-number">0.6179</span>, <span class="hljs-number">0.6217</span>, <span class="hljs-number">0.6255</span>, <span class="hljs-number">0.6293</span>, <span class="hljs-number">0.6331</span>, <span class="hljs-number">0.6368</span>, <span class="hljs-number">0.6406</span>, <span class="hljs-number">0.6443</span>, <span class="hljs-number">0.6480</span>, <span class="hljs-number">0.6517</span>,
<span class="hljs-comment">/* 0.4 */</span>
<span class="hljs-number">0.6554</span>, <span class="hljs-number">0.6591</span>, <span class="hljs-number">0.6628</span>, <span class="hljs-number">0.6664</span>, <span class="hljs-number">0.6700</span>, <span class="hljs-number">0.6736</span>, <span class="hljs-number">0.6772</span>, <span class="hljs-number">0.6808</span>, <span class="hljs-number">0.6844</span>, <span class="hljs-number">0.6879</span>,
<span class="hljs-comment">/* 0.5 */</span>
<span class="hljs-number">0.6915</span>, <span class="hljs-number">0.6950</span>, <span class="hljs-number">0.6985</span>, <span class="hljs-number">0.7019</span>, <span class="hljs-number">0.7054</span>, <span class="hljs-number">0.7088</span>, <span class="hljs-number">0.7123</span>, <span class="hljs-number">0.7157</span>, <span class="hljs-number">0.7190</span>, <span class="hljs-number">0.7224</span>,
<span class="hljs-comment">/* 0.6 */</span>
<span class="hljs-number">0.7257</span>, <span class="hljs-number">0.7291</span>, <span class="hljs-number">0.7324</span>, <span class="hljs-number">0.7357</span>, <span class="hljs-number">0.7389</span>, <span class="hljs-number">0.7422</span>, <span class="hljs-number">0.7454</span>, <span class="hljs-number">0.7486</span>, <span class="hljs-number">0.7517</span>, <span class="hljs-number">0.7549</span>,
<span class="hljs-comment">/* 0.7 */</span>
<span class="hljs-number">0.7580</span>, <span class="hljs-number">0.7611</span>, <span class="hljs-number">0.7642</span>, <span class="hljs-number">0.7673</span>, <span class="hljs-number">0.7704</span>, <span class="hljs-number">0.7734</span>, <span class="hljs-number">0.7764</span>, <span class="hljs-number">0.7794</span>, <span class="hljs-number">0.7823</span>, <span class="hljs-number">0.7852</span>,
<span class="hljs-comment">/* 0.8 */</span>
<span class="hljs-number">0.7881</span>, <span class="hljs-number">0.7910</span>, <span class="hljs-number">0.7939</span>, <span class="hljs-number">0.7967</span>, <span class="hljs-number">0.7995</span>, <span class="hljs-number">0.8023</span>, <span class="hljs-number">0.8051</span>, <span class="hljs-number">0.8078</span>, <span class="hljs-number">0.8106</span>, <span class="hljs-number">0.8133</span>,
<span class="hljs-comment">/* 0.9 */</span>
<span class="hljs-number">0.8159</span>, <span class="hljs-number">0.8186</span>, <span class="hljs-number">0.8212</span>, <span class="hljs-number">0.8238</span>, <span class="hljs-number">0.8264</span>, <span class="hljs-number">0.8289</span>, <span class="hljs-number">0.8315</span>, <span class="hljs-number">0.8340</span>, <span class="hljs-number">0.8365</span>, <span class="hljs-number">0.8389</span>,
<span class="hljs-comment">/* 1.0 */</span>
<span class="hljs-number">0.8413</span>, <span class="hljs-number">0.8438</span>, <span class="hljs-number">0.8461</span>, <span class="hljs-number">0.8485</span>, <span class="hljs-number">0.8508</span>, <span class="hljs-number">0.8531</span>, <span class="hljs-number">0.8554</span>, <span class="hljs-number">0.8577</span>, <span class="hljs-number">0.8599</span>, <span class="hljs-number">0.8621</span>,
<span class="hljs-comment">/* 1.1 */</span>
<span class="hljs-number">0.8643</span>, <span class="hljs-number">0.8665</span>, <span class="hljs-number">0.8686</span>, <span class="hljs-number">0.8708</span>, <span class="hljs-number">0.8729</span>, <span class="hljs-number">0.8749</span>, <span class="hljs-number">0.8770</span>, <span class="hljs-number">0.8790</span>, <span class="hljs-number">0.8810</span>, <span class="hljs-number">0.8830</span>,
<span class="hljs-comment">/* 1.2 */</span>
<span class="hljs-number">0.8849</span>, <span class="hljs-number">0.8869</span>, <span class="hljs-number">0.8888</span>, <span class="hljs-number">0.8907</span>, <span class="hljs-number">0.8925</span>, <span class="hljs-number">0.8944</span>, <span class="hljs-number">0.8962</span>, <span class="hljs-number">0.8980</span>, <span class="hljs-number">0.8997</span>, <span class="hljs-number">0.9015</span>,
<span class="hljs-comment">/* 1.3 */</span>
<span class="hljs-number">0.9032</span>, <span class="hljs-number">0.9049</span>, <span class="hljs-number">0.9066</span>, <span class="hljs-number">0.9082</span>, <span class="hljs-number">0.9099</span>, <span class="hljs-number">0.9115</span>, <span class="hljs-number">0.9131</span>, <span class="hljs-number">0.9147</span>, <span class="hljs-number">0.9162</span>, <span class="hljs-number">0.9177</span>,
<span class="hljs-comment">/* 1.4 */</span>
<span class="hljs-number">0.9192</span>, <span class="hljs-number">0.9207</span>, <span class="hljs-number">0.9222</span>, <span class="hljs-number">0.9236</span>, <span class="hljs-number">0.9251</span>, <span class="hljs-number">0.9265</span>, <span class="hljs-number">0.9279</span>, <span class="hljs-number">0.9292</span>, <span class="hljs-number">0.9306</span>, <span class="hljs-number">0.9319</span>,
<span class="hljs-comment">/* 1.5 */</span>
<span class="hljs-number">0.9332</span>, <span class="hljs-number">0.9345</span>, <span class="hljs-number">0.9357</span>, <span class="hljs-number">0.9370</span>, <span class="hljs-number">0.9382</span>, <span class="hljs-number">0.9394</span>, <span class="hljs-number">0.9406</span>, <span class="hljs-number">0.9418</span>, <span class="hljs-number">0.9429</span>, <span class="hljs-number">0.9441</span>,
<span class="hljs-comment">/* 1.6 */</span>
<span class="hljs-number">0.9452</span>, <span class="hljs-number">0.9463</span>, <span class="hljs-number">0.9474</span>, <span class="hljs-number">0.9484</span>, <span class="hljs-number">0.9495</span>, <span class="hljs-number">0.9505</span>, <span class="hljs-number">0.9515</span>, <span class="hljs-number">0.9525</span>, <span class="hljs-number">0.9535</span>, <span class="hljs-number">0.9545</span>,
<span class="hljs-comment">/* 1.7 */</span>
<span class="hljs-number">0.9554</span>, <span class="hljs-number">0.9564</span>, <span class="hljs-number">0.9573</span>, <span class="hljs-number">0.9582</span>, <span class="hljs-number">0.9591</span>, <span class="hljs-number">0.9599</span>, <span class="hljs-number">0.9608</span>, <span class="hljs-number">0.9616</span>, <span class="hljs-number">0.9625</span>, <span class="hljs-number">0.9633</span>,
