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  25 <div class="section">
  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;15.&#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;15.&#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 X<sub>i</sub> are &#957; independent, normally distributed random variables with means
  70           &#956;<sub>i</sub> and variances &#963;<sub>i</sub><sup>2</sup>, then the random variable
  71         </p>
  72 <p>
  73           <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref1.svg"></span>
  74         </p>
  75 <p>
  76           is distributed according to the noncentral chi-squared distribution.
  77         </p>
  78 <p>
  79           The noncentral chi-squared distribution has two parameters: &#957; which specifies
  80           the number of degrees of freedom (i.e. the number of X<sub>i</sub>), and &#955; which is
  81           related to the mean of the random variables X<sub>i</sub> by:
  82         </p>
  83 <p>
  84           <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref2.svg"></span>
  85         </p>
  86 <p>
  87           (Note that some references define &#955; as one half of the above sum).
  88         </p>
  89 <p>
  90           This leads to a PDF of:
  91         </p>
  92 <p>
  93           <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span>
  94         </p>
  95 <p>
  96           where <span class="emphasis"><em>f(x;k)</em></span> is the central chi-squared distribution
  97           PDF, and <span class="emphasis"><em>I<sub>v</sub>(x)</em></span> is a modified Bessel function of the
  98           first kind.
  99         </p>
 100 <p>
 101           The following graph illustrates how the distribution changes for different
 102           values of &#955;:
 103         </p>
 104 <p>
 105           <span class="inlinemediaobject"><img src="../../../../graphs/nccs_pdf.svg" align="middle"></span>
 106         </p>
 107 <h5>
 108 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h0"></a>
 109           <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
 110           Functions</a>
 111         </h5>
 112 <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>
 113 </pre>
 114 <p>
 115           Constructs a Chi-Squared distribution with <span class="emphasis"><em>v</em></span> degrees
 116           of freedom and non-centrality parameter <span class="emphasis"><em>lambda</em></span>.
 117         </p>
 118 <p>
 119           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>.
 120         </p>
 121 <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>
 122 </pre>
 123 <p>
 124           Returns the parameter <span class="emphasis"><em>v</em></span> from which this object was
 125           constructed.
 126         </p>
 127 <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>
 128 </pre>
 129 <p>
 130           Returns the parameter <span class="emphasis"><em>lambda</em></span> from which this object
 131           was constructed.
 132         </p>
 133 <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>
 134 </pre>
 135 <p>
 136           This function returns the number of degrees of freedom <span class="emphasis"><em>v</em></span>
 137           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>
 138         </p>
 139 <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>
 140 <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>
 141 </pre>
 142 <p>
 143           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>
 144           <span class="identifier">q</span><span class="special">)</span></code>
 145           this function returns the number of degrees of freedom <span class="emphasis"><em>v</em></span>
 146           such that:
 147         </p>
 148 <p>
 149           <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>.
 150         </p>
 151 <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>
 152 </pre>
 153 <p>
 154           This function returns the non centrality parameter <span class="emphasis"><em>lambda</em></span>
 155           such that:
 156         </p>
 157 <p>
 158           <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>
 159         </p>
 160 <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>
 161 <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>
 162 </pre>
 163 <p>
 164           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>
 165           <span class="identifier">x</span><span class="special">,</span>
 166           <span class="identifier">q</span><span class="special">)</span></code>
 167           this function returns the non centrality parameter <span class="emphasis"><em>lambda</em></span>
 168           such that:
 169         </p>
 170 <p>
 171           <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>.
 172         </p>
 173 <h5>
 174 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h1"></a>
 175           <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
 176           Accessors</a>
 177         </h5>
 178 <p>
 179           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 180           functions</a> that are generic to all distributions are supported:
 181           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 182           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 183           <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>,
 184           <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>,
 185           <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>,
 186           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 187           <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>,
 188           <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>.
 189         </p>
 190 <p>
 191           The domain of the random variable is [0, +&#8734;].
 192         </p>
 193 <h5>
 194 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h2"></a>
 195           <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>
 196         </h5>
 197 <p>
 198           There is a <a class="link" href="../../stat_tut/weg/nccs_eg.html" title="Non Central Chi Squared Example">worked example</a>
 199           for the noncentral chi-squared distribution.
 200         </p>
 201 <h5>
 202 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h3"></a>
 203           <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>
 204         </h5>
 205 <p>
 206           The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>)
 207           found on various platforms with various floating point types. The failures
 208           in the comparison to the <a href="http://www.r-project.org/" target="_top">R Math
 209           library</a>, seem to be mostly in the corner cases when the probablity
 210           would be very small. Unless otherwise specified any floating-point type
 211           that is narrower than the one shown will have <a class="link" href="../../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 212           zero error</a>.
