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+ <a name="page_title"></a><section id="page_title">
+<h2></h2>
+
+<h1>Apache Commons Statistics User Guide</h1>
+ </section>
+
+ <a name="toc"></a><section id="toc">
+<h2><a name="Contents"></a>Contents</h2>
+
+<ul>
+
+<li>
+ <a href="#overview">Overview</a>
+ </li>
+
+<li>
+ <a href="#example-modules">Example Modules</a>
+ </li>
+
+<li>
+ <a href="#distributions">Probability Distributions</a>
+
+<ul>
+
+<li>
+ <a href="#dist_overview">Overview</a>
+ </li>
+
+<li>
+ <a href="#dist_api">API</a>
+ </li>
+
+<li>
+ <a href="#dist_imp_details">Implementation Details</a>
+ </li>
+
+<li>
+ <a href="#dist_complements">Complementary Probabilities</a>
+ </li>
+ </ul>
+ </li>
+
+ </ul>
+ </section>
+
+ <a name="overview"></a><section id="overview">
+<h2><a name="Overview"></a>Overview</h2>
+
+<p>
+ Apache Commons Statistics provides utilities for statistical applications. The code
+ originated in the <code><a class="externalLink" href="https://commons.apache.org/proper/commons-math/">
+ commons-math</a></code> project but was pulled out into a separate project for better
+ maintainability and has since undergone numerous improvements.
+ </p>
+
+
+<p>
+ Commons Statistics is divided into a number of submodules:
+ </p>
+
+<ul>
+
+<li>
+ <code><a href="../commons-statistics-distribution/index.html">
+ commons-statistics-distribution</a></code> - Provides interfaces
+ and classes for probability distributions.
+ </li>
+ </ul>
+ </section>
+
+ <a name="example-modules"></a><section id="example-modules">
+<h2><a name="Example_Modules"></a>Example Modules</h2>
+
+<p>
+ In addition to the modules above, the Commons Statistics
+ <a class="externalLink" href="https://commons.apache.org/statistics/download_statistics.cgi">source distribution</a>
+ contains example code demonstrating library functionality and/or providing useful
+ development utilities. These modules are not part of the public API of the library and no
+ guarantees are made concerning backwards compatibility. The
+ <a href="../commons-statistics-examples/modules.html">example module parent page</a>
+ contains a listing of available modules.
+ </p>
+ <hr />
+ </section>
+
+ <a name="distributions"></a><section id="distributions">
+<h2><a name="Probability_Distributions"></a>Probability Distributions</h2>
+ <a name="dist_overview"></a><section id="dist_overview">
+<h3><a name="Overview"></a>Overview</h3>
+
+<p>
+ The <code>commons-statistics-distribution</code> module provides a framework and implementations for some commonly used
+ probability distributions. Continuous univariate distributions are represented by
+ implementations of the
+ <a href="../commons-statistics-distribution/apidocs/org/apache/commons/statistics/distribution/ContinuousDistribution.html">ContinuousDistribution</a>
+ interface. Discrete distributions implement
+ <a href="../commons-statistics-distribution/apidocs/org/apache/commons/statistics/distribution/DiscreteDistribution.html">DiscreteDistribution</a>
+ (values must be mapped to integers).
+ </p>
+ </section>
+ <a name="dist_api"></a><section id="dist_api">
+<h3><a name="API"></a>API</h3>
+
+<p>
+ The distribution framework provides the means to compute probability density,
+ probability mass and cumulative probability functions for several well-known
+ discrete (integer-valued) and continuous probability distributions.
+ The API also allows for the computation of inverse cumulative probabilities
+ and sampling from distributions.
+ </p>
+
+<p>
+ For an instance <code>f</code> of a distribution <code>F</code>,
+ and a domain value, <code>x</code>, <code>f.cumulativeProbability(x)</code>
+ computes <code>P(X <= x)</code> where <code>X</code> is a random variable distributed
+ as <code>F</code>. The complement of the cumulative probability,
+ <code>f.survivalProbability(x)</code> computes <code>P(X > x)</code>. Note that
+ the survival probability is approximately equal to <code>1 - P(X <= x)</code> but
+ does not suffer from cancellation error as the cumulative probability approaches 1.