<span class="hljs-comment">/* 1.8 */</span>
<span class="hljs-number">0.9641</span>, <span class="hljs-number">0.9649</span>, <span class="hljs-number">0.9656</span>, <span class="hljs-number">0.9664</span>, <span class="hljs-number">0.9671</span>, <span class="hljs-number">0.9678</span>, <span class="hljs-number">0.9686</span>, <span class="hljs-number">0.9693</span>, <span class="hljs-number">0.9699</span>, <span class="hljs-number">0.9706</span>,
<span class="hljs-comment">/* 1.9 */</span>
<span class="hljs-number">0.9713</span>, <span class="hljs-number">0.9719</span>, <span class="hljs-number">0.9726</span>, <span class="hljs-number">0.9732</span>, <span class="hljs-number">0.9738</span>, <span class="hljs-number">0.9744</span>, <span class="hljs-number">0.9750</span>, <span class="hljs-number">0.9756</span>, <span class="hljs-number">0.9761</span>, <span class="hljs-number">0.9767</span>,
<span class="hljs-comment">/* 2.0 */</span>
<span class="hljs-number">0.9772</span>, <span class="hljs-number">0.9778</span>, <span class="hljs-number">0.9783</span>, <span class="hljs-number">0.9788</span>, <span class="hljs-number">0.9793</span>, <span class="hljs-number">0.9798</span>, <span class="hljs-number">0.9803</span>, <span class="hljs-number">0.9808</span>, <span class="hljs-number">0.9812</span>, <span class="hljs-number">0.9817</span>,
<span class="hljs-comment">/* 2.1 */</span>
<span class="hljs-number">0.9821</span>, <span class="hljs-number">0.9826</span>, <span class="hljs-number">0.9830</span>, <span class="hljs-number">0.9834</span>, <span class="hljs-number">0.9838</span>, <span class="hljs-number">0.9842</span>, <span class="hljs-number">0.9846</span>, <span class="hljs-number">0.9850</span>, <span class="hljs-number">0.9854</span>, <span class="hljs-number">0.9857</span>,
<span class="hljs-comment">/* 2.2 */</span>
<span class="hljs-number">0.9861</span>, <span class="hljs-number">0.9864</span>, <span class="hljs-number">0.9868</span>, <span class="hljs-number">0.9871</span>, <span class="hljs-number">0.9875</span>, <span class="hljs-number">0.9878</span>, <span class="hljs-number">0.9881</span>, <span class="hljs-number">0.9884</span>, <span class="hljs-number">0.9887</span>, <span class="hljs-number">0.9890</span>,
<span class="hljs-comment">/* 2.3 */</span>
<span class="hljs-number">0.9893</span>, <span class="hljs-number">0.9896</span>, <span class="hljs-number">0.9898</span>, <span class="hljs-number">0.9901</span>, <span class="hljs-number">0.9904</span>, <span class="hljs-number">0.9906</span>, <span class="hljs-number">0.9909</span>, <span class="hljs-number">0.9911</span>, <span class="hljs-number">0.9913</span>, <span class="hljs-number">0.9916</span>,
<span class="hljs-comment">/* 2.4 */</span>
<span class="hljs-number">0.9918</span>, <span class="hljs-number">0.9920</span>, <span class="hljs-number">0.9922</span>, <span class="hljs-number">0.9925</span>, <span class="hljs-number">0.9927</span>, <span class="hljs-number">0.9929</span>, <span class="hljs-number">0.9931</span>, <span class="hljs-number">0.9932</span>, <span class="hljs-number">0.9934</span>, <span class="hljs-number">0.9936</span>,
<span class="hljs-comment">/* 2.5 */</span>
<span class="hljs-number">0.9938</span>, <span class="hljs-number">0.9940</span>, <span class="hljs-number">0.9941</span>, <span class="hljs-number">0.9943</span>, <span class="hljs-number">0.9945</span>, <span class="hljs-number">0.9946</span>, <span class="hljs-number">0.9948</span>, <span class="hljs-number">0.9949</span>, <span class="hljs-number">0.9951</span>, <span class="hljs-number">0.9952</span>,
<span class="hljs-comment">/* 2.6 */</span>
<span class="hljs-number">0.9953</span>, <span class="hljs-number">0.9955</span>, <span class="hljs-number">0.9956</span>, <span class="hljs-number">0.9957</span>, <span class="hljs-number">0.9959</span>, <span class="hljs-number">0.9960</span>, <span class="hljs-number">0.9961</span>, <span class="hljs-number">0.9962</span>, <span class="hljs-number">0.9963</span>, <span class="hljs-number">0.9964</span>,
<span class="hljs-comment">/* 2.7 */</span>
<span class="hljs-number">0.9965</span>, <span class="hljs-number">0.9966</span>, <span class="hljs-number">0.9967</span>, <span class="hljs-number">0.9968</span>, <span class="hljs-number">0.9969</span>, <span class="hljs-number">0.9970</span>, <span class="hljs-number">0.9971</span>, <span class="hljs-number">0.9972</span>, <span class="hljs-number">0.9973</span>, <span class="hljs-number">0.9974</span>,
<span class="hljs-comment">/* 2.8 */</span>
<span class="hljs-number">0.9974</span>, <span class="hljs-number">0.9975</span>, <span class="hljs-number">0.9976</span>, <span class="hljs-number">0.9977</span>, <span class="hljs-number">0.9977</span>, <span class="hljs-number">0.9978</span>, <span class="hljs-number">0.9979</span>, <span class="hljs-number">0.9979</span>, <span class="hljs-number">0.9980</span>, <span class="hljs-number">0.9981</span>,
<span class="hljs-comment">/* 2.9 */</span>
<span class="hljs-number">0.9981</span>, <span class="hljs-number">0.9982</span>, <span class="hljs-number">0.9982</span>, <span class="hljs-number">0.9983</span>, <span class="hljs-number">0.9984</span>, <span class="hljs-number">0.9984</span>, <span class="hljs-number">0.9985</span>, <span class="hljs-number">0.9985</span>, <span class="hljs-number">0.9986</span>, <span class="hljs-number">0.9986</span>,
<span class="hljs-comment">/* 3.0 */</span>
<span class="hljs-number">0.9987</span>, <span class="hljs-number">0.9987</span>, <span class="hljs-number">0.9987</span>, <span class="hljs-number">0.9988</span>, <span class="hljs-number">0.9988</span>, <span class="hljs-number">0.9989</span>, <span class="hljs-number">0.9989</span>, <span class="hljs-number">0.9989</span>, <span class="hljs-number">0.9990</span>, <span class="hljs-number">0.9990</span>
];</pre></div></div>
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<h1 id="-cumulative-standard-normal-probability-http-en-wikipedia-org-wiki-standard_normal_table-"><a href="http://en.wikipedia.org/wiki/Standard_normal_table">Cumulative Standard Normal Probability</a></h1>
<p>Since probability tables cannot be
printed for every normal distribution, as there are an infinite variety
of normal distributions, it is common practice to convert a normal to a
