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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist"></a><a class="link" href="nc_chi_squared_dist.html" title="Noncentral Chi-Squared Distribution">Noncentral
  28         Chi-Squared Distribution</a>
  29 </h4></div></div></div>
  30 <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">distributions</span><span class="special">/</span><span class="identifier">non_central_chi_squared</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <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>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  34           <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">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">non_central_chi_squared_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">non_central_chi_squared_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">non_central_chi_squared</span><span class="special">;</span>
  38
  39 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</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>
  40 <span class="keyword">class</span> <span class="identifier">non_central_chi_squared_distribution</span>
  41 <span class="special">{</span>
  42 <span class="keyword">public</span><span class="special">:</span>
  43    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  44    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  45
  46    <span class="comment">// Constructor:</span>
  47    <span class="identifier">non_central_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
  48
  49    <span class="comment">// Accessor to degrees of freedom parameter v:</span>
  50    <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  51
  52    <span class="comment">// Accessor to non centrality parameter lambda:</span>
  53    <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  54
  55    <span class="comment">// Parameter finders:</span>
  56    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
  57    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">A</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">B</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">C</span><span class="special">&gt;</span>
  58    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">complemented3_type</span><span class="special">&lt;</span><span class="identifier">A</span><span class="special">,</span><span class="identifier">B</span><span class="special">,</span><span class="identifier">C</span><span class="special">&gt;&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  59
  60    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_non_centrality</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
  61    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">A</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">B</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">C</span><span class="special">&gt;</span>
  62    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_non_centrality</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">complemented3_type</span><span class="special">&lt;</span><span class="identifier">A</span><span class="special">,</span><span class="identifier">B</span><span class="special">,</span><span class="identifier">C</span><span class="special">&gt;&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  63 <span class="special">};</span>
  64
  65 <span class="special">}}</span> <span class="comment">// namespaces</span>
  66 </pre>
  67 <p>
  68           The noncentral chi-squared distribution is a generalization of the <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi Squared Distribution</a>.
  69           If <span class="emphasis"><em>X<sub>i</sub></em></span> are /&#957;/ independent, normally distributed random
  70           variables with means /&#956;<sub>i</sub>/ and variances <span class="emphasis"><em>&#963;<sub>i</sub><sup>2</sup></em></span>, then the
  71           random variable
  72         </p>
  73 <div class="blockquote"><blockquote class="blockquote"><p>
  74             <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref1.svg"></span>
  75
  76           </p></blockquote></div>
  77 <p>
  78           is distributed according to the noncentral chi-squared distribution.
  79         </p>
  80 <p>
  81           The noncentral chi-squared distribution has two parameters: /&#957;/ which specifies
  82           the number of degrees of freedom (i.e. the number of <span class="emphasis"><em>X<sub>i</sub>)</em></span>,
  83           and &#955; which is related to the mean of the random variables <span class="emphasis"><em>X<sub>i</sub></em></span>
  84           by:
  85         </p>
  86 <div class="blockquote"><blockquote class="blockquote"><p>
  87             <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref2.svg"></span>
  88
  89           </p></blockquote></div>
  90 <p>
  91           (Note that some references define &#955; as one half of the above sum).
  92         </p>
  93 <p>
  94           This leads to a PDF of:
  95         </p>
  96 <div class="blockquote"><blockquote class="blockquote"><p>
  97             <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span>
  98
  99           </p></blockquote></div>
 100 <p>
 101           where <span class="emphasis"><em>f(x;k)</em></span> is the central chi-squared distribution
 102           PDF, and <span class="emphasis"><em>I<sub>v</sub>(x)</em></span> is a modified Bessel function of the
 103           first kind.