+ The cancellation error may cause a (total) loss of accuracy when
+ <code>P(X <= x) ~ 1</code>
+ (see <a href="#complements">complementary probabilities</a>).
+ </p>
+
+<div class="source">
+<pre class="prettyprint">
+TDistribution t = TDistribution.of(29);
+double lowerTail = t.cumulativeProbability(-2.656); // P(T(29) <= -2.656)
+double upperTail = t.survivalProbability(2.75); // P(T(29) > 2.75)
+</pre></div>
+
+<p>
+ For <a href="../commons-statistics-distribution/apidocs/org/apache/commons/statistics/distribution/DiscreteDistribution.html">discrete</a>
+ <code>F</code>, the probability mass function is given by <code>f.probability(x)</code>.
+ For <a href="../commons-statistics-distribution/apidocs/org/apache/commons/statistics/distribution/ContinuousDistribution.html">continuous</a>
+ <code>F</code>, the probability density function is given by <code>f.density(x)</code>.
+ Distributions also implement <code>f.probability(x1, x2)</code> for computing
+ <code>P(x1 < X <= x2)</code>.
+ </p>
+
+<div class="source">
+<pre class="prettyprint">
+PoissonDistribution pd = PoissonDistribution.of(1.23);
+double p1 = pd.probability(5);
+double p2 = pd.probability(5, 5);
+double p3 = pd.probability(4, 5);
+// p2 == 0
+// p1 == p3
+</pre></div>
+
+<p>
+ Inverse distribution functions can be computed using the
+ <code>inverseCumulativeProbability</code> and <code>inverseSurvivalProbability</code>
+ methods. For continuous <code>f</code> and <code>p</code> a probability,
+ <code>f.inverseCumulativeProbability(p)</code> returns
+ </p>
+
+<p>
+ \[ x = \begin{cases}
+ \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\
+ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0
+ \end{cases} \]
+ </p>
+
+<p>
+ where <code>X</code> is distributed as <code>F</code>.<br />
+ Likewise <code>f.inverseSurvivalProbability(p)</code> returns
+ </p>
+
+<p>
+ \[ x = \begin{cases}
+ \inf \{ x \in \mathbb R : P(X \ge x) \le p\} & \text{for } 0 \le p \lt 1 \\
+ \inf \{ x \in \mathbb R : P(X \ge x) \lt 1 \} & \text{for } p = 1
+ \end{cases} \]
+ </p>
+
+<div class="source">
+<pre class="prettyprint">
+NormalDistribution n = NormalDistribution.of(0, 1);
+double x1 = n.inverseCumulativeProbability(1e-300);
+double x2 = n.inverseSurvivalProbability(1e-300);
+// x1 == -x2 ~ -37.0471
+</pre></div>
+
+<p>
+ For discrete <code>F</code>, the definition is the same, with \( \mathbb Z \)
+ (the integers) in place of \( \mathbb R \) (but note that, in the discrete case,
+ the ≥ in the definition can make a difference when <code>p</code> is an attained
+ value of the distribution).
+ </p>
+
+<p>
+ All distributions provide accessors for the parameters used to create the distribution,
+ and a mean and variance. The return value when the mean or variance
+ is undefined is noted in the class javadoc.
+ </p>
+
+<div class="source">
+<pre class="prettyprint">
+ChiSquaredDistribution chi2 = ChiSquaredDistribution.of(42);
+double df = chi2.getDegreesOfFreedom(); // 42
+double mean = chi2.getMean(); // 42
+double variance = chi2.getVariance(); // 84
+
+CauchyDistribution cauchy = CauchyDistribution.of(1.23, 4.56);
+double location = cauchy.getLocation(); // 1.23
+double scale = cauchy.getScale(); // 4.56
+double undefined1 = cauchy.getMean(); // NaN
+double undefined2 = cauchy.getVariance(); // NaN
+</pre></div>
+
+<p>
+ The supported domain of the distribution is provided by the
+ <code>getSupportLowerBound</code> and <code>getSupportUpperBound</code> methods.
