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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist"></a><a class="link" href="nc_chi_squared_dist.html" title="Noncentral Chi-Squared Distribution"> |
| Noncentral Chi-Squared Distribution</a> |
| </h5></div></div></div> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">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">></span></pre> |
| <p> |
| </p> |
| <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> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> |
| <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">non_central_chi_squared_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">non_central_chi_squared_distribution</span><span class="special"><></span> <span class="identifier">non_central_chi_squared</span><span class="special">;</span> |
| |
| <span class="keyword">template</span> <span class="special"><</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="Policies">Policy</a><span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">non_central_chi_squared_distribution</span> |
| <span class="special">{</span> |
| <span class="keyword">public</span><span class="special">:</span> |
| <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> |
| |
| <span class="comment">// Constructor: |
| </span> <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> |
| |
| <span class="comment">// Accessor to degrees of freedom parameter v: |
| </span> <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> |
| |
| <span class="comment">// Accessor to non centrality parameter lambda: |
| </span> <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| |
| <span class="comment">// Parameter finders: |
| </span> <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> |
| <span class="keyword">template</span> <span class="special"><</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">></span> |
| <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"><</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">>&</span> <span class="identifier">c</span><span class="special">);</span> |
| |
| <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> |
| <span class="keyword">template</span> <span class="special"><</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">></span> |
| <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"><</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">>&</span> <span class="identifier">c</span><span class="special">);</span> |
| <span class="special">};</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| 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>. If X<sub>i</sub> are ν independent, normally distributed random |
| variables with means μ<sub>i</sub> and variances σ<sub>i</sub><sup>2</sup>, then the random variable |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_chi_squ_ref1.png"></span> |
| </p> |
| <p> |
| is distributed according to the noncentral chi-squared distribution. |
| </p> |
| <p> |
| The noncentral chi-squared distribution has two parameters: ν which specifies |
| the number of degrees of freedom (i.e. the number of X<sub>i</sub>), and λ which is |
| related to the mean of the random variables X<sub>i</sub> by: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_chi_squ_ref2.png"></span> |
| </p> |
| <p> |
| (Note that some references define λ as one half of the above sum). |
| </p> |
| <p> |
| This leads to a PDF of: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_chi_squ_ref3.png"></span> |
| </p> |
| <p> |
| where <span class="emphasis"><em>f(x;k)</em></span> is the central chi-squared distribution |
| PDF, and <span class="emphasis"><em>I<sub>v</sub>(x)</em></span> is a modified Bessel function of |
| the first kind. |
| </p> |
| <p> |
| The following graph illustrates how the distribution changes for different |
| values of λ: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/nccs_pdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.member_functions"></a><h5> |
| <a name="id1045615"></a> |
| <a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <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> |
| </pre> |
| <p> |
| Constructs a Chi-Squared distribution with <span class="emphasis"><em>v</em></span> degrees |
| of freedom and non-centrality parameter <span class="emphasis"><em>lambda</em></span>. |
| </p> |
| <p> |
| Requires v > 0 and lambda >= 0, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </p> |
| <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> |
| </pre> |
| <p> |
| Returns the parameter <span class="emphasis"><em>v</em></span> from which this object was |
| constructed. |
| </p> |
| <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> |
| </pre> |
| <p> |
| Returns the parameter <span class="emphasis"><em>lambda</em></span> from which this object |
| was constructed. |
| </p> |
| <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> |
| </pre> |
| <p> |
| This function returns the number of degrees of freedom <span class="emphasis"><em>v</em></span> |
| 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"><</span><span class="identifier">RealType</span><span class="special">,</span> |
| <span class="identifier">Policy</span><span class="special">>(</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> |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> |
| <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"><</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">>&</span> <span class="identifier">c</span><span class="special">);</span> |
| </pre> |
| <p> |
| 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> <span class="identifier">q</span><span class="special">)</span></code> this function returns the number of |
