You are viewing a plain text version of this content. The canonical link for it is here.
Posted to commits@commons.apache.org by ah...@apache.org on 2022/12/01 16:47:13 UTC
svn commit: r58392 [44/50] - in /dev/commons/statistics/1.0-RC1: ./ binaries/ site/ site/css/ site/images/ site/images/logos/ site/img/ site/js/ site/release-notes/ site/style/ site/userguide/ site/xref-test/ site/xref-test/org/ site/xref-test/org/apac...
Added: dev/commons/statistics/1.0-RC1/site/xref/org/apache/commons/statistics/distribution/TDistribution.html
==============================================================================
--- dev/commons/statistics/1.0-RC1/site/xref/org/apache/commons/statistics/distribution/TDistribution.html (added)
+++ dev/commons/statistics/1.0-RC1/site/xref/org/apache/commons/statistics/distribution/TDistribution.html Thu Dec 1 16:47:12 2022
@@ -0,0 +1,347 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
+<head><meta http-equiv="content-type" content="text/html; charset=UTF-8" />
+<title>TDistribution xref</title>
+<link type="text/css" rel="stylesheet" href="../../../../../stylesheet.css" />
+</head>
+<body>
+<div id="overview"><a href="../../../../../../apidocs/org/apache/commons/statistics/distribution/TDistribution.html">View Javadoc</a></div><pre>
+<a class="jxr_linenumber" name="L1" href="#L1">1</a> <em class="jxr_comment">/*</em>
+<a class="jxr_linenumber" name="L2" href="#L2">2</a> <em class="jxr_comment"> * Licensed to the Apache Software Foundation (ASF) under one or more</em>
+<a class="jxr_linenumber" name="L3" href="#L3">3</a> <em class="jxr_comment"> * contributor license agreements. See the NOTICE file distributed with</em>
+<a class="jxr_linenumber" name="L4" href="#L4">4</a> <em class="jxr_comment"> * this work for additional information regarding copyright ownership.</em>
+<a class="jxr_linenumber" name="L5" href="#L5">5</a> <em class="jxr_comment"> * The ASF licenses this file to You under the Apache License, Version 2.0</em>
+<a class="jxr_linenumber" name="L6" href="#L6">6</a> <em class="jxr_comment"> * (the "License"); you may not use this file except in compliance with</em>
+<a class="jxr_linenumber" name="L7" href="#L7">7</a> <em class="jxr_comment"> * the License. You may obtain a copy of the License at</em>
+<a class="jxr_linenumber" name="L8" href="#L8">8</a> <em class="jxr_comment"> *</em>
+<a class="jxr_linenumber" name="L9" href="#L9">9</a> <em class="jxr_comment"> * <a href="http://www.apache.org/licenses/LICENSE-2.0" target="alexandria_uri">http://www.apache.org/licenses/LICENSE-2.0</a></em>
+<a class="jxr_linenumber" name="L10" href="#L10">10</a> <em class="jxr_comment"> *</em>
+<a class="jxr_linenumber" name="L11" href="#L11">11</a> <em class="jxr_comment"> * Unless required by applicable law or agreed to in writing, software</em>
+<a class="jxr_linenumber" name="L12" href="#L12">12</a> <em class="jxr_comment"> * distributed under the License is distributed on an "AS IS" BASIS,</em>
+<a class="jxr_linenumber" name="L13" href="#L13">13</a> <em class="jxr_comment"> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</em>
+<a class="jxr_linenumber" name="L14" href="#L14">14</a> <em class="jxr_comment"> * See the License for the specific language governing permissions and</em>
+<a class="jxr_linenumber" name="L15" href="#L15">15</a> <em class="jxr_comment"> * limitations under the License.</em>
+<a class="jxr_linenumber" name="L16" href="#L16">16</a> <em class="jxr_comment"> */</em>
+<a class="jxr_linenumber" name="L17" href="#L17">17</a> <strong class="jxr_keyword">package</strong> org.apache.commons.statistics.distribution;
+<a class="jxr_linenumber" name="L18" href="#L18">18</a>
+<a class="jxr_linenumber" name="L19" href="#L19">19</a> <strong class="jxr_keyword">import</strong> org.apache.commons.numbers.gamma.RegularizedBeta;
+<a class="jxr_linenumber" name="L20" href="#L20">20</a> <strong class="jxr_keyword">import</strong> org.apache.commons.rng.UniformRandomProvider;
+<a class="jxr_linenumber" name="L21" href="#L21">21</a> <strong class="jxr_keyword">import</strong> org.apache.commons.rng.sampling.distribution.TSampler;
+<a class="jxr_linenumber" name="L22" href="#L22">22</a> <strong class="jxr_keyword">import</strong> org.apache.commons.numbers.gamma.Beta;
+<a class="jxr_linenumber" name="L23" href="#L23">23</a> <strong class="jxr_keyword">import</strong> org.apache.commons.numbers.gamma.LogBeta;
+<a class="jxr_linenumber" name="L24" href="#L24">24</a>
+<a class="jxr_linenumber" name="L25" href="#L25">25</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L26" href="#L26">26</a> <em class="jxr_javadoccomment"> * Implementation of Student's t-distribution.</em>
+<a class="jxr_linenumber" name="L27" href="#L27">27</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L28" href="#L28">28</a> <em class="jxr_javadoccomment"> * <p>The probability density function of \( X \) is:</em>
