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| 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 15. 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 15. 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 15. 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 15. Policies: Controlling Precision, Error Handling etc">policy documentation | |
| 57 | for more details</a>. | |
| 58 | </p> | |
| 59 | <p> | |
| 60 |     <span class="inlinemediaobject"><img src="../../equations/owens_t.svg"></span> | |
| 61 | </p> | |
| 62 | <p> | |
| 63 | <span class="inlinemediaobject"><img src="../../graphs/plot_owens_t.png"></span> | |
| 64 | </p> | |
| 65 | <p> | |
| 66 | 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 | |
| 67 | > h and 0 < Y < a * X)</em></span>, where <span class="emphasis"><em>X</em></span> and | |
| 68 | <span class="emphasis"><em>Y</em></span> are independent standard normal random variables. | |
| 69 | </p> | |
| 70 | <p> | |
| 71 | For h and a > 0, T(h,a), gives the volume of an uncorrelated bivariate normal | |
| 72 | distribution with zero means and unit variances over the area between <span class="emphasis"><em>y | |
| 73 | = ax</em></span> and <span class="emphasis"><em>y = 0</em></span> and to the right of <span class="emphasis"><em>x | |
| 74 | = h</em></span>. | |
| 75 | </p> | |
| 76 | <p> | |
| 77 | That is the area shaded in the figure below (Owens 1956). | |
| 78 | </p> | |
| 79 | <p> | |
| 80 | <span class="inlinemediaobject"><img src="../../graphs/owens_integration_area.svg" align="middle"></span> | |
| 81 | </p> | |
| 82 | <p> | |
| 83 | and is also illustrated by a 3D plot. | |
| 84 | </p> | |
| 85 | <p> | |
| 86 | <span class="inlinemediaobject"><img src="../../graphs/plot_owens_3d_xyp.png"></span> | |
| 87 | </p> | |
| 88 | <p> | |
| 89 | 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 | |
| 90 | Normal Distribution</a>. It is also used in the computation of bivariate | |
| 91 | and multivariate normal distribution probabilities. The return type of this | |
| 92 | 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 | |
| 93 | 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 | |
| 94 | otherwise. | |
| 95 | </p> | |
| 96 | <p> | |
| 97 | Owen's original paper (page 1077) provides some additional corner cases. | |
| 98 | </p> | |
| 99 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 100 | <span class="emphasis"><em>T(h, 0) = 0</em></span> | |
| 101 | </p></blockquote></div> | |
| 102 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 103 | <span class="emphasis"><em>T(0, a) = ½π arctan(a)</em></span> | |
| 104 | </p></blockquote></div> | |
| 105 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 106 | <span class="emphasis"><em>T(h, 1) = ½ G(h) [1 - G(h)]</em></span> | |
| 107 | </p></blockquote></div> | |
| 108 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 109 | <span class="emphasis"><em>T(h, ∞) = G(|h|)</em></span> | |
| 110 | </p></blockquote></div> | |
| 111 | <p> | |
| 112 | where G(h) is the univariate normal with zero mean and unit variance integral | |
| 113 | from -∞ to h. | |
| 114 | </p> | |
| 115 | <h5> | |
| 116 | <a name="math_toolkit.owens_t.h2"></a> | |
| 117 | <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> | |
| 118 | </h5> | |
| 119 | <p> | |
| 120 | Over the built-in types and range tested, errors are less than 10 * std::numeric_limits<RealType>::epsilon(). | |
| 121 | </p> | |
| 122 | <div class="table"> | |
| 123 | <a name="math_toolkit.owens_t.table_owens_t"></a><p class="title"><b>Table 6.83. Error rates for owens_t</b></p> | |
| 124 | <div class="table-contents"><table class="table" summary="Error rates for owens_t"> | |
| 125 | <colgroup> | |
| 126 | <col> | |
| 127 | <col> | |
| 128 | <col> | |
| 129 | <col> | |
| 130 | <col> | |
| 131 | </colgroup> | |
| 132 | <thead><tr> | |
| 133 | <th> | |
| 134 | </th> | |
| 135 | <th> | |
| 136 | <p> | |
| 137 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
| 138 | </p> | |
| 139 | </th> | |
| 140 | <th> | |
| 141 | <p> | |
| 142 | GNU C++ version 5.1.0<br> linux<br> double | |
| 143 | </p> | |
| 144 | </th> | |
| 145 | <th> | |
| 146 | <p> | |
| 147 | GNU C++ version 5.1.0<br> linux<br> long double | |
| 148 | </p> | |
| 149 | </th> | |
| 150 | <th> | |
| 151 | <p> | |
| 152 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
| 153 | </p> | |
| 154 | </th> | |
| 155 | </tr></thead> | |
| 156 | <tbody> | |
| 157 | <tr> | |
| 158 | <td> | |
| 159 | <p> | |
| 160 | Owens T (medium small values) | |
| 161 | </p> | |
| 162 | </td> | |
| 163 | <td> | |
| 164 | <p> | |
| 165 | <span class="blue">Max = 4.37ε (Mean = 0.973ε)</span> | |
| 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 = 0.942ε)</span> | |
| 176 | </p> | |
| 177 | </td> | |
| 178 | <td> | |
| 179 | <p> | |
| 180 | <span class="blue">Max = 3.34ε (Mean = 0.91ε)</span> | |
| 181 | </p> | |
| 182 | </td> | |
| 183 | </tr> | |
| 184 | <tr> | |
| 185 | <td> | |
| 186 | <p> | |
| 187 | Owens T (large and diverse values) | |
| 188 | </p> | |
| 189 | </td> | |
| 190 | <td> | |
| 191 | <p> | |
| 192 | <span class="blue">Max = 3.78ε (Mean = 0.621ε)</span> | |
| 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 = 1.38ε)</span> | |