| degrees of freedom <span class="emphasis"><em>v</em></span> such that: |
| </p> |
| <p> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> |
| <span class="identifier">Policy</span><span class="special">>(</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>. |
| </p> |
| <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> |
| </pre> |
| <p> |
| This function returns the non centrality parameter <span class="emphasis"><em>lambda</em></span> |
| such that: |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">non_central_chi_squared</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> |
| <span class="identifier">Policy</span><span class="special">>(</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> |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> |
| <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"><</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">>&</span> <span class="identifier">c</span><span class="special">);</span> |
| </pre> |
| <p> |
| 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> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">q</span><span class="special">)</span></code> this function returns the non centrality |
| parameter <span class="emphasis"><em>lambda</em></span> such that: |
| </p> |
| <p> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> |
| <span class="identifier">Policy</span><span class="special">>(</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>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.non_member_accessors"></a><h5> |
| <a name="id1046516"></a> |
| <a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.non_member_accessors">Non-member |
| Accessors</a> |
| </h5> |
| <p> |
| All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member |
| accessor functions</a> that are generic to all distributions are supported: |
| <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard |
| Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, |
| <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, |
| <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, |
| <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, |
| <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. |
| </p> |
| <p> |
| The domain of the random variable is [0, +∞]. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.examples"></a><h5> |
| <a name="id1047344"></a> |
| <a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.examples">Examples</a> |
| </h5> |
| <p> |
| There is a <a class="link" href="../../stat_tut/weg/nccs_eg.html" title="Non Central Chi Squared Example">worked |
| example</a> for the noncentral chi-squared distribution. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.accuracy"></a><h5> |
| <a name="id1047371"></a> |
| <a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>) |
| found on various platforms with various floating-point types, along with |
| comparisons to the <a href="http://www.r-project.org/" target="_top">R-2.5.1 Math |
| library</a>. Unless otherwise specified, any floating-point type |
| that is narrower than the one shown will have <a class="link" href="../../../backgrounders/relative_error.html#zero_error">effectively |
| zero error</a>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.errors_in_cdf_of_the_noncentral_chi_squared"></a><p class="title"><b>Table 14. Errors In CDF of the Noncentral Chi-Squared</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In CDF of the Noncentral Chi-Squared"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| ν,λ < 200 |
| </p> |
| </th> |
| <th> |
| <p> |
| ν,λ > 200 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=50 Mean=9.9 |
| </p> |
| <p> |
| R Peak=685 Mean=109 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=9780 Mean=718 |
| </p> |
| <p> |
| R Peak=3x10<sup>8</sup> Mean=2x10<sup>7</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA32, gcc-4.1.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=270 Mean=27 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=7900 Mean=900 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=107 Mean=17 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=5000 Mean=630 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=270 Mean=20 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4600 Mean=560 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><p> |
| Error rates for the complement of the CDF and for the quantile functions |
| are broadly similar. Special mention should go to the <code class="computeroutput"><span class="identifier">mode</span></code> |
| function: there is no closed form for this function, so it is evaluated |
| numerically by finding the maxima of the PDF: in principal this can not |
| produce an accuracy greater than the square root of the machine epsilon. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.tests"></a><h5> |
| <a name="id1047632"></a> |
| <a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.tests">Tests</a> |
| </h5> |
| <p> |
| There are two sets of test data used to verify this implementation: firstly |
| we can compare with published data, for example with Table 6 of "Self-Validating |
| Computations of Probabilities for Selected Central and Noncentral Univariate |
| Probability Functions", Morgan C. Wang and William J. Kennedy, Journal |
| of the American Statistical Association, Vol. 89, No. 427. (Sep., 1994), |
| pp. 878-887. Secondly, we have tables of test data, computed with this |
| implementation and using interval arithmetic - this data should be accurate |