+<a class="jxr_linenumber" name="L29" href="#L29">29</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L30" href="#L30">30</a> <em class="jxr_javadoccomment"> * <p>\[ f(x; v) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{\!-\frac{\nu+1}{2}} \]</em>
+<a class="jxr_linenumber" name="L31" href="#L31">31</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L32" href="#L32">32</a> <em class="jxr_javadoccomment"> * <p>for \( v &gt; 0 \) the degrees of freedom,</em>
+<a class="jxr_linenumber" name="L33" href="#L33">33</a> <em class="jxr_javadoccomment"> * \( \Gamma \) is the gamma function, and</em>
+<a class="jxr_linenumber" name="L34" href="#L34">34</a> <em class="jxr_javadoccomment"> * \( x \in (-\infty, \infty) \).</em>
+<a class="jxr_linenumber" name="L35" href="#L35">35</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L36" href="#L36">36</a> <em class="jxr_javadoccomment"> * @see <a href="<a href="https://en.wikipedia.org/wiki/Student%27s_t-distribution" target="alexandria_uri">https://en.wikipedia.org/wiki/Student%27s_t-distribution</a>">Student&#39;s t-distribution (Wikipedia)</a></em>
+<a class="jxr_linenumber" name="L37" href="#L37">37</a> <em class="jxr_javadoccomment"> * @see <a href="<a href="https://mathworld.wolfram.com/Studentst-Distribution.html" target="alexandria_uri">https://mathworld.wolfram.com/Studentst-Distribution.html</a>">Student&#39;s t-distribution (MathWorld)</a></em>
+<a class="jxr_linenumber" name="L38" href="#L38">38</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L39" href="#L39">39</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">abstract</strong> <strong class="jxr_keyword">class</strong> <a name="TDistribution" href="../../../../../org/apache/commons/statistics/distribution/TDistribution.html#TDistribution">TDistribution</a> <strong class="jxr_keyword">extends</strong> <a name="AbstractContinuousDistribution" href="../../../../../org/apache/commons/statistics/distribution/AbstractContinuousDistribution.html#AbstractContinuousDistribution">AbstractContinuousDistribution</a> {
+<a class="jxr_linenumber" name="L40" href="#L40">40</a> <em class="jxr_javadoccomment">/** A standard normal distribution used for calculations.</em>
+<a class="jxr_linenumber" name="L41" href="#L41">41</a> <em class="jxr_javadoccomment"> * This is immutable and thread-safe and can be used across instances. */</em>
+<a class="jxr_linenumber" name="L42" href="#L42">42</a> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">final</strong> <a name="NormalDistribution" href="../../../../../org/apache/commons/statistics/distribution/NormalDistribution.html#NormalDistribution">NormalDistribution</a> STANDARD_NORMAL = NormalDistribution.of(0, 1);
+<a class="jxr_linenumber" name="L43" href="#L43">43</a>
+<a class="jxr_linenumber" name="L44" href="#L44">44</a> <em class="jxr_javadoccomment">/** The degrees of freedom. */</em>
+<a class="jxr_linenumber" name="L45" href="#L45">45</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> degreesOfFreedom;
+<a class="jxr_linenumber" name="L46" href="#L46">46</a>
+<a class="jxr_linenumber" name="L47" href="#L47">47</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L48" href="#L48">48</a> <em class="jxr_javadoccomment"> * Specialisation of the T-distribution used when there are infinite degrees of freedom.</em>
+<a class="jxr_linenumber" name="L49" href="#L49">49</a> <em class="jxr_javadoccomment"> * In this case the distribution matches a normal distribution. This is used when the</em>
+<a class="jxr_linenumber" name="L50" href="#L50">50</a> <em class="jxr_javadoccomment"> * variance is not different from 1.0.</em>
+<a class="jxr_linenumber" name="L51" href="#L51">51</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L52" href="#L52">52</a> <em class="jxr_javadoccomment"> * <p>This delegates all methods to the standard normal distribution. Instances are</em>
+<a class="jxr_linenumber" name="L53" href="#L53">53</a> <em class="jxr_javadoccomment"> * allowed to provide access to the degrees of freedom used during construction.</em>
+<a class="jxr_linenumber" name="L54" href="#L54">54</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L55" href="#L55">55</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">class</strong> NormalTDistribution <strong class="jxr_keyword">extends</strong> <a name="TDistribution" href="../../../../../org/apache/commons/statistics/distribution/TDistribution.html#TDistribution">TDistribution</a> {
+<a class="jxr_linenumber" name="L56" href="#L56">56</a>
+<a class="jxr_linenumber" name="L57" href="#L57">57</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L58" href="#L58">58</a> <em class="jxr_javadoccomment"> * @param degreesOfFreedom Degrees of freedom.</em>
+<a class="jxr_linenumber" name="L59" href="#L59">59</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L60" href="#L60">60</a> NormalTDistribution(<strong class="jxr_keyword">double</strong> degreesOfFreedom) {
+<a class="jxr_linenumber" name="L61" href="#L61">61</a> <strong class="jxr_keyword">super</strong>(degreesOfFreedom);