 213         </p>
 214 <div class="table">
 215 <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>
 216 <div class="table-contents"><table class="table" summary="Error rates for non central chi squared CDF">
 217 <colgroup>
 218 <col>
 219 <col>
 220 <col>
 221 <col>
 222 <col>
 223 </colgroup>
 224 <thead><tr>
 225 <th>
 226                 </th>
 227 <th>
 228                   <p>
 229                     Microsoft Visual C++ version 12.0<br> Win32<br> double
 230                   </p>
 231                 </th>
 232 <th>
 233                   <p>
 234                     GNU C++ version 5.1.0<br> linux<br> double
 235                   </p>
 236                 </th>
 237 <th>
 238                   <p>
 239                     GNU C++ version 5.1.0<br> linux<br> long double
 240                   </p>
 241                 </th>
 242 <th>
 243                   <p>
 244                     Sun compiler version 0x5130<br> Sun Solaris<br> long double
 245                   </p>
 246                 </th>
 247 </tr></thead>
 248 <tbody>
 249 <tr>
 250 <td>
 251                   <p>
 252                     Non Central Chi Squared, medium parameters
 253                   </p>
 254                 </td>
 255 <td>
 256                   <p>
 257                     <span class="blue">Max = 48.9&#949; (Mean = 10&#949;)</span>
 258                   </p>
 259                 </td>
 260 <td>
 261                   <p>
 262                     <span class="blue">Max = 0.99&#949; (Mean = 0.0529&#949;)</span><br>
 263                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 727&#949; (Mean = 121&#949;))
 264                   </p>
 265                 </td>
 266 <td>
 267                   <p>
 268                     <span class="blue">Max = 46.5&#949; (Mean = 10.3&#949;)</span>
 269                   </p>
 270                 </td>
 271 <td>
 272                   <p>
 273                     <span class="blue">Max = 115&#949; (Mean = 13.9&#949;)</span>
 274                   </p>
 275                 </td>
 276 </tr>
 277 <tr>
 278 <td>
 279                   <p>
 280                     Non Central Chi Squared, large parameters
 281                   </p>
 282                 </td>
 283 <td>
 284                   <p>
 285                     <span class="blue">Max = 9.79e+003&#949; (Mean = 723&#949;)</span>
 286                   </p>
 287                 </td>
 288 <td>
 289                   <p>
 290                     <span class="blue">Max = 1.07&#949; (Mean = 0.102&#949;)</span><br>
 291                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
 292                     = 3.27e+08&#949; (Mean = 2.23e+07&#949;))</span>
 293                   </p>
 294                 </td>
 295 <td>
 296                   <p>
 297                     <span class="blue">Max = 3.07e+03&#949; (Mean = 336&#949;)</span>
 298                   </p>
 299                 </td>
 300 <td>
 301                   <p>
 302                     <span class="blue">Max = 6.17e+03&#949; (Mean = 677&#949;)</span>
 303                   </p>
 304                 </td>
 305 </tr>
 306 </tbody>
 307 </table></div>
 308 </div>
 309 <br class="table-break"><div class="table">
 310 <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>
 311 <div class="table-contents"><table class="table" summary="Error rates for non central chi squared CDF complement">
 312 <colgroup>
 313 <col>
 314 <col>
 315 <col>
 316 <col>
 317 <col>
 318 </colgroup>
 319 <thead><tr>
 320 <th>
 321                 </th>
 322 <th>
 323                   <p>
 324                     Microsoft Visual C++ version 12.0<br> Win32<br> double
 325                   </p>
 326                 </th>
 327 <th>
 328                   <p>
 329                     GNU C++ version 5.1.0<br> linux<br> double
 330                   </p>
 331                 </th>
 332 <th>
 333                   <p>
 334                     GNU C++ version 5.1.0<br> linux<br> long double
 335                   </p>
 336                 </th>
 337 <th>
 338                   <p>
 339                     Sun compiler version 0x5130<br> Sun Solaris<br> long double
 340                   </p>
 341                 </th>
 342 </tr></thead>
 343 <tbody>
 344 <tr>
 345 <td>
 346                   <p>
 347                     Non Central Chi Squared, medium parameters
 348                   </p>
 349                 </td>
 350 <td>
 351                   <p>
 352                     <span class="blue">Max = 98.6&#949; (Mean = 15.8&#949;)</span>
 353                   </p>
 354                 </td>
 355 <td>
 356                   <p>
 357                     <span class="blue">Max = 0.96&#949; (Mean = 0.0635&#949;)</span><br>
 358                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
 359                     = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_non_central_chi_squared_CDF_complement_Rmath_3_0_2_Non_Central_Chi_Squared_medium_parameters">And
