1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Owen's T function</title> 5<link rel="stylesheet" href="../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../special.html" title="Chapter 8. Special Functions"> 9<link rel="prev" href="inv_hyper/atanh.html" title="atanh"> 10<link rel="next" href="daubechies.html" title="Daubechies Wavelets and Scaling Functions"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<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> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.owens_t"></a><a class="link" href="owens_t.html" title="Owen's T function">Owen's T function</a> 28</h2></div></div></div> 29<h5> 30<a name="math_toolkit.owens_t.h0"></a> 31 <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> 32 </h5> 33<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> 34</pre> 35<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> 36 37<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 38<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> 39 40<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> 41<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> 42 43<span class="special">}}</span> <span class="comment">// namespaces</span> 44</pre> 45<h5> 46<a name="math_toolkit.owens_t.h1"></a> 47 <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> 48 </h5> 49<p> 50 Returns the <a href="http://en.wikipedia.org/wiki/Owen%27s_T_function" target="_top">Owens_t 51 function</a> of <span class="emphasis"><em>h</em></span> and <span class="emphasis"><em>a</em></span>. 52 </p> 53<p> 54 The final <a class="link" href="../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 55 be used to control the behaviour of the function: how it handles errors, what 56 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 57 for more details</a>. 58 </p> 59<p> 60 61 </p> 62<div class="blockquote"><blockquote class="blockquote"><p> 63 <span class="inlinemediaobject"><img src="../../equations/owens_t.svg"></span> 64 65 </p></blockquote></div> 66<p> 67 <span class="inlinemediaobject"><img src="../../graphs/plot_owens_t.png"></span> 68 </p> 69<p> 70 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 71 > h and 0 < Y < a * X)</em></span>, where <span class="emphasis"><em>X</em></span> and 72 <span class="emphasis"><em>Y</em></span> are independent standard normal random variables. 73 </p> 74<p> 75 For h and a > 0, T(h,a), gives the volume of an uncorrelated bivariate normal 76 distribution with zero means and unit variances over the area between <span class="emphasis"><em>y 77 = ax</em></span> and <span class="emphasis"><em>y = 0</em></span> and to the right of <span class="emphasis"><em>x 78 = h</em></span>. 79 </p> 80<p> 81 That is the area shaded in the figure below (Owens 1956). 82 </p> 83<div class="blockquote"><blockquote class="blockquote"><p> 84 <span class="inlinemediaobject"><img src="../../graphs/owens_integration_area.svg" align="middle"></span> 85 86 </p></blockquote></div> 87<p> 88 and is also illustrated by a 3D plot. 89 </p> 90<p> 91 <span class="inlinemediaobject"><img src="../../graphs/plot_owens_3d_xyp.png"></span> 92 </p> 93<p> 94 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 95 Normal Distribution</a>. It is also used in the computation of bivariate 96 and multivariate normal distribution probabilities. The return type of this 97 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 98 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 99 otherwise. 100 </p> 101<p> 102 Owen's original paper (page 1077) provides some additional corner cases. 103 </p> 104<div class="blockquote"><blockquote class="blockquote"><p> 105 <span class="serif_italic"><span class="emphasis"><em>T(h, 0) = 0</em></span></span> 106 </p></blockquote></div> 107<div class="blockquote"><blockquote class="blockquote"><p> 108 <span class="serif_italic"><span class="emphasis"><em>T(0, a) = ½π arctan(a)</em></span></span> 109 </p></blockquote></div> 110<div class="blockquote"><blockquote class="blockquote"><p> 111 <span class="serif_italic"><span class="emphasis"><em>T(h, 1) = ½ G(h) [1 - G(h)]</em></span></span> 112 </p></blockquote></div> 113<div class="blockquote"><blockquote class="blockquote"><p> 114 <span class="serif_italic"><span class="emphasis"><em>T(h, ∞) = G(|h|)</em></span></span> 115 </p></blockquote></div> 116<p> 117 where G(h) is the univariate normal with zero mean and unit variance integral 118 from -∞ to h. 119 </p> 120<h5> 121<a name="math_toolkit.owens_t.h2"></a> 122 <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> 123 </h5> 124<p> 125 Over the built-in types and range tested, errors are less than 10 * std::numeric_limits<RealType>::epsilon(). 