 104         </p>
 105 <p>
 106           The following graph illustrates how the distribution changes for different
 107           values of &#955;:
 108         </p>
 109 <div class="blockquote"><blockquote class="blockquote"><p>
 110             <span class="inlinemediaobject"><img src="../../../../graphs/nccs_pdf.svg" align="middle"></span>
 111
 112           </p></blockquote></div>
 113 <h5>
 114 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h0"></a>
 115           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.member_functions"></a></span><a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist_ref.dists.nc_chi_squared_dist.member_functions">Member
 116           Functions</a>
 117         </h5>
 118 <pre class="programlisting"><span class="identifier">non_central_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
 119 </pre>
 120 <p>
 121           Constructs a Chi-Squared distribution with <span class="emphasis"><em>v</em></span> degrees
 122           of freedom and non-centrality parameter <span class="emphasis"><em>lambda</em></span>.
 123         </p>
 124 <p>
 125           Requires v &gt; 0 and lambda &gt;= 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 126         </p>
 127 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 128 </pre>
 129 <p>
 130           Returns the parameter <span class="emphasis"><em>v</em></span> from which this object was
 131           constructed.
 132         </p>
 133 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 134 </pre>
 135 <p>
 136           Returns the parameter <span class="emphasis"><em>lambda</em></span> from which this object
 137           was constructed.
 138         </p>
 139 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
 140 </pre>
 141 <p>
 142           This function returns the number of degrees of freedom <span class="emphasis"><em>v</em></span>
 143           such that: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">non_central_chi_squared</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Policy</span><span class="special">&gt;(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">lambda</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">p</span></code>
 144         </p>
 145 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">A</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">B</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">C</span><span class="special">&gt;</span>
 146 <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">complemented3_type</span><span class="special">&lt;</span><span class="identifier">A</span><span class="special">,</span><span class="identifier">B</span><span class="special">,</span><span class="identifier">C</span><span class="special">&gt;&amp;</span> <span class="identifier">c</span><span class="special">);</span>
 147 </pre>
 148 <p>
 149           When called with argument <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">lambda</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span>
 150           <span class="identifier">q</span><span class="special">)</span></code>
 151           this function returns the number of degrees of freedom <span class="emphasis"><em>v</em></span>
 152           such that:
 153         </p>
 154 <p>
 155           <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">non_central_chi_squared</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Policy</span><span class="special">&gt;(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">lambda</span><span class="special">),</span> <span class="identifier">x</span><span class="special">))</span> <span class="special">==</span> <span class="identifier">q</span></code>.
 156         </p>
 157 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_non_centrality</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
 158 </pre>
 159 <p>
 160           This function returns the non centrality parameter <span class="emphasis"><em>lambda</em></span>
 161           such that:
 162         </p>
 163 <p>
 164           <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">non_central_chi_squared</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Policy</span><span class="special">&gt;(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">lambda</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">p</span></code>
 165         </p>
 166 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">A</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">B</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">C</span><span class="special">&gt;</span>
 167 <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_non_centrality</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">complemented3_type</span><span class="special">&lt;</span><span class="identifier">A</span><span class="special">,</span><span class="identifier">B</span><span class="special">,</span><span class="identifier">C</span><span class="special">&gt;&amp;</span> <span class="identifier">c</span><span class="special">);</span>
 168 </pre>
 169 <p>
 170           When called with argument <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span>
 171           <span class="identifier">x</span><span class="special">,</span>
 172           <span class="identifier">q</span><span class="special">)</span></code>
 173           this function returns the non centrality parameter <span class="emphasis"><em>lambda</em></span>
 174           such that:
 175         </p>
 176 <p>
 177           <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">non_central_chi_squared</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Policy</span><span class="special">&gt;(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">lambda</span><span class="special">),</span> <span class="identifier">x</span><span class="special">))</span> <span class="special">==</span> <span class="identifier">q</span></code>.
