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  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">&lt;</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">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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&#160;20.&#160;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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy documentation
  57       for more details</a>.
  58     </p>
  59 <p>
  60       &#8198; &#8198;
  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       &gt; h and 0 &lt; Y &lt; 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 &gt; 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) = &#189;&#960; 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) = &#189; 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, &#8734;) = 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 -&#8734; 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&lt;RealType&gt;::epsilon().
 126     </p>
 127 <div class="table">
 128 <a name="math_toolkit.owens_t.table_owens_t"></a><p class="title"><b>Table&#160;8.86.&#160;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 = 0&#949; (Mean = 0&#949;)</span>
 171               </p>
 172             </td>
 173 <td>
 174               <p>
 175                 <span class="blue">Max = 3.34&#949; (Mean = 0.944&#949;)</span>
 176               </p>
 177             </td>
 178 <td>
 179               <p>
 180                 <span class="blue">Max = 3.34&#949; (Mean = 0.911&#949;)</span>
 181               </p>
 182             </td>
 183 <td>
 184               <p>
 185                 <span class="blue">Max = 4.37&#949; (Mean = 0.98&#949;)</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 = 0&#949; (Mean = 0&#949;)</span>
 198               </p>
 199             </td>
 200 <td>
 201               <p>
 202                 <span class="blue">Max = 49&#949; (Mean = 2.16&#949;)</span>
 203               </p>
 204             </td>
 205 <td>
 206               <p>
 207                 <span class="blue">Max = 24.5&#949; (Mean = 1.39&#949;)</span>
 208               </p>
 209             </td>
 210 <td>
 211               <p>
 212                 <span class="blue">Max = 3.78&#949; (Mean = 0.621&#949;)</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&lt;RealType&gt;::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&amp;version=1.0&amp;verb=Display&amp;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 evalualution (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 &lt; 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 &gt; 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 &#169; 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       R&#229;de, 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>
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