325 lines
17 KiB
HTML
325 lines
17 KiB
HTML
<html>
|
||
<head>
|
||
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
|
||
<title>Owen's T function</title>
|
||
<link rel="stylesheet" href="../math.css" type="text/css">
|
||
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
||
<link rel="home" href="../index.html" title="Math Toolkit 3.0.0">
|
||
<link rel="up" href="../special.html" title="Chapter 8. Special Functions">
|
||
<link rel="prev" href="inv_hyper/atanh.html" title="atanh">
|
||
<link rel="next" href="daubechies.html" title="Daubechies Wavelets and Scaling Functions">
|
||
</head>
|
||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||
<table cellpadding="2" width="100%"><tr>
|
||
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td>
|
||
<td align="center"><a href="../../../../../index.html">Home</a></td>
|
||
<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
|
||
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
||
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
||
<td align="center"><a href="../../../../../more/index.htm">More</a></td>
|
||
</tr></table>
|
||
<hr>
|
||
<div class="spirit-nav">
|
||
<a accesskey="p" href="inv_hyper/atanh.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../special.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="daubechies.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||
<a name="math_toolkit.owens_t"></a><a class="link" href="owens_t.html" title="Owen's T function">Owen's T function</a>
|
||
</h2></div></div></div>
|
||
<h5>
|
||
<a name="math_toolkit.owens_t.h0"></a>
|
||
<span class="phrase"><a name="math_toolkit.owens_t.synopsis"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.synopsis">Synopsis</a>
|
||
</h5>
|
||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">owens_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||
</pre>
|
||
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
|
||
|
||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||
<a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">owens_t</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">h</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">);</span>
|
||
|
||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||
<a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">owens_t</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">h</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||
|
||
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||
</pre>
|
||
<h5>
|
||
<a name="math_toolkit.owens_t.h1"></a>
|
||
<span class="phrase"><a name="math_toolkit.owens_t.description"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.description">Description</a>
|
||
</h5>
|
||
<p>
|
||
Returns the <a href="http://en.wikipedia.org/wiki/Owen%27s_T_function" target="_top">Owens_t
|
||
function</a> of <span class="emphasis"><em>h</em></span> and <span class="emphasis"><em>a</em></span>.
|
||
</p>
|
||
<p>
|
||
The final <a class="link" href="../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
|
||
be used to control the behaviour of the function: how it handles errors, what
|
||
level of precision to use etc. Refer to the <a class="link" href="../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy documentation
|
||
for more details</a>.
|
||
</p>
|
||
<p>
|
||
|
||
</p>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="inlinemediaobject"><img src="../../equations/owens_t.svg"></span>
|
||
|
||
</p></blockquote></div>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../graphs/plot_owens_t.png"></span>
|
||
</p>
|
||
<p>
|
||
The function <code class="computeroutput"><span class="identifier">owens_t</span><span class="special">(</span><span class="identifier">h</span><span class="special">,</span> <span class="identifier">a</span><span class="special">)</span></code> gives the probability of the event <span class="emphasis"><em>(X
|
||
> h and 0 < Y < a * X)</em></span>, where <span class="emphasis"><em>X</em></span> and
|
||
<span class="emphasis"><em>Y</em></span> are independent standard normal random variables.
|
||
</p>
|
||
<p>
|
||
For h and a > 0, T(h,a), gives the volume of an uncorrelated bivariate normal
|
||
distribution with zero means and unit variances over the area between <span class="emphasis"><em>y
|
||
= ax</em></span> and <span class="emphasis"><em>y = 0</em></span> and to the right of <span class="emphasis"><em>x
|
||
= h</em></span>.
|
||
</p>
|
||
<p>
|
||
That is the area shaded in the figure below (Owens 1956).
|
||
</p>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="inlinemediaobject"><img src="../../graphs/owens_integration_area.svg" align="middle"></span>
|
||
|
||
</p></blockquote></div>
|
||
<p>
|
||
and is also illustrated by a 3D plot.