standard normal and then use the standard normal table to find probabilities</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">cumulative_std_normal_probability</span><span class="hljs-params">(z)</span> {</span></pre></div></div>
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<p>Calculate the position of this value.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> absZ = <span class="hljs-built_in">Math</span>.abs(z),</pre></div></div>
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<p>Each row begins with a different
significant digit: 0.5, 0.6, 0.7, and so on. So the row is simply
this values significant digit: 0.567 will be in row 0, so row=0,
0.643 will be in row 1, so row=10.</p>
</div>
<div class="content"><div class='highlight'><pre> row = <span class="hljs-built_in">Math</span>.floor(absZ * <span class="hljs-number">10</span>),
column = <span class="hljs-number">10</span> * (<span class="hljs-built_in">Math</span>.floor(absZ * <span class="hljs-number">100</span>) / <span class="hljs-number">10</span> - <span class="hljs-built_in">Math</span>.floor(absZ * <span class="hljs-number">100</span> / <span class="hljs-number">10</span>)),
index = <span class="hljs-built_in">Math</span>.min((row * <span class="hljs-number">10</span>) + column, standard_normal_table.length - <span class="hljs-number">1</span>);</pre></div></div>
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<p>The index we calculate must be in the table as a positive value,
but we still pay attention to whether the input is positive
or negative, and flip the output value as a last step.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (z &gt;= <span class="hljs-number">0</span>) {
<span class="hljs-keyword">return</span> standard_normal_table[index];
} <span class="hljs-keyword">else</span> {</pre></div></div>
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<p>due to floating-point arithmetic, values in the table with
4 significant figures can nevertheless end up as repeating
fractions when theyre computed here.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> +(<span class="hljs-number">1</span> - standard_normal_table[index]).toFixed(<span class="hljs-number">4</span>);
}
}</pre></div></div>
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<h1 id="-z-score-or-standard-score-http-en-wikipedia-org-wiki-standard_score-"><a href="http://en.wikipedia.org/wiki/Standard_score">Z-Score, or Standard Score</a></h1>
<p>The standard score is the number of standard deviations an observation
or datum is above or below the mean. Thus, a positive standard score
represents a datum above the mean, while a negative standard score
represents a datum below the mean. It is a dimensionless quantity
obtained by subtracting the population mean from an individual raw
score and then dividing the difference by the population standard
deviation.</p>
<p>The z-score is only defined if one knows the population parameters;
if one only has a sample set, then the analogous computation with
sample mean and sample standard deviation yields the
Students t-statistic.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">z_score</span><span class="hljs-params">(x, mean, standard_deviation)</span> {</span>
<span class="hljs-keyword">return</span> (x - mean) / standard_deviation;
}</pre></div></div>
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<p>We use <code>ε</code>, epsilon, as a stopping criterion when we want to iterate
until were “close enough”.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> epsilon = <span class="hljs-number">0.0001</span>;</pre></div></div>
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<h1 id="-factorial-https-en-wikipedia-org-wiki-factorial-"><a href="https://en.wikipedia.org/wiki/Factorial">Factorial</a></h1>
<p>A factorial, usually written n!, is the product of all positive
integers less than or equal to n. Often factorial is implemented
recursively, but this iterative approach is significantly faster
and simpler.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">factorial</span><span class="hljs-params">(n)</span> {</span></pre></div></div>
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<p>factorial is mathematically undefined for negative numbers</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (n &lt; <span class="hljs-number">0</span> ) { <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>; }</pre></div></div>
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<p>typically youll expand the factorial function going down, like
5! = 5 <em> 4 </em> 3 <em> 2 </em> 1. This is going in the opposite direction,
counting from 2 up to the number in question, and since anything
multiplied by 1 is itself, the loop only needs to start at 2.</p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> accumulator = <span class="hljs-number">1</span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">2</span>; i &lt;= n; i++) {</pre></div></div>
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<p>for each number up to and including the number <code>n</code>, multiply
the accumulator my that number.</p>
</div>
<div class="content"><div class='highlight'><pre> accumulator *= i;
}
<span class="hljs-keyword">return</span> accumulator;
}</pre></div></div>
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<h1 id="bernoulli-distribution">Bernoulli Distribution</h1>
<p>The <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution">Bernoulli distribution</a>
is the probability discrete
distribution of a random variable which takes value 1 with success
probability <code>p</code> and value 0 with failure
probability <code>q</code> = 1 - <code>p</code>. It can be used, for example, to represent the
toss of a coin, where “1” is defined to mean “heads” and “0” is defined
to mean “tails” (or vice versa). It is
a special case of a Binomial Distribution
where <code>n</code> = 1.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">bernoulli_distribution</span><span class="hljs-params">(p)</span> {</span></pre></div></div>
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<p>Check that <code>p</code> is a valid probability (0 ≤ p ≤ 1)</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (p &lt; <span class="hljs-number">0</span> || p &gt; <span class="hljs-number">1</span> ) { <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>; }
<span class="hljs-keyword">return</span> binomial_distribution(<span class="hljs-number">1</span>, p);
}</pre></div></div>
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<h1 id="binomial-distribution">Binomial Distribution</h1>
