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4 | <title>Noncentral Chi-Squared Distribution</title> | |
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24 | </div> | |
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"><</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> | |
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"><</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 15. 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<></a> <span class="special">></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"><></span> <span class="identifier">non_central_chi_squared</span><span class="special">;</span> | |
38 | ||
39 | <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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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"><</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> | |
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"><</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> | |
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"><</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> | |
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"><</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> | |
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 ν independent, normally distributed random variables with means | |
70 | μ<sub>i</sub> and variances σ<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: ν which specifies | |
80 | the number of degrees of freedom (i.e. the number of X<sub>i</sub>), and λ 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 λ 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 λ: | |
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 > 0 and lambda >= 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"><</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> | |
138 | </p> | |
139 | <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> | |
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"><</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> | |
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"><</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>. | |
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"><</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> | |
159 | </p> | |
160 | <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> | |
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"><</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> | |
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"><</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>. | |
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, +∞]. | |
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 5.6. 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 = 48.9ε (Mean = 10ε)</span> | |
258 | </p> | |
259 | </td> | |
260 | <td> | |
261 | <p> | |
262 | <span class="blue">Max = 0.99ε (Mean = 0.0529ε)</span><br> | |
263 | <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 727ε (Mean = 121ε)) | |
264 | </p> | |
265 | </td> | |
266 | <td> | |
267 | <p> | |
268 | <span class="blue">Max = 46.5ε (Mean = 10.3ε)</span> | |
269 | </p> | |
270 | </td> | |
271 | <td> | |
272 | <p> | |
273 | <span class="blue">Max = 115ε (Mean = 13.9ε)</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+003ε (Mean = 723ε)</span> | |
286 | </p> | |
287 | </td> | |
288 | <td> | |
289 | <p> | |
290 | <span class="blue">Max = 1.07ε (Mean = 0.102ε)</span><br> | |
291 | <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max | |
292 | = 3.27e+08ε (Mean = 2.23e+07ε))</span> | |
293 | </p> | |
294 | </td> | |
295 | <td> | |
296 | <p> | |
297 | <span class="blue">Max = 3.07e+03ε (Mean = 336ε)</span> | |
298 | </p> | |
299 | </td> | |
300 | <td> | |
301 | <p> | |
302 | <span class="blue">Max = 6.17e+03ε (Mean = 677ε)</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 5.7. 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 = 98.6ε (Mean = 15.8ε)</span> | |
353 | </p> | |
354 | </td> | |
355 | <td> | |
356 | <p> | |
357 | <span class="blue">Max = 0.96ε (Mean = 0.0635ε)</span><br> | |
358 | <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max | |
359 | = +INFε (Mean = +INFε) <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 = 107ε (Mean = 17.1ε)</span> | |
366 | </p> | |
367 | </td> | |
368 | <td> | |
369 | <p> | |
370 | <span class="blue">Max = 171ε (Mean = 22.8ε)</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+003ε (Mean = 705ε)</span> | |
383 | </p> | |
384 | </td> | |
385 | <td> | |
386 | <p> | |
387 | <span class="blue">Max = 2.11ε (Mean = 0.278ε)</span><br> | |
388 | <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max | |
389 | = +INFε (Mean = +INFε) <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+03ε (Mean = 630ε)</span> | |
396 | </p> | |
397 | </td> | |
398 | <td> | |
399 | <p> | |
400 | <span class="blue">Max = 5.1e+03ε (Mean = 577ε)</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(ν,λ;ν+λ) ≈ 0.5 | |
444 | </p> | |
445 | <p> | |
446 | and so compute the CDF when the random variable is less than ν+λ, and its | |
447 | complement when the random variable is greater than ν+λ. 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 & 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 λth 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 λ'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 & 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 © 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 Råde, 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> | |
568 | <div class="spirit-nav"> | |
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