 178         </p>
 179 <h5>
 180 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h1"></a>
 181           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.non_member_accessors"></a></span><a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist_ref.dists.nc_chi_squared_dist.non_member_accessors">Non-member
 182           Accessors</a>
 183         </h5>
 184 <p>
 185           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 186           functions</a> that are generic to all distributions are supported:
 187           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 188           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 189           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 190           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
 191           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 192           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 193           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
 194           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 195         </p>
 196 <p>
 197           The domain of the random variable is [0, +&#8734;].
 198         </p>
 199 <h5>
 200 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h2"></a>
 201           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.examples"></a></span><a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist_ref.dists.nc_chi_squared_dist.examples">Examples</a>
 202         </h5>
 203 <p>
 204           There is a <a class="link" href="../../stat_tut/weg/nccs_eg.html" title="Non Central Chi Squared Example">worked example</a>
 205           for the noncentral chi-squared distribution.
 206         </p>
 207 <h5>
 208 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h3"></a>
 209           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.accuracy"></a></span><a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist_ref.dists.nc_chi_squared_dist.accuracy">Accuracy</a>
 210         </h5>
 211 <p>
 212           The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>)
 213           found on various platforms with various floating point types. The failures
 214           in the comparison to the <a href="http://www.r-project.org/" target="_top">R Math
 215           library</a>, seem to be mostly in the corner cases when the probablity
 216           would be very small. Unless otherwise specified any floating-point type
 217           that is narrower than the one shown will have <a class="link" href="../../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 218           zero error</a>.
 219         </p>
 220 <div class="table">
 221 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.table_non_central_chi_squared_CDF"></a><p class="title"><b>Table&#160;5.6.&#160;Error rates for non central chi squared CDF</b></p>
 222 <div class="table-contents"><table class="table" summary="Error rates for non central chi squared CDF">
 223 <colgroup>
 224 <col>
 225 <col>
 226 <col>
 227 <col>
 228 <col>
 229 </colgroup>
 230 <thead><tr>
 231 <th>
 232                 </th>
 233 <th>
 234                   <p>
 235                     GNU C++ version 7.1.0<br> linux<br> double
 236                   </p>
 237                 </th>
 238 <th>
 239                   <p>
 240                     GNU C++ version 7.1.0<br> linux<br> long double
 241                   </p>
 242                 </th>
 243 <th>
 244                   <p>
 245                     Sun compiler version 0x5150<br> Sun Solaris<br> long double
 246                   </p>
 247                 </th>
 248 <th>
 249                   <p>
 250                     Microsoft Visual C++ version 14.1<br> Win32<br> double
 251                   </p>
 252                 </th>
 253 </tr></thead>
 254 <tbody>
 255 <tr>
 256 <td>
 257                   <p>
 258                     Non Central Chi Squared, medium parameters
 259                   </p>
 260                 </td>
 261 <td>
 262                   <p>
 263                     <span class="blue">Max = 0.99&#949; (Mean = 0.0544&#949;)</span><br>
 264                     <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 727&#949; (Mean = 121&#949;))
 265                   </p>
 266                 </td>
 267 <td>
 268                   <p>
 269                     <span class="blue">Max = 46.5&#949; (Mean = 10.3&#949;)</span>
 270                   </p>
 271                 </td>
 272 <td>
 273                   <p>
 274                     <span class="blue">Max = 115&#949; (Mean = 13.9&#949;)</span>
 275                   </p>
 276                 </td>
 277 <td>
 278                   <p>
 279                     <span class="blue">Max = 48.9&#949; (Mean = 10&#949;)</span>
 280                   </p>
 281                 </td>
 282 </tr>
 283 <tr>
 284 <td>
 285                   <p>
 286                     Non Central Chi Squared, large parameters
 287                   </p>
 288                 </td>
 289 <td>
 290                   <p>