<p>The <a href="http://en.wikipedia.org/wiki/Binomial_distribution">Binomial Distribution</a> is the discrete probability
distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields
success with probability <code>probability</code>. Such a success/failure experiment is also called a Bernoulli experiment or
Bernoulli trial; when trials = 1, the Binomial Distribution is a Bernoulli Distribution.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">binomial_distribution</span><span class="hljs-params">(trials, probability)</span> {</span></pre></div></div>
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<p>Check that <code>p</code> is a valid probability (0 ≤ p ≤ 1),
that <code>n</code> is an integer, strictly positive.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (probability &lt; <span class="hljs-number">0</span> || probability &gt; <span class="hljs-number">1</span> ||
trials &lt;= <span class="hljs-number">0</span> || trials % <span class="hljs-number">1</span> !== <span class="hljs-number">0</span>) {
<span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>;
}</pre></div></div>
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<p>a <a href="https://en.wikipedia.org/wiki/Probability_mass_function">probability mass function</a></p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">probability_mass</span><span class="hljs-params">(x, trials, probability)</span> {</span>
<span class="hljs-keyword">return</span> factorial(trials) /
(factorial(x) * factorial(trials - x)) *
(<span class="hljs-built_in">Math</span>.pow(probability, x) * <span class="hljs-built_in">Math</span>.pow(<span class="hljs-number">1</span> - probability, trials - x));
}</pre></div></div>
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<p>We initialize <code>x</code>, the random variable, and <code>accumulator</code>, an accumulator
for the cumulative distribution function to 0. <code>distribution_functions</code>
is the object well return with the <code>probability_of_x</code> and the
<code>cumulative_probability_of_x</code>, as well as the calculated mean &amp;
variance. We iterate until the <code>cumulative_probability_of_x</code> is
within <code>epsilon</code> of 1.0.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> x = <span class="hljs-number">0</span>,
cumulative_probability = <span class="hljs-number">0</span>,
cells = {};</pre></div></div>
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<p>This algorithm iterates through each potential outcome,
until the <code>cumulative_probability</code> is very close to 1, at
which point weve defined the vast majority of outcomes</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">do</span> {
cells[x] = probability_mass(x, trials, probability);
cumulative_probability += cells[x];
x++;</pre></div></div>
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<p>when the cumulative_probability is nearly 1, weve calculated
the useful range of this distribution</p>
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<div class="content"><div class='highlight'><pre> } <span class="hljs-keyword">while</span> (cumulative_probability &lt; <span class="hljs-number">1</span> - epsilon);
<span class="hljs-keyword">return</span> cells;
}</pre></div></div>
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<h1 id="poisson-distribution">Poisson Distribution</h1>
<p>The <a href="http://en.wikipedia.org/wiki/Poisson_distribution">Poisson Distribution</a>
is a discrete probability distribution that expresses the probability
of a given number of events occurring in a fixed interval of time
and/or space if these events occur with a known average rate and
independently of the time since the last event.</p>
<p>The Poisson Distribution is characterized by the strictly positive
mean arrival or occurrence rate, <code>λ</code>.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">poisson_distribution</span><span class="hljs-params">(lambda)</span> {</span></pre></div></div>
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<p>Check that lambda is strictly positive</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">if</span> (lambda &lt;= <span class="hljs-number">0</span>) { <span class="hljs-keyword">return</span> <span class="hljs-literal">null</span>; }</pre></div></div>
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<p>our current place in the distribution</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> x = <span class="hljs-number">0</span>,</pre></div></div>
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<p>and we keep track of the current cumulative probability, in
order to know when to stop calculating chances.</p>
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<div class="content"><div class='highlight'><pre> cumulative_probability = <span class="hljs-number">0</span>,</pre></div></div>
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<p>the calculated cells to be returned</p>
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<div class="content"><div class='highlight'><pre> cells = {};</pre></div></div>
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<p>a <a href="https://en.wikipedia.org/wiki/Probability_mass_function">probability mass function</a></p>
</div>
<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">probability_mass</span><span class="hljs-params">(x, lambda)</span> {</span>
<span class="hljs-keyword">return</span> (<span class="hljs-built_in">Math</span>.pow(<span class="hljs-built_in">Math</span>.E, -lambda) * <span class="hljs-built_in">Math</span>.pow(lambda, x)) /
factorial(x);
}</pre></div></div>
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<p>This algorithm iterates through each potential outcome,
until the <code>cumulative_probability</code> is very close to 1, at
which point weve defined the vast majority of outcomes</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">do</span> {
cells[x] = probability_mass(x, lambda);
cumulative_probability += cells[x];
x++;</pre></div></div>
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<p>when the cumulative_probability is nearly 1, weve calculated
the useful range of this distribution</p>
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<div class="content"><div class='highlight'><pre> } <span class="hljs-keyword">while</span> (cumulative_probability &lt; <span class="hljs-number">1</span> - epsilon);
<span class="hljs-keyword">return</span> cells;
}</pre></div></div>
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<h1 id="percentage-points-of-the-2-chi-squared-distribution">Percentage Points of the χ2 (Chi-Squared) Distribution</h1>