+<a class="jxr_linenumber" name="L62" href="#L62">62</a> }
+<a class="jxr_linenumber" name="L63" href="#L63">63</a>
+<a class="jxr_linenumber" name="L64" href="#L64">64</a> @Override
+<a class="jxr_linenumber" name="L65" href="#L65">65</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> density(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L66" href="#L66">66</a> <strong class="jxr_keyword">return</strong> STANDARD_NORMAL.density(x);
+<a class="jxr_linenumber" name="L67" href="#L67">67</a> }
+<a class="jxr_linenumber" name="L68" href="#L68">68</a>
+<a class="jxr_linenumber" name="L69" href="#L69">69</a> @Override
+<a class="jxr_linenumber" name="L70" href="#L70">70</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> probability(<strong class="jxr_keyword">double</strong> x0, <strong class="jxr_keyword">double</strong> x1) {
+<a class="jxr_linenumber" name="L71" href="#L71">71</a> <strong class="jxr_keyword">return</strong> STANDARD_NORMAL.probability(x0, x1);
+<a class="jxr_linenumber" name="L72" href="#L72">72</a> }
+<a class="jxr_linenumber" name="L73" href="#L73">73</a>
+<a class="jxr_linenumber" name="L74" href="#L74">74</a> @Override
+<a class="jxr_linenumber" name="L75" href="#L75">75</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> logDensity(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L76" href="#L76">76</a> <strong class="jxr_keyword">return</strong> STANDARD_NORMAL.logDensity(x);
+<a class="jxr_linenumber" name="L77" href="#L77">77</a> }
+<a class="jxr_linenumber" name="L78" href="#L78">78</a>
+<a class="jxr_linenumber" name="L79" href="#L79">79</a> @Override
+<a class="jxr_linenumber" name="L80" href="#L80">80</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> cumulativeProbability(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L81" href="#L81">81</a> <strong class="jxr_keyword">return</strong> STANDARD_NORMAL.cumulativeProbability(x);
+<a class="jxr_linenumber" name="L82" href="#L82">82</a> }
+<a class="jxr_linenumber" name="L83" href="#L83">83</a>
+<a class="jxr_linenumber" name="L84" href="#L84">84</a> @Override
+<a class="jxr_linenumber" name="L85" href="#L85">85</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> inverseCumulativeProbability(<strong class="jxr_keyword">double</strong> p) {
+<a class="jxr_linenumber" name="L86" href="#L86">86</a> <strong class="jxr_keyword">return</strong> STANDARD_NORMAL.inverseCumulativeProbability(p);
+<a class="jxr_linenumber" name="L87" href="#L87">87</a> }
+<a class="jxr_linenumber" name="L88" href="#L88">88</a>
+<a class="jxr_linenumber" name="L89" href="#L89">89</a> <em class="jxr_comment">// Survival probability functions inherit the symmetry operations from the TDistribution</em>
+<a class="jxr_linenumber" name="L90" href="#L90">90</a>
+<a class="jxr_linenumber" name="L91" href="#L91">91</a> @Override
+<a class="jxr_linenumber" name="L92" href="#L92">92</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getMean() {
+<a class="jxr_linenumber" name="L93" href="#L93">93</a> <strong class="jxr_keyword">return</strong> 0;
+<a class="jxr_linenumber" name="L94" href="#L94">94</a> }
+<a class="jxr_linenumber" name="L95" href="#L95">95</a>
+<a class="jxr_linenumber" name="L96" href="#L96">96</a> @Override
+<a class="jxr_linenumber" name="L97" href="#L97">97</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getVariance() {
+<a class="jxr_linenumber" name="L98" href="#L98">98</a> <strong class="jxr_keyword">return</strong> 1.0;
+<a class="jxr_linenumber" name="L99" href="#L99">99</a> }
+<a class="jxr_linenumber" name="L100" href="#L100">100</a>
+<a class="jxr_linenumber" name="L101" href="#L101">101</a> @Override
+<a class="jxr_linenumber" name="L102" href="#L102">102</a> <strong class="jxr_keyword">public</strong> Sampler createSampler(UniformRandomProvider rng) {
+<a class="jxr_linenumber" name="L103" href="#L103">103</a> <strong class="jxr_keyword">return</strong> STANDARD_NORMAL.createSampler(rng);
+<a class="jxr_linenumber" name="L104" href="#L104">104</a> }
+<a class="jxr_linenumber" name="L105" href="#L105">105</a> }
+<a class="jxr_linenumber" name="L106" href="#L106">106</a>
+<a class="jxr_linenumber" name="L107" href="#L107">107</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L108" href="#L108">108</a> <em class="jxr_javadoccomment"> * Implementation of Student's T-distribution.</em>
+<a class="jxr_linenumber" name="L109" href="#L109">109</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L110" href="#L110">110</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">class</strong> StudentsTDistribution <strong class="jxr_keyword">extends</strong> <a name="TDistribution" href="../../../../../org/apache/commons/statistics/distribution/TDistribution.html#TDistribution">TDistribution</a> {
+<a class="jxr_linenumber" name="L111" href="#L111">111</a> <em class="jxr_javadoccomment">/** 2. */</em>
+<a class="jxr_linenumber" name="L112" href="#L112">112</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> TWO = 2;