| 208 | </p> | |
| 209 | </td> | |
| 210 | </tr> | |
| 211 | </tbody> | |
| 212 | </table></div> | |
| 213 | </div> | |
| 214 | <br class="table-break"><h5> | |
| 215 | <a name="math_toolkit.owens_t.h3"></a> | |
| 216 | <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> | |
| 217 | </h5> | |
| 218 | <p> | |
| 219 | Test data was generated by Patefield and Tandy algorithms T1 and T4, and also | |
| 220 | the suggested reference routine T7. | |
| 221 | </p> | |
| 222 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
| 223 | <li class="listitem"> | |
| 224 | 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> | |
| 225 | (ie cancellation), | |
| 226 | </li> | |
| 227 | <li class="listitem"> | |
| 228 | T4 was rejected if there was no convergence, | |
| 229 | </li> | |
| 230 | <li class="listitem"> | |
| 231 | Both were rejected if they didn't agree. | |
| 232 | </li> | |
| 233 | </ul></div> | |
| 234 | <p> | |
| 235 | Over the built-in types and range tested, errors are less than 10 std::numeric_limits<RealType>::epsilon(). | |
| 236 | </p> | |
| 237 | <p> | |
| 238 | However, that there was a whole domain (large <span class="emphasis"><em>h</em></span>, small | |
| 239 | <span class="emphasis"><em>a</em></span>) where it was not possible to generate any reliable | |
| 240 | test values (all the methods got rejected for one reason or another). | |
| 241 | </p> | |
| 242 | <p> | |
| 243 | 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> | |
| 244 | and <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>. | |
| 245 | </p> | |
| 246 | <h5> | |
| 247 | <a name="math_toolkit.owens_t.h4"></a> | |
| 248 | <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> | |
| 249 | </h5> | |
| 250 | <p> | |
| 251 | 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. | |
| 252 | B. Owen, Tables for computing bivariate normal probabilities, Ann. Math. Statist., | |
| 253 | 27, 1075-1090 (1956)</a>. | |
| 254 | </p> | |
| 255 | <p> | |
| 256 | The algorithms of Patefield, M. and Tandy, D. "Fast and accurate Calculation | |
| 257 | of Owen's T-Function", Journal of Statistical Software, 5 (5), 1 - 25 | |
| 258 | (2000) are adapted for C++ with arbitrary RealType. | |
| 259 | </p> | |
| 260 | <p> | |
| 261 | The Patefield-Tandy algorithm provides six methods of evalualution (T1 to T6); | |
| 262 | the best method is selected according to the values of <span class="emphasis"><em>a</em></span> | |
| 263 | 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> | |
| 264 | for details. | |
| 265 | </p> | |
| 266 | <p> | |
| 267 | The Patefield-Tandy algorithm is accurate to approximately 20 decimal places, | |
| 268 | so for types with greater precision we use: | |
| 269 | </p> | |
| 270 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
| 271 | <li class="listitem"> | |
| 272 | A modified version of T1 which folds the calculation of <span class="emphasis"><em>atan(h)</em></span> | |
| 273 | into the T1 series (to avoid subtracting two values similar in magnitude), | |
| 274 | and then accelerates the resulting alternating series using method 1 from | |
| 275 | H. Cohen, F. Rodriguez Villegas, D. Zagier, "Convergence acceleration | |
| 276 | of alternating series", Bonn, (1991). The result is valid everywhere, | |
| 277 | but doesn't always converge, or may become too divergent in the first few | |
| 278 | terms to sum accurately. This is used for <span class="emphasis"><em>ah < 1</em></span>. | |
| 279 | </li> | |
| 280 | <li class="listitem"> | |
| 281 | A modified version of T2 which is accelerated in the same manner as T1. | |
| 282 | This is used for <span class="emphasis"><em>h > 1</em></span>. | |
| 283 | </li> | |
| 284 | <li class="listitem"> | |
| 285 | A version of T4 only when both T1 and T2 have failed to produce an accurate | |
| 286 | answer. | |
| 287 | </li> | |
| 288 | <li class="listitem"> | |
| 289 | Fallback to the Patefiled Tandy algorithm when all the above methods fail: | |
| 290 | this happens not at all for our test data at 100 decimal digits precision. | |
| 291 | However, there is a difficult area when <span class="emphasis"><em>a</em></span> is very | |
| 292 | close to 1 and the precision increases which may cause this to happen in | |
| 293 | very exceptional circumstances. | |
| 294 | </li> | |
| 295 | </ul></div> | |
| 296 | <p> | |
| 297 | Using the above algorithm and a 100-decimal digit type, results accurate to | |
| 298 | 80 decimal places were obtained in the difficult area where <span class="emphasis"><em>a</em></span> | |
| 299 | is close to 1, and greater than 95 decimal places elsewhere. | |
| 300 | </p> | |
| 301 | </div> | |
| 302 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 303 | <td align="left"></td> | |
| 304 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 305 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 306 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 307 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 308 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 309 | 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>) | |
| 310 | </p> | |
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