<p>The <a href="http://en.wikipedia.org/wiki/Chi-squared_distribution">χ2 (Chi-Squared) Distribution</a> is used in the common
chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two
criteria of classification of qualitative data, and in confidence interval estimation for a population standard
deviation of a normal distribution from a sample standard deviation.</p>
<p>Values from Appendix 1, Table III of William W. Hines &amp; Douglas C. Montgomery, “Probability and Statistics in
Engineering and Management Science”, Wiley (1980).</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> chi_squared_distribution_table = {
<span class="hljs-number">1</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">0.00</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">0.00</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">0.00</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">0.00</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">0.02</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">0.45</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">2.71</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">3.84</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">5.02</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">6.63</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">7.88</span> },
<span class="hljs-number">2</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">0.01</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">0.02</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">0.05</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">0.10</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">0.21</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">1.39</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">4.61</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">5.99</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">7.38</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">9.21</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">10.60</span> },
<span class="hljs-number">3</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">0.07</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">0.11</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">0.22</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">0.35</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">0.58</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">2.37</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">6.25</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">7.81</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">9.35</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">11.34</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">12.84</span> },
<span class="hljs-number">4</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">0.21</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">0.30</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">0.48</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">0.71</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">1.06</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">3.36</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">7.78</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">9.49</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">11.14</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">13.28</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">14.86</span> },
<span class="hljs-number">5</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">0.41</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">0.55</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">0.83</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">1.15</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">1.61</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">4.35</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">9.24</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">11.07</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">12.83</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">15.09</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">16.75</span> },
<span class="hljs-number">6</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">0.68</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">0.87</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">1.24</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">1.64</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">2.20</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">5.35</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">10.65</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">12.59</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">14.45</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">16.81</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">18.55</span> },
<span class="hljs-number">7</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">0.99</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">1.25</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">1.69</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">2.17</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">2.83</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">6.35</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">12.02</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">14.07</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">16.01</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">18.48</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">20.28</span> },
<span class="hljs-number">8</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">1.34</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">1.65</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">2.18</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">2.73</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">3.49</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">7.34</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">13.36</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">15.51</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">17.53</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">20.09</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">21.96</span> },