+ </p>
+
+<div class="source">
+<pre class="prettyprint">
+BinomialDistribution b = BinomialDistribution.of(13, 0.15);
+int lower = b.getSupportLowerBound(); // 0
+int upper = b.getSupportUpperBound(); // 13
+</pre></div>
+
+<p>
+ All distributions implement a <code>createSampler(UniformRandomProvider rng)</code>
+ method to support random sampling from the distribution, where <code>UniformRandomProvider</code>
+ is an interface defined in <a class="externalLink" href="https://commons.apache.org/proper/commons-rng/">Commons RNG</a>.
+ The sampler is a functional interface whose functional method is <code>sample()</code>,
+ suitable for generation of <code>double</code> or <code>int</code> samples.
+ Default <code>samples()</code> methods are provided to create a
+ <code>DoubleStream</code> or <code>IntStream</code>.
+ </p>
+
+<div class="source">
+<pre class="prettyprint">
+// From Commons RNG Simple
+UniformRandomProvider rng = RandomSource.KISS.create(123L);
+
+NormalDistribution n = NormalDistribution.of(0, 1);
+double x = n.createSampler(rng).sample();
+
+// Generate a number of samples
+GeometricDistribution g = GeometricDistribution.of(0.75);
+int[] k = g.createSampler(rng).samples(100).toArray();
+// k.length == 100
+</pre></div>
+
+<p>
+ Note that even when distributions are immutable, the sampler is not immutable as it
+ depends on the instance of the mutable <code>UniformRandomProvider</code>. Generation of
+ many samples in a multi-threaded application should use a separate instance of
+ <code>UniformRandomProvider</code> per thread. Any synchronization should be avoided
+ for best performance. By default the streams returned from the <code>samples()</code>
+ methods are sequential.
+ </p>
+ </section>
+ <a name="dist_imp_details"></a><section id="dist_imp_details">
+<h3><a name="Implementation_Details"></a>Implementation Details</h3>
+
+<p>
+ Instances are constructed using factory methods, typically a static method in the
+ distribution class named <code>of</code>. This allows the returned instance
+ to be specialised to the distribution parameters.
+ </p>
+
+<p>
+ Exceptions will be raised by the factory method when constructing the distribution
+ using invalid parameters. See the class javadoc for exception conditions.
+ </p>
+
+<p>
+ Unless otherwise noted, distribution instances are immutable. This allows sharing
+ an instance between threads for computations.
+ </p>
+
+<p>
+ Exceptions will not be raised by distributions for an invalid <code>x</code> argument
+ to probability functions. Typically the cumulative probability functions will return
+ 0 or 1 for an out-of-domain argument, depending on which the side of the domain bound
+ the argument falls, and the density or probability mass functions return 0.
+ Return values for <code>x</code> arguments when the result is
+ undefined should be documented in the class javadoc. For example the beta distribution
+ is undefined for <code>x = 0, alpha < 1</code> or <code>x = 1, beta < 1</code>.
+ Note: This out-of-domain behaviour may be different from distributions in the
+ <code>org.apache.commons.math3.distribution</code> package. Users upgrading from
+ <code><a class="externalLink" href="https://commons.apache.org/proper/commons-math/">commons-math</a></code>
+ should check the appropriate class javadoc.
+ </p>
+
+<p>
+ An exception will be raised by distributions for an invalid <code>p</code> argument
+ to inverse probability functions. The argument must be in the range <code>[0, 1]</code>.
+ </p>
+ </section>
+ <a name="dist_complements"></a><section id="dist_complements">
+<h3><a name="Complementary_Probabilities"></a>Complementary Probabilities</h3>
+
+<p>
+ The distributions provide the cumulative probability <code>p</code> and its complement,
+ the survival probability, <code>q = 1 - p</code>. When the probability
+ <code>q</code> is small use of the cumulative probability to compute <code>q</code> can
+ result in dramatic loss of accuracy. This is due to the distribution of floating-point
+ numbers having a
+ <a class="externalLink" href="https://en.wikipedia.org/wiki/Reciprocal_distribution">log-uniform</a>
+ distribution as the limiting distribution. There are far more
+ representable numbers as the probability value approaches zero than when it approaches
+ one.