+<a class="jxr_linenumber" name="L113" href="#L113">113</a> <em class="jxr_javadoccomment">/** The threshold for the density function where the</em>
+<a class="jxr_linenumber" name="L114" href="#L114">114</a> <em class="jxr_javadoccomment"> * power function base minus 1 is close to zero. */</em>
+<a class="jxr_linenumber" name="L115" href="#L115">115</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> CLOSE_TO_ZERO = 0.25;
+<a class="jxr_linenumber" name="L116" href="#L116">116</a>
+<a class="jxr_linenumber" name="L117" href="#L117">117</a> <em class="jxr_javadoccomment">/** -(v + 1) / 2, where v = degrees of freedom. */</em>
+<a class="jxr_linenumber" name="L118" href="#L118">118</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> mvp1Over2;
+<a class="jxr_linenumber" name="L119" href="#L119">119</a> <em class="jxr_javadoccomment">/** Density normalisation factor, sqrt(v) * beta(1/2, v/2), where v = degrees of freedom. */</em>
+<a class="jxr_linenumber" name="L120" href="#L120">120</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> densityNormalisation;
+<a class="jxr_linenumber" name="L121" href="#L121">121</a> <em class="jxr_javadoccomment">/** Log density normalisation term, 0.5 * log(v) + log(beta(1/2, v/2)), where v = degrees of freedom. */</em>
+<a class="jxr_linenumber" name="L122" href="#L122">122</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> logDensityNormalisation;
+<a class="jxr_linenumber" name="L123" href="#L123">123</a> <em class="jxr_javadoccomment">/** Cached value for inverse probability function. */</em>
+<a class="jxr_linenumber" name="L124" href="#L124">124</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> mean;
+<a class="jxr_linenumber" name="L125" href="#L125">125</a> <em class="jxr_javadoccomment">/** Cached value for inverse probability function. */</em>
+<a class="jxr_linenumber" name="L126" href="#L126">126</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> variance;
+<a class="jxr_linenumber" name="L127" href="#L127">127</a>
+<a class="jxr_linenumber" name="L128" href="#L128">128</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L129" href="#L129">129</a> <em class="jxr_javadoccomment"> * @param degreesOfFreedom Degrees of freedom.</em>
+<a class="jxr_linenumber" name="L130" href="#L130">130</a> <em class="jxr_javadoccomment"> * @param variance Precomputed variance</em>
+<a class="jxr_linenumber" name="L131" href="#L131">131</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L132" href="#L132">132</a> StudentsTDistribution(<strong class="jxr_keyword">double</strong> degreesOfFreedom, <strong class="jxr_keyword">double</strong> variance) {
+<a class="jxr_linenumber" name="L133" href="#L133">133</a> <strong class="jxr_keyword">super</strong>(degreesOfFreedom);
+<a class="jxr_linenumber" name="L134" href="#L134">134</a>
+<a class="jxr_linenumber" name="L135" href="#L135">135</a> mvp1Over2 = -0.5 * (degreesOfFreedom + 1);
+<a class="jxr_linenumber" name="L136" href="#L136">136</a> densityNormalisation = Math.sqrt(degreesOfFreedom) * Beta.value(0.5, degreesOfFreedom / 2);
+<a class="jxr_linenumber" name="L137" href="#L137">137</a> logDensityNormalisation = 0.5 * Math.log(degreesOfFreedom) + LogBeta.value(0.5, degreesOfFreedom / 2);
+<a class="jxr_linenumber" name="L138" href="#L138">138</a> mean = degreesOfFreedom > 1 ? 0 : Double.NaN;
+<a class="jxr_linenumber" name="L139" href="#L139">139</a> <strong class="jxr_keyword">this</strong>.variance = variance;
+<a class="jxr_linenumber" name="L140" href="#L140">140</a> }
+<a class="jxr_linenumber" name="L141" href="#L141">141</a>
+<a class="jxr_linenumber" name="L142" href="#L142">142</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L143" href="#L143">143</a> <em class="jxr_javadoccomment"> * @param degreesOfFreedom Degrees of freedom.</em>
+<a class="jxr_linenumber" name="L144" href="#L144">144</a> <em class="jxr_javadoccomment"> * @return the variance</em>
+<a class="jxr_linenumber" name="L145" href="#L145">145</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L146" href="#L146">146</a> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">double</strong> computeVariance(<strong class="jxr_keyword">double</strong> degreesOfFreedom) {
+<a class="jxr_linenumber" name="L147" href="#L147">147</a> <strong class="jxr_keyword">if</strong> (degreesOfFreedom == Double.POSITIVE_INFINITY) {
+<a class="jxr_linenumber" name="L148" href="#L148">148</a> <strong class="jxr_keyword">return</strong> 1;
+<a class="jxr_linenumber" name="L149" href="#L149">149</a> } <strong class="jxr_keyword">else</strong> <strong class="jxr_keyword">if</strong> (degreesOfFreedom > TWO) {
+<a class="jxr_linenumber" name="L150" href="#L150">150</a> <strong class="jxr_keyword">return</strong> degreesOfFreedom / (degreesOfFreedom - 2);
+<a class="jxr_linenumber" name="L151" href="#L151">151</a> } <strong class="jxr_keyword">else</strong> <strong class="jxr_keyword">if</strong> (degreesOfFreedom > 1) {