<span class="hljs-number">9</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">1.73</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">2.09</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">2.70</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">3.33</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">4.17</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">8.34</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">14.68</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">16.92</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">19.02</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">21.67</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">23.59</span> },
<span class="hljs-number">10</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">2.16</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">2.56</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">3.25</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">3.94</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">4.87</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">9.34</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">15.99</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">18.31</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">20.48</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">23.21</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">25.19</span> },
<span class="hljs-number">11</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">2.60</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">3.05</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">3.82</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">4.57</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">5.58</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">10.34</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">17.28</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">19.68</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">21.92</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">24.72</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">26.76</span> },
<span class="hljs-number">12</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">3.07</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">3.57</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">4.40</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">5.23</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">6.30</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">11.34</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">18.55</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">21.03</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">23.34</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">26.22</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">28.30</span> },
<span class="hljs-number">13</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">3.57</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">4.11</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">5.01</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">5.89</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">7.04</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">12.34</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">19.81</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">22.36</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">24.74</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">27.69</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">29.82</span> },
<span class="hljs-number">14</span>: { <span class="hljs-number">0.995</span>: <span class="hljs-number">4.07</span>, <span class="hljs-number">0.99</span>: <span class="hljs-number">4.66</span>, <span class="hljs-number">0.975</span>: <span class="hljs-number">5.63</span>, <span class="hljs-number">0.95</span>: <span class="hljs-number">6.57</span>, <span class="hljs-number">0.9</span>: <span class="hljs-number">7.79</span>, <span class="hljs-number">0.5</span>: <span class="hljs-number">13.34</span>, <span class="hljs-number">0.1</span>: <span class="hljs-number">21.06</span>, <span class="hljs-number">0.05</span>: <span class="hljs-number">23.68</span>, <span class="hljs-number">0.025</span>: <span class="hljs-number">26.12</span>, <span class="hljs-number">0.01</span>: <span class="hljs-number">29.14</span>, <span class="hljs-number">0.005</span>: <span class="hljs-number">31.32</span> },
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};</pre></div></div>
</li>
<li id="section-197">
<div class="annotation">
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<h1 id="-2-chi-squared-goodness-of-fit-test">χ2 (Chi-Squared) Goodness-of-Fit Test</h1>
<p>The <a href="http://en.wikipedia.org/wiki/Goodness_of_fit#Pearson.27s_chi-squared_test">χ2 (Chi-Squared) Goodness-of-Fit Test</a>
uses a measure of goodness of fit which is the sum of differences between observed and expected outcome frequencies
(that is, counts of observations), each squared and divided by the number of observations expected given the
hypothesized distribution. The resulting χ2 statistic, <code>chi_squared</code>, can be compared to the chi-squared distribution
to determine the goodness of fit. In order to determine the degrees of freedom of the chi-squared distribution, one
takes the total number of observed frequencies and subtracts the number of estimated parameters. The test statistic
follows, approximately, a chi-square distribution with (k c) degrees of freedom where <code>k</code> is the number of non-empty
cells and <code>c</code> is the number of estimated parameters for the distribution.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">chi_squared_goodness_of_fit</span><span class="hljs-params">(data, distribution_type, significance)</span> {</span></pre></div></div>
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<p>Estimate from the sample data, a weighted mean.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> input_mean = mean(data),</pre></div></div>
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<p>Calculated value of the χ2 statistic.</p>
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<div class="content"><div class='highlight'><pre> chi_squared = <span class="hljs-number">0</span>,</pre></div></div>
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<p>Degrees of freedom, calculated as (number of class intervals -
number of hypothesized distribution parameters estimated - 1)</p>
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<div class="content"><div class='highlight'><pre> degrees_of_freedom,</pre></div></div>
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<p>Number of hypothesized distribution parameters estimated, expected to be supplied in the distribution test.