 291                     <span class="blue">Max = 1.07&#949; (Mean = 0.102&#949;)</span><br>
 292                     <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
 293                     = 3.27e+08&#949; (Mean = 2.23e+07&#949;))</span>
 294                   </p>
 295                 </td>
 296 <td>
 297                   <p>
 298                     <span class="blue">Max = 3.07e+03&#949; (Mean = 336&#949;)</span>
 299                   </p>
 300                 </td>
 301 <td>
 302                   <p>
 303                     <span class="blue">Max = 6.17e+03&#949; (Mean = 677&#949;)</span>
 304                   </p>
 305                 </td>
 306 <td>
 307                   <p>
 308                     <span class="blue">Max = 9.79e+03&#949; (Mean = 723&#949;)</span>
 309                   </p>
 310                 </td>
 311 </tr>
 312 </tbody>
 313 </table></div>
 314 </div>
 315 <br class="table-break"><div class="table">
 316 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.table_non_central_chi_squared_CDF_complement"></a><p class="title"><b>Table&#160;5.7.&#160;Error rates for non central chi squared CDF complement</b></p>
 317 <div class="table-contents"><table class="table" summary="Error rates for non central chi squared CDF complement">
 318 <colgroup>
 319 <col>
 320 <col>
 321 <col>
 322 <col>
 323 <col>
 324 </colgroup>
 325 <thead><tr>
 326 <th>
 327                 </th>
 328 <th>
 329                   <p>
 330                     GNU C++ version 7.1.0<br> linux<br> double
 331                   </p>
 332                 </th>
 333 <th>
 334                   <p>
 335                     GNU C++ version 7.1.0<br> linux<br> long double
 336                   </p>
 337                 </th>
 338 <th>
 339                   <p>
 340                     Sun compiler version 0x5150<br> Sun Solaris<br> long double
 341                   </p>
 342                 </th>
 343 <th>
 344                   <p>
 345                     Microsoft Visual C++ version 14.1<br> Win32<br> double
 346                   </p>
 347                 </th>
 348 </tr></thead>
 349 <tbody>
 350 <tr>
 351 <td>
 352                   <p>
 353                     Non Central Chi Squared, medium parameters
 354                   </p>
 355                 </td>
 356 <td>
 357                   <p>
 358                     <span class="blue">Max = 0.96&#949; (Mean = 0.0635&#949;)</span><br>
 359                     <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
 360                     = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_chi_squared_CDF_complement_Rmath_3_2_3_Non_Central_Chi_Squared_medium_parameters">And
 361                     other failures.</a>)</span>
 362                   </p>
 363                 </td>
 364 <td>
 365                   <p>
 366                     <span class="blue">Max = 107&#949; (Mean = 17.2&#949;)</span>
 367                   </p>
 368                 </td>
 369 <td>
 370                   <p>
 371                     <span class="blue">Max = 171&#949; (Mean = 22.8&#949;)</span>
 372                   </p>
 373                 </td>
 374 <td>
 375                   <p>
 376                     <span class="blue">Max = 98.6&#949; (Mean = 15.8&#949;)</span>
 377                   </p>
 378                 </td>
 379 </tr>
 380 <tr>
 381 <td>
 382                   <p>
 383                     Non Central Chi Squared, large parameters
 384                   </p>
 385                 </td>
 386 <td>
 387                   <p>
 388                     <span class="blue">Max = 2.11&#949; (Mean = 0.278&#949;)</span><br>
 389                     <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max
 390                     = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_chi_squared_CDF_complement_Rmath_3_2_3_Non_Central_Chi_Squared_large_parameters">And
 391                     other failures.</a>)</span>
 392                   </p>
 393                 </td>
 394 <td>
 395                   <p>
 396                     <span class="blue">Max = 5.02e+03&#949; (Mean = 630&#949;)</span>
 397                   </p>
 398                 </td>
 399 <td>
 400                   <p>
 401                     <span class="blue">Max = 5.1e+03&#949; (Mean = 577&#949;)</span>
 402                   </p>
 403                 </td>
 404 <td>
 405                   <p>
 406                     <span class="blue">Max = 5.43e+03&#949; (Mean = 705&#949;)</span>
 407                   </p>
 408                 </td>
 409 </tr>
 410 </tbody>
 411 </table></div>
 412 </div>
 413 <br class="table-break"><p>
 414           Error rates for the quantile functions are broadly similar. Special mention
 415           should go to the <code class="computeroutput"><span class="identifier">mode</span></code> function:
 416           there is no closed form for this function, so it is evaluated numerically
 417           by finding the maxima of the PDF: in principal this can not produce an
 418           accuracy greater than the square root of the machine epsilon.