+<a class="jxr_linenumber" name="L152" href="#L152">152</a> <strong class="jxr_keyword">return</strong> Double.POSITIVE_INFINITY;
+<a class="jxr_linenumber" name="L153" href="#L153">153</a> } <strong class="jxr_keyword">else</strong> {
+<a class="jxr_linenumber" name="L154" href="#L154">154</a> <strong class="jxr_keyword">return</strong> Double.NaN;
+<a class="jxr_linenumber" name="L155" href="#L155">155</a> }
+<a class="jxr_linenumber" name="L156" href="#L156">156</a> }
+<a class="jxr_linenumber" name="L157" href="#L157">157</a>
+<a class="jxr_linenumber" name="L158" href="#L158">158</a> @Override
+<a class="jxr_linenumber" name="L159" href="#L159">159</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> density(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L160" href="#L160">160</a> <em class="jxr_comment">// 1 -(v+1)/2</em>
+<a class="jxr_linenumber" name="L161" href="#L161">161</a> <em class="jxr_comment">// ------------------- * (1 + t^2/v)</em>
+<a class="jxr_linenumber" name="L162" href="#L162">162</a> <em class="jxr_comment">// sqrt(v) B(1/2, v/2)</em>
+<a class="jxr_linenumber" name="L163" href="#L163">163</a>
+<a class="jxr_linenumber" name="L164" href="#L164">164</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> t2OverV = x * x / getDegreesOfFreedom();
+<a class="jxr_linenumber" name="L165" href="#L165">165</a> <strong class="jxr_keyword">if</strong> (t2OverV < CLOSE_TO_ZERO) {
+<a class="jxr_linenumber" name="L166" href="#L166">166</a> <em class="jxr_comment">// Avoid power function when the base is close to 1</em>
+<a class="jxr_linenumber" name="L167" href="#L167">167</a> <strong class="jxr_keyword">return</strong> Math.exp(Math.log1p(t2OverV) * mvp1Over2) / densityNormalisation;
+<a class="jxr_linenumber" name="L168" href="#L168">168</a> }
+<a class="jxr_linenumber" name="L169" href="#L169">169</a> <strong class="jxr_keyword">return</strong> Math.pow(1 + t2OverV, mvp1Over2) / densityNormalisation;
+<a class="jxr_linenumber" name="L170" href="#L170">170</a> }
+<a class="jxr_linenumber" name="L171" href="#L171">171</a>
+<a class="jxr_linenumber" name="L172" href="#L172">172</a> @Override
+<a class="jxr_linenumber" name="L173" href="#L173">173</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> logDensity(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L174" href="#L174">174</a> <strong class="jxr_keyword">return</strong> Math.log1p(x * x / getDegreesOfFreedom()) * mvp1Over2 - logDensityNormalisation;
+<a class="jxr_linenumber" name="L175" href="#L175">175</a> }
+<a class="jxr_linenumber" name="L176" href="#L176">176</a>
+<a class="jxr_linenumber" name="L177" href="#L177">177</a> @Override
+<a class="jxr_linenumber" name="L178" href="#L178">178</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> cumulativeProbability(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L179" href="#L179">179</a> <strong class="jxr_keyword">if</strong> (x == 0) {
+<a class="jxr_linenumber" name="L180" href="#L180">180</a> <strong class="jxr_keyword">return</strong> 0.5;
+<a class="jxr_linenumber" name="L181" href="#L181">181</a> }
+<a class="jxr_linenumber" name="L182" href="#L182">182</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> v = getDegreesOfFreedom();
+<a class="jxr_linenumber" name="L183" href="#L183">183</a>
+<a class="jxr_linenumber" name="L184" href="#L184">184</a> <em class="jxr_comment">// cdf(t) = 1 - 0.5 * I_x(t)(v/2, 1/2)</em>
+<a class="jxr_linenumber" name="L185" href="#L185">185</a> <em class="jxr_comment">// where x(t) = v / (v + t^2)</em>
+<a class="jxr_linenumber" name="L186" href="#L186">186</a> <em class="jxr_comment">//</em>
+<a class="jxr_linenumber" name="L187" href="#L187">187</a> <em class="jxr_comment">// When t^2 << v using the regularized beta results</em>
+<a class="jxr_linenumber" name="L188" href="#L188">188</a> <em class="jxr_comment">// in loss of precision. Use the complement instead:</em>
+<a class="jxr_linenumber" name="L189" href="#L189">189</a> <em class="jxr_comment">// I[x](a,b) = 1 - I[1-x](b,a)</em>
+<a class="jxr_linenumber" name="L190" href="#L190">190</a> <em class="jxr_comment">// x = v / (v + t^2)</em>
+<a class="jxr_linenumber" name="L191" href="#L191">191</a> <em class="jxr_comment">// 1-x = t^2 / (v + t^2)</em>
+<a class="jxr_linenumber" name="L192" href="#L192">192</a> <em class="jxr_comment">// Use the threshold documented in the Boost t-distribution:</em>
+<a class="jxr_linenumber" name="L193" href="#L193">193</a> <em class="jxr_comment">// https://www.boost.org/doc/libs/1_78_0/libs/math/doc/html/math_toolkit/dist_ref/dists/students_t_dist.html</em>
+<a class="jxr_linenumber" name="L194" href="#L194">194</a>
+<a class="jxr_linenumber" name="L195" href="#L195">195</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> t2 = x * x;
+<a class="jxr_linenumber" name="L196" href="#L196">196</a> <strong class="jxr_keyword">double</strong> z;