Lose one degree of freedom for estimating <code>lambda</code> from the sample data.</p>
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<div class="content"><div class='highlight'><pre> c = <span class="hljs-number">1</span>,</pre></div></div>
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<p>The hypothesized distribution.
Generate the hypothesized distribution.</p>
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<div class="content"><div class='highlight'><pre> hypothesized_distribution = distribution_type(input_mean),
observed_frequencies = [],
expected_frequencies = [],
k;</pre></div></div>
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<p>Create an array holding a histogram from the sample data, of
the form <code>{ value: numberOfOcurrences }</code></p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; data.length; i++) {
<span class="hljs-keyword">if</span> (observed_frequencies[data[i]] === <span class="hljs-literal">undefined</span>) {
observed_frequencies[data[i]] = <span class="hljs-number">0</span>;
}
observed_frequencies[data[i]]++;
}</pre></div></div>
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<p>The histogram we created might be sparse - there might be gaps
between values. So we iterate through the histogram, making
sure that instead of undefined, gaps have 0 values.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i &lt; observed_frequencies.length; i++) {
<span class="hljs-keyword">if</span> (observed_frequencies[i] === <span class="hljs-literal">undefined</span>) {
observed_frequencies[i] = <span class="hljs-number">0</span>;
}
}</pre></div></div>
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<p>Create an array holding a histogram of expected data given the
sample size and hypothesized distribution.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (k <span class="hljs-keyword">in</span> hypothesized_distribution) {
<span class="hljs-keyword">if</span> (k <span class="hljs-keyword">in</span> observed_frequencies) {
expected_frequencies[k] = hypothesized_distribution[k] * data.length;
}
}</pre></div></div>
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<p>Working backward through the expected frequencies, collapse classes
if less than three observations are expected for a class.
This transformation is applied to the observed frequencies as well.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (k = expected_frequencies.length - <span class="hljs-number">1</span>; k &gt;= <span class="hljs-number">0</span>; k--) {
<span class="hljs-keyword">if</span> (expected_frequencies[k] &lt; <span class="hljs-number">3</span>) {
expected_frequencies[k - <span class="hljs-number">1</span>] += expected_frequencies[k];
expected_frequencies.pop();
observed_frequencies[k - <span class="hljs-number">1</span>] += observed_frequencies[k];
observed_frequencies.pop();
}
}</pre></div></div>
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<p>Iterate through the squared differences between observed &amp; expected
frequencies, accumulating the <code>chi_squared</code> statistic.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (k = <span class="hljs-number">0</span>; k &lt; observed_frequencies.length; k++) {
chi_squared += <span class="hljs-built_in">Math</span>.pow(
observed_frequencies[k] - expected_frequencies[k], <span class="hljs-number">2</span>) /
expected_frequencies[k];
}</pre></div></div>
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<p>Calculate degrees of freedom for this test and look it up in the
<code>chi_squared_distribution_table</code> in order to
accept or reject the goodness-of-fit of the hypothesized distribution.</p>
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<div class="content"><div class='highlight'><pre> degrees_of_freedom = observed_frequencies.length - c - <span class="hljs-number">1</span>;
<span class="hljs-keyword">return</span> chi_squared_distribution_table[degrees_of_freedom][significance] &lt; chi_squared;
}</pre></div></div>
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<h1 id="mixin">Mixin</h1>
<p>Mixin simple_statistics to a single Array instance if provided
or the Array native object if not. This is an optional
feature that lets you treat simple_statistics as a native feature
of Javascript.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">mixin</span><span class="hljs-params">(array)</span> {</span>
<span class="hljs-keyword">var</span> support = !!(<span class="hljs-built_in">Object</span>.defineProperty &amp;&amp; <span class="hljs-built_in">Object</span>.defineProperties);
<span class="hljs-keyword">if</span> (!support) <span class="hljs-keyword">throw</span> <span class="hljs-keyword">new</span> <span class="hljs-built_in">Error</span>(<span class="hljs-string">'without defineProperty, simple-statistics cannot be mixed in'</span>);</pre></div></div>
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<p>only methods which work on basic arrays in a single step
are supported</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> arrayMethods = [<span class="hljs-string">'median'</span>, <span class="hljs-string">'standard_deviation'</span>, <span class="hljs-string">'sum'</span>,
<span class="hljs-string">'sample_skewness'</span>,
<span class="hljs-string">'mean'</span>, <span class="hljs-string">'min'</span>, <span class="hljs-string">'max'</span>, <span class="hljs-string">'quantile'</span>, <span class="hljs-string">'geometric_mean'</span>,