 360                     other failures.</a>)</span>
 361                   </p>
 362                 </td>
 363 <td>
 364                   <p>
 365                     <span class="blue">Max = 107&#949; (Mean = 17.1&#949;)</span>
 366                   </p>
 367                 </td>
 368 <td>
 369                   <p>
 370                     <span class="blue">Max = 171&#949; (Mean = 22.8&#949;)</span>
 371                   </p>
 372                 </td>
 373 </tr>
 374 <tr>
 375 <td>
 376                   <p>
 377                     Non Central Chi Squared, large parameters
 378                   </p>
 379                 </td>
 380 <td>
 381                   <p>
 382                     <span class="blue">Max = 5.43e+003&#949; (Mean = 705&#949;)</span>
 383                   </p>
 384                 </td>
 385 <td>
 386                   <p>
 387                     <span class="blue">Max = 2.11&#949; (Mean = 0.278&#949;)</span><br>
 388                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
 389                     = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_non_central_chi_squared_CDF_complement_Rmath_3_0_2_Non_Central_Chi_Squared_large_parameters">And
 390                     other failures.</a>)</span>
 391                   </p>
 392                 </td>
 393 <td>
 394                   <p>
 395                     <span class="blue">Max = 5.02e+03&#949; (Mean = 630&#949;)</span>
 396                   </p>
 397                 </td>
 398 <td>
 399                   <p>
 400                     <span class="blue">Max = 5.1e+03&#949; (Mean = 577&#949;)</span>
 401                   </p>
 402                 </td>
 403 </tr>
 404 </tbody>
 405 </table></div>
 406 </div>
 407 <br class="table-break"><p>
 408           Error rates for the quantile functions are broadly similar. Special mention
 409           should go to the <code class="computeroutput"><span class="identifier">mode</span></code> function:
 410           there is no closed form for this function, so it is evaluated numerically
 411           by finding the maxima of the PDF: in principal this can not produce an
 412           accuracy greater than the square root of the machine epsilon.
 413         </p>
 414 <h5>
 415 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h4"></a>
 416           <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>
 417         </h5>
 418 <p>
 419           There are two sets of test data used to verify this implementation: firstly
 420           we can compare with published data, for example with Table 6 of "Self-Validating
 421           Computations of Probabilities for Selected Central and Noncentral Univariate
 422           Probability Functions", Morgan C. Wang and William J. Kennedy, Journal
 423           of the American Statistical Association, Vol. 89, No. 427. (Sep., 1994),
 424           pp. 878-887. Secondly, we have tables of test data, computed with this
 425           implementation and using interval arithmetic - this data should be accurate
 426           to at least 50 decimal digits - and is the used for our accuracy tests.
 427         </p>
 428 <h5>
 429 <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h5"></a>
 430           <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>
 431         </h5>
 432 <p>
 433           The CDF and its complement are evaluated as follows:
 434         </p>
 435 <p>
 436           First we determine which of the two values (the CDF or its complement)
 437           is likely to be the smaller: for this we can use the relation due to Temme
 438           (see "Asymptotic and Numerical Aspects of the Noncentral Chi-Square
 439           Distribution", N. M. Temme, Computers Math. Applic. Vol 25, No. 5,
 440           55-63, 1993) that:
 441         </p>
 442 <p>
 443           F(&#957;,&#955;;&#957;+&#955;) &#8776; 0.5
 444         </p>
 445 <p>
 446           and so compute the CDF when the random variable is less than &#957;+&#955;, and its
 447           complement when the random variable is greater than &#957;+&#955;. If necessary the
 448           computed result is then subtracted from 1 to give the desired result (the
 449           CDF or its complement).
 450         </p>
 451 <p>
 452           For small values of the non centrality parameter, the CDF is computed using
 453           the method of Ding (see "Algorithm AS 275: Computing the Non-Central
 454           #2 Distribution Function", Cherng G. Ding, Applied Statistics, Vol.
 455           41, No. 2. (1992), pp. 478-482). This uses the following series representation:
 456         </p>
 457 <p>
 458           <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref4.svg"></span>
 459         </p>
 460 <p>
 461           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>
 462           with the subsequent terms being computed by recursion as shown above.