| to at least 50 decimal digits - and is the used for our accuracy tests. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.implementation"></a><h5> |
| <a name="id1047651"></a> |
| <a class="link" href="nc_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| The CDF and its complement are evaluated as follows: |
| </p> |
| <p> |
| First we determine which of the two values (the CDF or its complement) |
| is likely to be the smaller: for this we can use the relation due to |
| Temme (see "Asymptotic and Numerical Aspects of the Noncentral Chi-Square |
| Distribution", N. M. Temme, Computers Math. Applic. Vol 25, No. |
| 5, 55-63, 1993) that: |
| </p> |
| <p> |
| F(ν,λ;ν+λ) ≈ 0.5 |
| </p> |
| <p> |
| and so compute the CDF when the random variable is less than ν+λ, and its |
| complement when the random variable is greater than ν+λ. If necessary the |
| computed result is then subtracted from 1 to give the desired result |
| (the CDF or its complement). |
| </p> |
| <p> |
| For small values of the non centrality parameter, the CDF is computed |
| using the method of Ding (see "Algorithm AS 275: Computing the Non-Central |
| #2 Distribution Function", Cherng G. Ding, Applied Statistics, Vol. |
| 41, No. 2. (1992), pp. 478-482). This uses the following series representation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_chi_squ_ref4.png"></span> |
| </p> |
| <p> |
| which requires just one call to <a class="link" href="../../../special/sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a> |
| with the subsequent terms being computed by recursion as shown above. |
| </p> |
| <p> |
| For larger values of the non-centrality parameter, Ding's method can |
| take an unreasonable number of terms before convergence is achieved. |
| Furthermore, the largest term is not the first term, so in extreme cases |
| the first term may be zero, leading to a zero result, even though the |
| true value may be non-zero. |
| </p> |
| <p> |
| Therefore, when the non-centrality parameter is greater than 200, the |
| method due to Krishnamoorthy (see "Computing discrete mixtures of |
| continuous distributions: noncentral chisquare, noncentral t and the |
| distribution of the square of the sample multiple correlation coefficient", |
| Denise Benton and K. Krishnamoorthy, Computational Statistics & Data |
| Analysis, 43, (2003), 249-267) is used. |
| </p> |
| <p> |
| This method uses the well known sum: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_chi_squ_ref5.png"></span> |
| </p> |
| <p> |
| Where P<sub>a</sub>(x) is the incomplete gamma function. |
| </p> |
| <p> |
| The method starts at the λth term, which is where the Poisson weighting |
| function achieves its maximum value, although this is not necessarily |
| the largest overall term. Subsequent terms are calculated via the normal |
| recurrence relations for the incomplete gamma function, and iteration |
| proceeds both forwards and backwards until sufficient precision has been |
| achieved. It should be noted that recurrence in the forwards direction |
| of P<sub>a</sub>(x) is numerically unstable. However, since we always start <span class="emphasis"><em>after</em></span> |
| the largest term in the series, numeric instability is introduced more |
| slowly than the series converges. |
| </p> |
| <p> |
| Computation of the complement of the CDF uses an extension of Krishnamoorthy's |
| method, given that: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_chi_squ_ref6.png"></span> |
| </p> |
| <p> |
| we can again start at the λ'th term and proceed in both directions from |
| there until the required precision is achieved. This time it is backwards |
| recursion on the incomplete gamma function Q<sub>a</sub>(x) which is unstable. However, |
| as long as we start well <span class="emphasis"><em>before</em></span> the largest term, |
| this is not an issue in practice. |
| </p> |
| <p> |
| The PDF is computed directly using the relation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_chi_squ_ref3.png"></span> |
| </p> |
| <p> |
| 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 |
| Squared Distribution</a> and <span class="emphasis"><em>I<sub>v</sub>(x)</em></span> is a modified |
| Bessel function, see <a class="link" href="../../../special/bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a>. |
| For small values of the non-centrality parameter the relation in terms |
| of <a class="link" href="../../../special/bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a> |
| is used. However, this method fails for large values of the non-centrality |
| parameter, so in that case the infinite sum is evaluated using the method |
| of Benton and Krishnamoorthy, and the usual recurrence relations for |
| successive terms. |
| </p> |
| <p> |
| The quantile functions are computed by numeric inversion of the CDF. |
| </p> |
| <p> |
| There is no <a href="http://en.wikipedia.org/wiki/Closed_form" target="_top">closed |
| form</a> for the mode of the noncentral chi-squared distribution: |
| it is computed numerically by finding the maximum of the PDF. Likewise, |
| the median is computed numerically via the quantile. |
| </p> |
| <p> |
| The remaining non-member functions use the following formulas: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_chi_squ_ref7.png"></span> |
| </p> |
| <p> |
| Some analytic properties of noncentral distributions (particularly unimodality, |
| and monotonicity of their modes) are surveyed and summarized by: |
| </p> |
| <p> |
| Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, |
| 141 (2003) 3-12. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
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