+<a class="jxr_linenumber" name="L197" href="#L197">197</a> <strong class="jxr_keyword">if</strong> (v < 2 * t2) {
+<a class="jxr_linenumber" name="L198" href="#L198">198</a> z = RegularizedBeta.value(v / (v + t2), v / 2, 0.5) / 2;
+<a class="jxr_linenumber" name="L199" href="#L199">199</a> } <strong class="jxr_keyword">else</strong> {
+<a class="jxr_linenumber" name="L200" href="#L200">200</a> z = RegularizedBeta.complement(t2 / (v + t2), 0.5, v / 2) / 2;
+<a class="jxr_linenumber" name="L201" href="#L201">201</a> }
+<a class="jxr_linenumber" name="L202" href="#L202">202</a> <strong class="jxr_keyword">return</strong> x > 0 ? 1 - z : z;
+<a class="jxr_linenumber" name="L203" href="#L203">203</a> }
+<a class="jxr_linenumber" name="L204" href="#L204">204</a>
+<a class="jxr_linenumber" name="L205" href="#L205">205</a> @Override
+<a class="jxr_linenumber" name="L206" href="#L206">206</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getMean() {
+<a class="jxr_linenumber" name="L207" href="#L207">207</a> <strong class="jxr_keyword">return</strong> mean;
+<a class="jxr_linenumber" name="L208" href="#L208">208</a> }
+<a class="jxr_linenumber" name="L209" href="#L209">209</a>
+<a class="jxr_linenumber" name="L210" href="#L210">210</a> @Override
+<a class="jxr_linenumber" name="L211" href="#L211">211</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getVariance() {
+<a class="jxr_linenumber" name="L212" href="#L212">212</a> <strong class="jxr_keyword">return</strong> variance;
+<a class="jxr_linenumber" name="L213" href="#L213">213</a> }
+<a class="jxr_linenumber" name="L214" href="#L214">214</a>
+<a class="jxr_linenumber" name="L215" href="#L215">215</a> @Override
+<a class="jxr_linenumber" name="L216" href="#L216">216</a> <strong class="jxr_keyword">double</strong> getMedian() {
+<a class="jxr_linenumber" name="L217" href="#L217">217</a> <em class="jxr_comment">// Overridden for the probability(double, double) method.</em>
+<a class="jxr_linenumber" name="L218" href="#L218">218</a> <em class="jxr_comment">// This is intentionally not a public method.</em>
+<a class="jxr_linenumber" name="L219" href="#L219">219</a> <strong class="jxr_keyword">return</strong> 0;
+<a class="jxr_linenumber" name="L220" href="#L220">220</a> }
+<a class="jxr_linenumber" name="L221" href="#L221">221</a>
+<a class="jxr_linenumber" name="L222" href="#L222">222</a> @Override
+<a class="jxr_linenumber" name="L223" href="#L223">223</a> <strong class="jxr_keyword">public</strong> Sampler createSampler(UniformRandomProvider rng) {
+<a class="jxr_linenumber" name="L224" href="#L224">224</a> <em class="jxr_comment">// T distribution sampler.</em>
+<a class="jxr_linenumber" name="L225" href="#L225">225</a> <strong class="jxr_keyword">return</strong> TSampler.of(rng, getDegreesOfFreedom())::sample;
+<a class="jxr_linenumber" name="L226" href="#L226">226</a> }
+<a class="jxr_linenumber" name="L227" href="#L227">227</a> }
+<a class="jxr_linenumber" name="L228" href="#L228">228</a>
+<a class="jxr_linenumber" name="L229" href="#L229">229</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L230" href="#L230">230</a> <em class="jxr_javadoccomment"> * @param degreesOfFreedom Degrees of freedom.</em>
+<a class="jxr_linenumber" name="L231" href="#L231">231</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L232" href="#L232">232</a> <strong class="jxr_keyword">private</strong> <a name="TDistribution" href="../../../../../org/apache/commons/statistics/distribution/TDistribution.html#TDistribution">TDistribution</a>(<strong class="jxr_keyword">double</strong> degreesOfFreedom) {
+<a class="jxr_linenumber" name="L233" href="#L233">233</a> <strong class="jxr_keyword">this</strong>.degreesOfFreedom = degreesOfFreedom;
+<a class="jxr_linenumber" name="L234" href="#L234">234</a> }
+<a class="jxr_linenumber" name="L235" href="#L235">235</a>
+<a class="jxr_linenumber" name="L236" href="#L236">236</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L237" href="#L237">237</a> <em class="jxr_javadoccomment"> * Creates a Student's t-distribution.</em>
+<a class="jxr_linenumber" name="L238" href="#L238">238</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L239" href="#L239">239</a> <em class="jxr_javadoccomment"> * @param degreesOfFreedom Degrees of freedom.</em>
+<a class="jxr_linenumber" name="L240" href="#L240">240</a> <em class="jxr_javadoccomment"> * @return the distribution</em>
+<a class="jxr_linenumber" name="L241" href="#L241">241</a> <em class="jxr_javadoccomment"> * @throws IllegalArgumentException if {@code degreesOfFreedom <= 0}</em>
+<a class="jxr_linenumber" name="L242" href="#L242">242</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L243" href="#L243">243</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">static</strong> <a name="TDistribution" href="../../../../../org/apache/commons/statistics/distribution/TDistribution.html#TDistribution">TDistribution</a> of(<strong class="jxr_keyword">double</strong> degreesOfFreedom) {