126 </p> 127<div class="table"> 128<a name="math_toolkit.owens_t.table_owens_t"></a><p class="title"><b>Table 8.86. Error rates for owens_t</b></p> 129<div class="table-contents"><table class="table" summary="Error rates for owens_t"> 130<colgroup> 131<col> 132<col> 133<col> 134<col> 135<col> 136</colgroup> 137<thead><tr> 138<th> 139 </th> 140<th> 141 <p> 142 GNU C++ version 7.1.0<br> linux<br> double 143 </p> 144 </th> 145<th> 146 <p> 147 GNU C++ version 7.1.0<br> linux<br> long double 148 </p> 149 </th> 150<th> 151 <p> 152 Sun compiler version 0x5150<br> Sun Solaris<br> long double 153 </p> 154 </th> 155<th> 156 <p> 157 Microsoft Visual C++ version 14.1<br> Win32<br> double 158 </p> 159 </th> 160</tr></thead> 161<tbody> 162<tr> 163<td> 164 <p> 165 Owens T (medium small values) 166 </p> 167 </td> 168<td> 169 <p> 170 <span class="blue">Max = 0ε (Mean = 0ε)</span> 171 </p> 172 </td> 173<td> 174 <p> 175 <span class="blue">Max = 3.34ε (Mean = 0.944ε)</span> 176 </p> 177 </td> 178<td> 179 <p> 180 <span class="blue">Max = 3.34ε (Mean = 0.911ε)</span> 181 </p> 182 </td> 183<td> 184 <p> 185 <span class="blue">Max = 4.37ε (Mean = 0.98ε)</span> 186 </p> 187 </td> 188</tr> 189<tr> 190<td> 191 <p> 192 Owens T (large and diverse values) 193 </p> 194 </td> 195<td> 196 <p> 197 <span class="blue">Max = 0ε (Mean = 0ε)</span> 198 </p> 199 </td> 200<td> 201 <p> 202 <span class="blue">Max = 49ε (Mean = 2.16ε)</span> 203 </p> 204 </td> 205<td> 206 <p> 207 <span class="blue">Max = 24.5ε (Mean = 1.39ε)</span> 208 </p> 209 </td> 210<td> 211 <p> 212 <span class="blue">Max = 3.78ε (Mean = 0.621ε)</span> 213 </p> 214 </td> 215</tr> 216</tbody> 217</table></div> 218</div> 219<br class="table-break"><h5> 220<a name="math_toolkit.owens_t.h3"></a> 221 <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> 222 </h5> 223<p> 224 Test data was generated by Patefield and Tandy algorithms T1 and T4, and also 225 the suggested reference routine T7. 226 </p> 227<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 228<li class="listitem"> 229 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> 230 (ie cancellation), 231 </li> 232<li class="listitem"> 233 T4 was rejected if there was no convergence, 234 </li> 235<li class="listitem"> 236 Both were rejected if they didn't agree. 237 </li> 238</ul></div> 239<p> 240 Over the built-in types and range tested, errors are less than 10 std::numeric_limits<RealType>::epsilon(). 241 </p> 242<p> 243 However, that there was a whole domain (large <span class="emphasis"><em>h</em></span>, small 244 <span class="emphasis"><em>a</em></span>) where it was not possible to generate any reliable 245 test values (all the methods got rejected for one reason or another). 246 </p> 247<p> 248 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> 249 and <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>. 250 </p> 251<h5> 252<a name="math_toolkit.owens_t.h4"></a> 253 <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> 254 </h5> 255<p> 256 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. 257 B. Owen, Tables for computing bivariate normal probabilities, Ann. Math. Statist., 258 27, 1075-1090 (1956)</a>. 259 </p> 260<p> 261 The algorithms of Patefield, M. and Tandy, D. "Fast and accurate Calculation 262 of Owen's T-Function", Journal of Statistical Software, 5 (5), 1 - 25 263 (2000) are adapted for C++ with arbitrary RealType. 264 </p> 265<p> 266 The Patefield-Tandy algorithm provides six methods of evaluation (T1 to T6); 267 the best method is selected according to the values of <span class="emphasis"><em>a</em></span> 268 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> 269 for details. 270 </p> 271<p> 272 The Patefield-Tandy algorithm is accurate to approximately 20 decimal places, 273 so for types with greater precision we use: 274 </p> 275<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 276<li class="listitem"> 277 A modified version of T1 which folds the calculation of <span class="emphasis"><em>atan(h)</em></span> 278 into the T1 series (to avoid subtracting two values similar in magnitude), 279 and then accelerates the resulting alternating series using method 1 from 280 H. Cohen, F. Rodriguez Villegas, D. Zagier, "Convergence acceleration 281 of alternating series", Bonn, (1991). The result is valid everywhere, 282 but doesn't always converge, or may become too divergent in the first few 283 terms to sum accurately. This is used for <span class="emphasis"><em>ah < 1</em></span>. 284 </li> 285<li class="listitem"> 286 A modified version of T2 which is accelerated in the same manner as T1. 287 This is used for <span class="emphasis"><em>h > 1</em></span>. 288 </li> 289<li class="listitem"> 290 A version of T4 only when both T1 and T2 have failed to produce an accurate 291 answer. 292 </li> 293<li class="listitem"> 294 Fallback to the Patefiled Tandy algorithm when all the above methods fail: 295 this happens not at all for our test data at 100 decimal digits precision. 296 However, there is a difficult area when <span class="emphasis"><em>a</em></span> is very 297 close to 1 and the precision increases which may cause this to happen in 298 very exceptional circumstances. 299 </li> 300</ul></div> 301<p> 302 Using the above algorithm and a 100-decimal digit type, results accurate to 303 80 decimal places were obtained in the difficult area where <span class="emphasis"><em>a</em></span> 304 is close to 1, and greater than 95 decimal places elsewhere. 305 </p> 306</div> 307<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 308<td align="left"></td> 309<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 310 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 311 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 312 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 313 Daryle Walker and Xiaogang Zhang<p> 314 Distributed under the Boost Software License, Version 1.0. (See accompanying 315 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>) 316 </p> 317</div></td> 318</tr></table> 319<hr> 320<div class="spirit-nav"> 321<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> 322</div> 323</body> 324</html> 325