 463         </p>
 464 <p>
 465           For larger values of the non-centrality parameter, Ding's method can take
 466           an unreasonable number of terms before convergence is achieved. Furthermore,
 467           the largest term is not the first term, so in extreme cases the first term
 468           may be zero, leading to a zero result, even though the true value may be
 469           non-zero.
 470         </p>
 471 <p>
 472           Therefore, when the non-centrality parameter is greater than 200, the method
 473           due to Krishnamoorthy (see "Computing discrete mixtures of continuous
 474           distributions: noncentral chisquare, noncentral t and the distribution
 475           of the square of the sample multiple correlation coefficient", Denise
 476           Benton and K. Krishnamoorthy, Computational Statistics &amp; Data Analysis,
 477           43, (2003), 249-267) is used.
 478         </p>
 479 <p>
 480           This method uses the well known sum:
 481         </p>
 482 <p>
 483           <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref5.svg"></span>
 484         </p>
 485 <p>
 486           Where P<sub>a</sub>(x) is the incomplete gamma function.
 487         </p>
 488 <p>
 489           The method starts at the &#955;th term, which is where the Poisson weighting
 490           function achieves its maximum value, although this is not necessarily the
 491           largest overall term. Subsequent terms are calculated via the normal recurrence
 492           relations for the incomplete gamma function, and iteration proceeds both
 493           forwards and backwards until sufficient precision has been achieved. It
 494           should be noted that recurrence in the forwards direction of P<sub>a</sub>(x) is numerically
 495           unstable. However, since we always start <span class="emphasis"><em>after</em></span> the
 496           largest term in the series, numeric instability is introduced more slowly
 497           than the series converges.
 498         </p>
 499 <p>
 500           Computation of the complement of the CDF uses an extension of Krishnamoorthy's
 501           method, given that:
 502         </p>
 503 <p>
 504           <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref6.svg"></span>
 505         </p>
 506 <p>
 507           we can again start at the &#955;'th term and proceed in both directions from
 508           there until the required precision is achieved. This time it is backwards
 509           recursion on the incomplete gamma function Q<sub>a</sub>(x) which is unstable. However,
 510           as long as we start well <span class="emphasis"><em>before</em></span> the largest term,
 511           this is not an issue in practice.
 512         </p>
 513 <p>
 514           The PDF is computed directly using the relation:
 515         </p>
 516 <p>
 517           <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span>
 518         </p>
 519 <p>
 520           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
 521           Squared Distribution</a> and <span class="emphasis"><em>I<sub>v</sub>(x)</em></span> is a modified
 522           Bessel function, see <a class="link" href="../../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a>.
 523           For small values of the non-centrality parameter the relation in terms
 524           of <a class="link" href="../../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a> is used.
 525           However, this method fails for large values of the non-centrality parameter,
 526           so in that case the infinite sum is evaluated using the method of Benton
 527           and Krishnamoorthy, and the usual recurrence relations for successive terms.
 528         </p>
 529 <p>
 530           The quantile functions are computed by numeric inversion of the CDF. An
 531           improve starting quess is from Thomas Luu, <a href="http://discovery.ucl.ac.uk/1482128/%2c" target="_top">Fast
 532           and accurate parallel computation of quantile functions for random number
 533           generation, Doctorial Thesis, 2016</a>.
 534         </p>
 535 <p>
 536           There is no <a href="http://en.wikipedia.org/wiki/Closed_form" target="_top">closed
 537           form</a> for the mode of the noncentral chi-squared distribution: it
 538           is computed numerically by finding the maximum of the PDF. Likewise, the
 539           median is computed numerically via the quantile.
 540         </p>
 541 <p>
 542           The remaining non-member functions use the following formulas:
 543         </p>
 544 <p>
 545           <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref7.svg"></span>
 546         </p>
 547 <p>
 548           Some analytic properties of noncentral distributions (particularly unimodality,
 549           and monotonicity of their modes) are surveyed and summarized by:
 550         </p>
 551 <p>
 552           Andrea van Aubel &amp; Wolfgang Gawronski, Applied Mathematics and Computation,
 553           141 (2003) 3-12.
 554         </p>
 555 </div>
 556 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 557 <td align="left"></td>
 558 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 559       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 560       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 561       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 562         Distributed under the Boost Software License, Version 1.0. (See accompanying
 563         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>)
 564       </p>
 565 </div></td>
 566 </tr></table>
 567 <hr>
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