+<a class="jxr_linenumber" name="L244" href="#L244">244</a> <strong class="jxr_keyword">if</strong> (degreesOfFreedom <= 0) {
+<a class="jxr_linenumber" name="L245" href="#L245">245</a> <strong class="jxr_keyword">throw</strong> <strong class="jxr_keyword">new</strong> <a name="DistributionException" href="../../../../../org/apache/commons/statistics/distribution/DistributionException.html#DistributionException">DistributionException</a>(DistributionException.NOT_STRICTLY_POSITIVE,
+<a class="jxr_linenumber" name="L246" href="#L246">246</a> degreesOfFreedom);
+<a class="jxr_linenumber" name="L247" href="#L247">247</a> }
+<a class="jxr_linenumber" name="L248" href="#L248">248</a> <em class="jxr_comment">// If the variance converges to 1 use a NormalDistribution.</em>
+<a class="jxr_linenumber" name="L249" href="#L249">249</a> <em class="jxr_comment">// Occurs at 2^55 = 3.60e16</em>
+<a class="jxr_linenumber" name="L250" href="#L250">250</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> variance = StudentsTDistribution.computeVariance(degreesOfFreedom);
+<a class="jxr_linenumber" name="L251" href="#L251">251</a> <strong class="jxr_keyword">if</strong> (variance == 1) {
+<a class="jxr_linenumber" name="L252" href="#L252">252</a> <strong class="jxr_keyword">return</strong> <strong class="jxr_keyword">new</strong> NormalTDistribution(degreesOfFreedom);
+<a class="jxr_linenumber" name="L253" href="#L253">253</a> }
+<a class="jxr_linenumber" name="L254" href="#L254">254</a> <strong class="jxr_keyword">return</strong> <strong class="jxr_keyword">new</strong> StudentsTDistribution(degreesOfFreedom, variance);
+<a class="jxr_linenumber" name="L255" href="#L255">255</a> }
+<a class="jxr_linenumber" name="L256" href="#L256">256</a>
+<a class="jxr_linenumber" name="L257" href="#L257">257</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L258" href="#L258">258</a> <em class="jxr_javadoccomment"> * Gets the degrees of freedom parameter of this distribution.</em>
+<a class="jxr_linenumber" name="L259" href="#L259">259</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L260" href="#L260">260</a> <em class="jxr_javadoccomment"> * @return the degrees of freedom.</em>
+<a class="jxr_linenumber" name="L261" href="#L261">261</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L262" href="#L262">262</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getDegreesOfFreedom() {
+<a class="jxr_linenumber" name="L263" href="#L263">263</a> <strong class="jxr_keyword">return</strong> degreesOfFreedom;
+<a class="jxr_linenumber" name="L264" href="#L264">264</a> }
+<a class="jxr_linenumber" name="L265" href="#L265">265</a>
+<a class="jxr_linenumber" name="L266" href="#L266">266</a> <em class="jxr_javadoccomment">/** {@inheritDoc} */</em>
+<a class="jxr_linenumber" name="L267" href="#L267">267</a> @Override
+<a class="jxr_linenumber" name="L268" href="#L268">268</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> survivalProbability(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L269" href="#L269">269</a> <em class="jxr_comment">// Exploit symmetry</em>
+<a class="jxr_linenumber" name="L270" href="#L270">270</a> <strong class="jxr_keyword">return</strong> cumulativeProbability(-x);
+<a class="jxr_linenumber" name="L271" href="#L271">271</a> }
+<a class="jxr_linenumber" name="L272" href="#L272">272</a>
+<a class="jxr_linenumber" name="L273" href="#L273">273</a> <em class="jxr_javadoccomment">/** {@inheritDoc} */</em>
+<a class="jxr_linenumber" name="L274" href="#L274">274</a> @Override
+<a class="jxr_linenumber" name="L275" href="#L275">275</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> inverseSurvivalProbability(<strong class="jxr_keyword">double</strong> p) {
+<a class="jxr_linenumber" name="L276" href="#L276">276</a> <em class="jxr_comment">// Exploit symmetry</em>
+<a class="jxr_linenumber" name="L277" href="#L277">277</a> <strong class="jxr_keyword">return</strong> -inverseCumulativeProbability(p);
+<a class="jxr_linenumber" name="L278" href="#L278">278</a> }
+<a class="jxr_linenumber" name="L279" href="#L279">279</a>
+<a class="jxr_linenumber" name="L280" href="#L280">280</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L281" href="#L281">281</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L282" href="#L282">282</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L283" href="#L283">283</a> <em class="jxr_javadoccomment"> * <p>For degrees of freedom parameter \( v \), the mean is:</em>
+<a class="jxr_linenumber" name="L284" href="#L284">284</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L285" href="#L285">285</a> <em class="jxr_javadoccomment"> * <p>\[ \mathbb{E}[X] = \begin{cases}</em>
+<a class="jxr_linenumber" name="L286" href="#L286">286</a> <em class="jxr_javadoccomment"> * 0 &amp; \text{for } v \gt 1 \\</em>
+<a class="jxr_linenumber" name="L287" href="#L287">287</a> <em class="jxr_javadoccomment"> * \text{undefined} &amp; \text{otherwise}</em>