|
||
</p>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../graphs/plot_owens_3d_xyp.png"></span>
|
||
</p>
|
||
<p>
|
||
This function is used in the computation of the <a class="link" href="dist_ref/dists/skew_normal_dist.html" title="Skew Normal Distribution">Skew
|
||
Normal Distribution</a>. It is also used in the computation of bivariate
|
||
and multivariate normal distribution probabilities. The return type of this
|
||
function is computed using the <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
|
||
type calculation rules</em></span></a>: the result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type T
|
||
otherwise.
|
||
</p>
|
||
<p>
|
||
Owen's original paper (page 1077) provides some additional corner cases.
|
||
</p>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>T(h, 0) = 0</em></span></span>
|
||
</p></blockquote></div>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>T(0, a) = ½π arctan(a)</em></span></span>
|
||
</p></blockquote></div>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>T(h, 1) = ½ G(h) [1 - G(h)]</em></span></span>
|
||
</p></blockquote></div>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>T(h, ∞) = G(|h|)</em></span></span>
|
||
</p></blockquote></div>
|
||
<p>
|
||
where G(h) is the univariate normal with zero mean and unit variance integral
|
||
from -∞ to h.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.owens_t.h2"></a>
|
||
<span class="phrase"><a name="math_toolkit.owens_t.accuracy"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.accuracy">Accuracy</a>
|
||
</h5>
|
||
<p>
|
||
Over the built-in types and range tested, errors are less than 10 * std::numeric_limits<RealType>::epsilon().
|
||
</p>
|
||
<div class="table">
|
||
<a name="math_toolkit.owens_t.table_owens_t"></a><p class="title"><b>Table 8.86. Error rates for owens_t</b></p>
|
||
<div class="table-contents"><table class="table" summary="Error rates for owens_t">
|
||
<colgroup>
|
||
<col>
|
||
<col>
|
||
<col>
|
||
<col>
|
||
<col>
|
||
</colgroup>
|
||
<thead><tr>
|
||
<th>
|
||
</th>
|
||
<th>
|
||
<p>
|
||
GNU C++ version 7.1.0<br> linux<br> double
|
||
</p>
|
||
</th>
|
||
<th>
|
||
<p>
|
||
GNU C++ version 7.1.0<br> linux<br> long double
|
||
</p>
|
||
</th>
|
||
<th>
|
||
<p>
|
||
Sun compiler version 0x5150<br> Sun Solaris<br> long double
|
||
</p>
|
||
</th>
|
||
<th>
|
||
<p>
|
||
Microsoft Visual C++ version 14.1<br> Win32<br> double
|
||
</p>
|
||
</th>
|
||
</tr></thead>
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
Owens T (medium small values)
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 0ε (Mean = 0ε)</span>
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 3.34ε (Mean = 0.944ε)</span>
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 3.34ε (Mean = 0.911ε)</span>
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 4.37ε (Mean = 0.98ε)</span>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
Owens T (large and diverse values)
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 0ε (Mean = 0ε)</span>
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 49ε (Mean = 2.16ε)</span>
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 24.5ε (Mean = 1.39ε)</span>
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 3.78ε (Mean = 0.621ε)</span>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
</div>
|
||
<br class="table-break"><h5>
|
||
<a name="math_toolkit.owens_t.h3"></a>
|
||
<span class="phrase"><a name="math_toolkit.owens_t.testing"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.testing">Testing</a>
|
||
</h5>
|
||
<p>
|
||
Test data was generated by Patefield and Tandy algorithms T1 and T4, and also
|
||
the suggested reference routine T7.
|
||
</p>
|
||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||
<li class="listitem">
|
||
T1 was rejected if the result was too small compared to <code class="computeroutput"><span class="identifier">atan</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span></code>
|
||
(ie cancellation),
|
||
</li>
|
||
<li class="listitem">
|
||
T4 was rejected if there was no convergence,
|
||
</li>
|
||
<li class="listitem">
|
||
Both were rejected if they didn't agree.
|
||
</li>
|
||
</ul></div>
|
||
<p>
|
||
Over the built-in types and range tested, errors are less than 10 std::numeric_limits<RealType>::epsilon().