<span class="hljs-string">'harmonic_mean'</span>];</pre></div></div>
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<p>create a closure with a method name so that a reference
like <code>arrayMethods[i]</code> doesnt follow the loop increment</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">wrap</span><span class="hljs-params">(method)</span> {</span>
<span class="hljs-keyword">return</span> <span class="hljs-function"><span class="hljs-keyword">function</span><span class="hljs-params">()</span> {</span></pre></div></div>
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<p>cast any arguments into an array, since theyre
natively objects</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> args = <span class="hljs-built_in">Array</span>.prototype.slice.apply(<span class="hljs-built_in">arguments</span>);</pre></div></div>
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<p>make the first argument the array itself</p>
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<div class="content"><div class='highlight'><pre> args.unshift(<span class="hljs-keyword">this</span>);</pre></div></div>
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<p>return the result of the ss method</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">return</span> ss[method].apply(ss, args);
};
}</pre></div></div>
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<p>select object to extend</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">var</span> extending;
<span class="hljs-keyword">if</span> (array) {</pre></div></div>
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<p>create a shallow copy of the array so that our internal
operations do not change it by reference</p>
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<div class="content"><div class='highlight'><pre> extending = array.slice();
} <span class="hljs-keyword">else</span> {
extending = <span class="hljs-built_in">Array</span>.prototype;
}</pre></div></div>
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<p>for each array function, define a function that gets
the array as the first argument.
We use <a href="https://developer.mozilla.org/en-US/docs/JavaScript/Reference/Global_Objects/Object/defineProperty">defineProperty</a>
because it allows these properties to be non-enumerable:
<code>for (var in x)</code> loops will not run into problems with this
implementation.</p>
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<div class="content"><div class='highlight'><pre> <span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; arrayMethods.length; i++) {
<span class="hljs-built_in">Object</span>.defineProperty(extending, arrayMethods[i], {
value: wrap(arrayMethods[i]),
configurable: <span class="hljs-literal">true</span>,
enumerable: <span class="hljs-literal">false</span>,
writable: <span class="hljs-literal">true</span>
});
}
<span class="hljs-keyword">return</span> extending;
}
ss.linear_regression = linear_regression;
ss.standard_deviation = standard_deviation;
ss.r_squared = r_squared;
ss.median = median;
ss.mean = mean;
ss.mode = mode;
ss.min = min;
ss.max = max;
ss.sum = sum;
ss.quantile = quantile;
ss.quantile_sorted = quantile_sorted;
ss.iqr = iqr;
ss.mad = mad;
ss.chunk = chunk;
ss.shuffle = shuffle;
ss.shuffle_in_place = shuffle_in_place;
ss.sample = sample;
ss.sample_covariance = sample_covariance;
ss.sample_correlation = sample_correlation;
ss.sample_variance = sample_variance;
ss.sample_standard_deviation = sample_standard_deviation;
ss.sample_skewness = sample_skewness;
ss.geometric_mean = geometric_mean;
ss.harmonic_mean = harmonic_mean;
ss.variance = variance;
ss.t_test = t_test;
ss.t_test_two_sample = t_test_two_sample;</pre></div></div>
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<p>jenks</p>
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<div class="content"><div class='highlight'><pre> ss.jenksMatrices = jenksMatrices;
ss.jenksBreaks = jenksBreaks;
ss.jenks = jenks;
ss.bayesian = bayesian;</pre></div></div>
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<p>Distribution-related methods</p>
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<div class="content"><div class='highlight'><pre> ss.epsilon = epsilon; <span class="hljs-comment">// We make ε available to the test suite.</span>
ss.factorial = factorial;
ss.bernoulli_distribution = bernoulli_distribution;
ss.binomial_distribution = binomial_distribution;
ss.poisson_distribution = poisson_distribution;
ss.chi_squared_goodness_of_fit = chi_squared_goodness_of_fit;</pre></div></div>
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<p>Normal distribution</p>
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<div class="content"><div class='highlight'><pre> ss.z_score = z_score;
ss.cumulative_std_normal_probability = cumulative_std_normal_probability;
ss.standard_normal_table = standard_normal_table;</pre></div></div>
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<p>Alias this into its common name</p>
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<div class="content"><div class='highlight'><pre> ss.average = mean;
ss.interquartile_range = iqr;
ss.mixin = mixin;
ss.median_absolute_deviation = mad;
})(<span class="hljs-keyword">this</span>);</pre></div></div>
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