 419         </p>
 420 <h5>
 421 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h4"></a>
 422           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.tests"></a></span><a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist_ref.dists.nc_chi_squared_dist.tests">Tests</a>
 423         </h5>
 424 <p>
 425           There are two sets of test data used to verify this implementation: firstly
 426           we can compare with published data, for example with Table 6 of "Self-Validating
 427           Computations of Probabilities for Selected Central and Noncentral Univariate
 428           Probability Functions", Morgan C. Wang and William J. Kennedy, Journal
 429           of the American Statistical Association, Vol. 89, No. 427. (Sep., 1994),
 430           pp. 878-887. Secondly, we have tables of test data, computed with this
 431           implementation and using interval arithmetic - this data should be accurate
 432           to at least 50 decimal digits - and is the used for our accuracy tests.
 433         </p>
 434 <h5>
 435 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h5"></a>
 436           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.implementation"></a></span><a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist_ref.dists.nc_chi_squared_dist.implementation">Implementation</a>
 437         </h5>
 438 <p>
 439           The CDF and its complement are evaluated as follows:
 440         </p>
 441 <p>
 442           First we determine which of the two values (the CDF or its complement)
 443           is likely to be the smaller: for this we can use the relation due to Temme
 444           (see "Asymptotic and Numerical Aspects of the Noncentral Chi-Square
 445           Distribution", N. M. Temme, Computers Math. Applic. Vol 25, No. 5,
 446           55-63, 1993) that:
 447         </p>
 448 <p>
 449           F(&#957;,&#955;;&#957;+&#955;) &#8776; 0.5
 450         </p>
 451 <p>
 452           and so compute the CDF when the random variable is less than &#957;+&#955;, and its
 453           complement when the random variable is greater than &#957;+&#955;. If necessary the
 454           computed result is then subtracted from 1 to give the desired result (the
 455           CDF or its complement).
 456         </p>
 457 <p>
 458           For small values of the non centrality parameter, the CDF is computed using
 459           the method of Ding (see "Algorithm AS 275: Computing the Non-Central
 460           #2 Distribution Function", Cherng G. Ding, Applied Statistics, Vol.
 461           41, No. 2. (1992), pp. 478-482). This uses the following series representation:
 462         </p>
 463 <div class="blockquote"><blockquote class="blockquote"><p>
 464             <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref4.svg"></span>
 465
 466           </p></blockquote></div>
 467 <p>
 468           which requires just one call to <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>
 469           with the subsequent terms being computed by recursion as shown above.
 470         </p>
 471 <p>
 472           For larger values of the non-centrality parameter, Ding's method can take
 473           an unreasonable number of terms before convergence is achieved. Furthermore,
 474           the largest term is not the first term, so in extreme cases the first term
 475           may be zero, leading to a zero result, even though the true value may be
 476           non-zero.
 477         </p>
 478 <p>
 479           Therefore, when the non-centrality parameter is greater than 200, the method
 480           due to Krishnamoorthy (see "Computing discrete mixtures of continuous
 481           distributions: noncentral chisquare, noncentral t and the distribution
 482           of the square of the sample multiple correlation coefficient", Denise
 483           Benton and K. Krishnamoorthy, Computational Statistics &amp; Data Analysis,
 484           43, (2003), 249-267) is used.