|
||
</p>
|
||
<p>
|
||
However, that there was a whole domain (large <span class="emphasis"><em>h</em></span>, small
|
||
<span class="emphasis"><em>a</em></span>) where it was not possible to generate any reliable
|
||
test values (all the methods got rejected for one reason or another).
|
||
</p>
|
||
<p>
|
||
There are also two sets of sanity tests: spot values are computed using <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram Mathematica</a>
|
||
and <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.owens_t.h4"></a>
|
||
<span class="phrase"><a name="math_toolkit.owens_t.implementation"></a></span><a class="link" href="owens_t.html#math_toolkit.owens_t.implementation">Implementation</a>
|
||
</h5>
|
||
<p>
|
||
The function was proposed and evaluated by <a href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177728074" target="_top">Donald.
|
||
B. Owen, Tables for computing bivariate normal probabilities, Ann. Math. Statist.,
|
||
27, 1075-1090 (1956)</a>.
|
||
</p>
|
||
<p>
|
||
The algorithms of Patefield, M. and Tandy, D. "Fast and accurate Calculation
|
||
of Owen's T-Function", Journal of Statistical Software, 5 (5), 1 - 25
|
||
(2000) are adapted for C++ with arbitrary RealType.
|
||
</p>
|
||
<p>
|
||
The Patefield-Tandy algorithm provides six methods of evaluation (T1 to T6);
|
||
the best method is selected according to the values of <span class="emphasis"><em>a</em></span>
|
||
and <span class="emphasis"><em>h</em></span>. See the original paper and the source in <a href="../../../../../boost/math/special_functions/owens_t.hpp" target="_top">owens_t.hpp</a>
|
||
for details.
|
||
</p>
|
||
<p>
|
||
The Patefield-Tandy algorithm is accurate to approximately 20 decimal places,
|
||
so for types with greater precision we use:
|
||
</p>
|
||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||
<li class="listitem">
|
||
A modified version of T1 which folds the calculation of <span class="emphasis"><em>atan(h)</em></span>
|
||
into the T1 series (to avoid subtracting two values similar in magnitude),
|
||
and then accelerates the resulting alternating series using method 1 from
|
||
H. Cohen, F. Rodriguez Villegas, D. Zagier, "Convergence acceleration
|
||
of alternating series", Bonn, (1991). The result is valid everywhere,
|
||
but doesn't always converge, or may become too divergent in the first few
|
||
terms to sum accurately. This is used for <span class="emphasis"><em>ah < 1</em></span>.
|
||
</li>
|
||
<li class="listitem">
|
||
A modified version of T2 which is accelerated in the same manner as T1.
|
||
This is used for <span class="emphasis"><em>h > 1</em></span>.
|
||
</li>
|
||
<li class="listitem">
|
||
A version of T4 only when both T1 and T2 have failed to produce an accurate
|
||
answer.
|
||
</li>
|
||
<li class="listitem">
|
||
Fallback to the Patefiled Tandy algorithm when all the above methods fail:
|
||
this happens not at all for our test data at 100 decimal digits precision.
|
||
However, there is a difficult area when <span class="emphasis"><em>a</em></span> is very
|
||
close to 1 and the precision increases which may cause this to happen in
|
||
very exceptional circumstances.
|
||
</li>
|
||
</ul></div>
|
||
<p>
|
||
Using the above algorithm and a 100-decimal digit type, results accurate to
|
||
80 decimal places were obtained in the difficult area where <span class="emphasis"><em>a</em></span>
|
||
is close to 1, and greater than 95 decimal places elsewhere.
|
||
</p>
|
||
</div>
|
||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||
<td align="left"></td>
|
||
<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||
Walker and Xiaogang Zhang<p>
|
||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||
</p>
|
||
</div></td>
|
||
</tr></table>
|
||
<hr>
|
||
<div class="spirit-nav">
|
||
<a accesskey="p" href="inv_hyper/atanh.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../special.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="daubechies.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
</div>
|
||
</body>
|
||
</html>
|