+ </p>
+
+<p>
+ The difference is illustrated with the result of computing the upper tail of a
+ probability distribution.
+ </p>
+
+<div class="source">
+<pre class="prettyprint">
+ChiSquaredDistribution chi2 = ChiSquaredDistribution.of(42);
+double q1 = 1 - chi2.cumulativeProbability(168);
+double q2 = chi2.survivalProbability(168);
+// q1 == 0
+// q2 != 0
+</pre></div>
+
+<p>
+ In this case the value <code>1 - p</code> has only a single bit of information as
+ <code>x</code> approaches 168. For example the value <code>1 - p(x=167)</code>
+ is <code>2<sup>-53</sup></code> (or approximately <code>1.11e-16</code>).
+ The complement <code>q</code> retains information
+ much further into the long tail as shown in the following table:
+ </p>
+
+<table class="bodyTable" border="1" style="width: auto">
+
+<tr class="a">
+<th colspan="3"><font size="+1">Chi-squared distribution, 42 degrees of freedom</font></th></tr>
+
+<tr class="b">
+<th>x</th>
+<th>1 - p</th>
+<th>q</th></tr>
+
+<tr class="a">
+<td align="left">166</td>
+<td>1.11e-16</td>
+<td>1.16e-16</td></tr>
+
+<tr class="b">
+<td align="left">167</td>
+<td>1.11e-16</td>
+<td>7.96e-17</td></tr>
+
+<tr class="a">
+<td align="left">168</td>
+<td>0</td>
+<td>5.43e-17</td></tr>
+
+<tr class="b">
+<td align="left">...</td>
+<td></td>
+<td></td></tr>
+
+<tr class="a">
+<td align="left">200</td>
+<td>0</td>
+<td>1.19e-22</td></tr>
+ </table>
+
+<p>
+ Probability computations should use the appropriate cumulative or survival function
+ to calculate the lower or upper tail respectively. The same care should be applied
+ when inverting probability distributions. It is preferred to compute either
+ <code>p ≤ 0.5</code> or <code>q ≤ 0.5</code> without loss of accuracy and then
+ invert respectively the cumulative probability using <code>p</code> or the survival
+ probabilty using <code>q</code> to obtain <code>x</code>.
+ </p>
+
+<div class="source">
+<pre class="prettyprint">
+ChiSquaredDistribution chi2 = ChiSquaredDistribution.of(42);
+double q = 5.43e-17;
+// Incorrect: p = 1 - q == 1.0 !!!
+double x1 = chi2.inverseCumulativeProbability(1 - q);
+// Correct: invert q
+double x2 = chi2.inverseSurvivalProbability(q);
+// x1 == +infinity
+// x2 ~ 168.0
+</pre></div>
+
+<p>
+ Note: The survival probability functions were not present in the
+ <code>org.apache.commons.math3.distribution</code> package. Users upgrading from
+ <code><a class="externalLink" href="https://commons.apache.org/proper/commons-math/">commons-math</a></code>
+ should update usage of the cumulative probability functions where appropriate.
+ </p>
+ </section>
+ </section>
+
+
+
+
+ </td>
+ </tr>
+ </table>
+ </div>
+
+ <div class="footer">
+ <p>Copyright © 2018-2022
+ <a href="https://www.apache.org/">The Apache Software Foundation</a>.
+ All Rights Reserved.</p>
+
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Added: dev/commons/statistics/1.0-RC1/site/xref-test/allclasses-frame.html
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--- dev/commons/statistics/1.0-RC1/site/xref-test/allclasses-frame.html (added)
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+ <body>
+ <h1 class="bar">All Classes</h1>
+ <div class="indexContainer">
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+ <li><a href="org/apache/commons/statistics/distribution/ZipfDistributionTest.html#ZipfDistributionTest" target="classFrame">ZipfDistributionTest</a></li>
+ </ul>
+ </div>
+ </body>
+</html>
Added: dev/commons/statistics/1.0-RC1/site/xref-test/index.html
==============================================================================
--- dev/commons/statistics/1.0-RC1/site/xref-test/index.html (added)
+++ dev/commons/statistics/1.0-RC1/site/xref-test/index.html Thu Dec 1 16:47:12 2022
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