 485         </p>
 486 <p>
 487           This method uses the well known sum:
 488         </p>
 489 <div class="blockquote"><blockquote class="blockquote"><p>
 490             <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref5.svg"></span>
 491
 492           </p></blockquote></div>
 493 <p>
 494           Where <span class="emphasis"><em>P<sub>a</sub>(x)</em></span> is the incomplete gamma function.
 495         </p>
 496 <p>
 497           The method starts at the &#955;th term, which is where the Poisson weighting
 498           function achieves its maximum value, although this is not necessarily the
 499           largest overall term. Subsequent terms are calculated via the normal recurrence
 500           relations for the incomplete gamma function, and iteration proceeds both
 501           forwards and backwards until sufficient precision has been achieved. It
 502           should be noted that recurrence in the forwards direction of P<sub>a</sub>(x) is numerically
 503           unstable. However, since we always start <span class="emphasis"><em>after</em></span> the
 504           largest term in the series, numeric instability is introduced more slowly
 505           than the series converges.
 506         </p>
 507 <p>
 508           Computation of the complement of the CDF uses an extension of Krishnamoorthy's
 509           method, given that:
 510         </p>
 511 <div class="blockquote"><blockquote class="blockquote"><p>
 512             <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref6.svg"></span>
 513
 514           </p></blockquote></div>
 515 <p>
 516           we can again start at the &#955;'th term and proceed in both directions from
 517           there until the required precision is achieved. This time it is backwards
 518           recursion on the incomplete gamma function Q<sub>a</sub>(x) which is unstable. However,
 519           as long as we start well <span class="emphasis"><em>before</em></span> the largest term,
 520           this is not an issue in practice.
 521         </p>
 522 <p>
 523           The PDF is computed directly using the relation:
 524         </p>
 525 <div class="blockquote"><blockquote class="blockquote"><p>
 526             <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span>
 527
 528           </p></blockquote></div>
 529 <p>
 530           Where <span class="emphasis"><em>f(x; v)</em></span> is the PDF of the central <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi
 531           Squared Distribution</a> and <span class="emphasis"><em>I<sub>v</sub>(x)</em></span> is a modified
 532           Bessel function, see <a class="link" href="../../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a>.
 533           For small values of the non-centrality parameter the relation in terms
 534           of <a class="link" href="../../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a> is used.
 535           However, this method fails for large values of the non-centrality parameter,
 536           so in that case the infinite sum is evaluated using the method of Benton
 537           and Krishnamoorthy, and the usual recurrence relations for successive terms.
 538         </p>
 539 <p>
 540           The quantile functions are computed by numeric inversion of the CDF. An
 541           improve starting quess is from Thomas Luu, <a href="http://discovery.ucl.ac.uk/1482128/%2c" target="_top">Fast
 542           and accurate parallel computation of quantile functions for random number
 543           generation, Doctorial Thesis, 2016</a>.
 544         </p>
 545 <p>
 546           There is no <a href="http://en.wikipedia.org/wiki/Closed_form" target="_top">closed
 547           form</a> for the mode of the noncentral chi-squared distribution: it
 548           is computed numerically by finding the maximum of the PDF. Likewise, the
 549           median is computed numerically via the quantile.
 550         </p>
 551 <p>
 552           The remaining non-member functions use the following formulas:
 553         </p>
 554 <div class="blockquote"><blockquote class="blockquote"><p>
 555             <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref7.svg"></span>
 556
 557           </p></blockquote></div>
 558 <p>
 559           Some analytic properties of noncentral distributions (particularly unimodality,
 560           and monotonicity of their modes) are surveyed and summarized by:
 561         </p>
 562 <p>
 563           Andrea van Aubel &amp; Wolfgang Gawronski, Applied Mathematics and Computation,
 564           141 (2003) 3-12.
 565         </p>
 566 </div>
 567 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 568 <td align="left"></td>
 569 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 570       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 571       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 572       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 573       Daryle Walker and Xiaogang Zhang<p>
 574         Distributed under the Boost Software License, Version 1.0. (See accompanying
 575         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>)
 576       </p>
 577 </div></td>
 578 </tr></table>
 579 <hr>
 580 <div class="spirit-nav">
 581 <a accesskey="p" href="nc_beta_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="nc_f_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
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