+<a class="jxr_linenumber" name="L288" href="#L288">288</a> <em class="jxr_javadoccomment"> * \end{cases} \]</em>
+<a class="jxr_linenumber" name="L289" href="#L289">289</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L290" href="#L290">290</a> <em class="jxr_javadoccomment"> * @return the mean, or {@link Double#NaN NaN} if it is not defined.</em>
+<a class="jxr_linenumber" name="L291" href="#L291">291</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L292" href="#L292">292</a> @Override
+<a class="jxr_linenumber" name="L293" href="#L293">293</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">abstract</strong> <strong class="jxr_keyword">double</strong> getMean();
+<a class="jxr_linenumber" name="L294" href="#L294">294</a>
+<a class="jxr_linenumber" name="L295" href="#L295">295</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L296" href="#L296">296</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L297" href="#L297">297</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L298" href="#L298">298</a> <em class="jxr_javadoccomment"> * <p>For degrees of freedom parameter \( v \), the variance is:</em>
+<a class="jxr_linenumber" name="L299" href="#L299">299</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L300" href="#L300">300</a> <em class="jxr_javadoccomment"> * <p>\[ \operatorname{var}[X] = \begin{cases}</em>
+<a class="jxr_linenumber" name="L301" href="#L301">301</a> <em class="jxr_javadoccomment"> * \frac{v}{v - 2} &amp; \text{for } v \gt 2 \\</em>
+<a class="jxr_linenumber" name="L302" href="#L302">302</a> <em class="jxr_javadoccomment"> * \infty &amp; \text{for } 1 \lt v \le 2 \\</em>
+<a class="jxr_linenumber" name="L303" href="#L303">303</a> <em class="jxr_javadoccomment"> * \text{undefined} &amp; \text{otherwise}</em>
+<a class="jxr_linenumber" name="L304" href="#L304">304</a> <em class="jxr_javadoccomment"> * \end{cases} \]</em>
+<a class="jxr_linenumber" name="L305" href="#L305">305</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L306" href="#L306">306</a> <em class="jxr_javadoccomment"> * @return the variance, or {@link Double#NaN NaN} if it is not defined.</em>
+<a class="jxr_linenumber" name="L307" href="#L307">307</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L308" href="#L308">308</a> @Override
+<a class="jxr_linenumber" name="L309" href="#L309">309</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">abstract</strong> <strong class="jxr_keyword">double</strong> getVariance();
+<a class="jxr_linenumber" name="L310" href="#L310">310</a>
+<a class="jxr_linenumber" name="L311" href="#L311">311</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L312" href="#L312">312</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L313" href="#L313">313</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L314" href="#L314">314</a> <em class="jxr_javadoccomment"> * <p>The lower bound of the support is always negative infinity.</em>
+<a class="jxr_linenumber" name="L315" href="#L315">315</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L316" href="#L316">316</a> <em class="jxr_javadoccomment"> * @return {@link Double#NEGATIVE_INFINITY negative infinity}.</em>
+<a class="jxr_linenumber" name="L317" href="#L317">317</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L318" href="#L318">318</a> @Override
+<a class="jxr_linenumber" name="L319" href="#L319">319</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getSupportLowerBound() {
+<a class="jxr_linenumber" name="L320" href="#L320">320</a> <strong class="jxr_keyword">return</strong> Double.NEGATIVE_INFINITY;
+<a class="jxr_linenumber" name="L321" href="#L321">321</a> }
+<a class="jxr_linenumber" name="L322" href="#L322">322</a>
+<a class="jxr_linenumber" name="L323" href="#L323">323</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L324" href="#L324">324</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L325" href="#L325">325</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L326" href="#L326">326</a> <em class="jxr_javadoccomment"> * <p>The upper bound of the support is always positive infinity.</em>
+<a class="jxr_linenumber" name="L327" href="#L327">327</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L328" href="#L328">328</a> <em class="jxr_javadoccomment"> * @return {@link Double#POSITIVE_INFINITY positive infinity}.</em>
+<a class="jxr_linenumber" name="L329" href="#L329">329</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L330" href="#L330">330</a> @Override
+<a class="jxr_linenumber" name="L331" href="#L331">331</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getSupportUpperBound() {
+<a class="jxr_linenumber" name="L332" href="#L332">332</a> <strong class="jxr_keyword">return</strong> Double.POSITIVE_INFINITY;
+<a class="jxr_linenumber" name="L333" href="#L333">333</a> }
+<a class="jxr_linenumber" name="L334" href="#L334">334</a> }
+</pre>
+<hr/>
+<div id="footer">Copyright © 2018–2022 <a href="https://www.apache.org/">The Apache Software Foundation</a>. All rights reserved.</div>
+</body>
+</html>