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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.negative_binomial_dist"></a><a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative | |
28 | Binomial 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">negative_binomial</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">negative_binomial_distribution</span><span class="special">;</span> | |
36 | ||
37 | <span class="keyword">typedef</span> <span class="identifier">negative_binomial_distribution</span><span class="special"><></span> <span class="identifier">negative_binomial</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">negative_binomial_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 | <span class="comment">// Constructor from successes and success_fraction:</span> | |
46 | <span class="identifier">negative_binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span> | |
47 | ||
48 | <span class="comment">// Parameter accessors:</span> | |
49 | <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> | |
50 | <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> | |
51 | ||
52 | <span class="comment">// Bounds on success fraction:</span> | |
53 | <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span> | |
54 | <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> | |
55 | <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span> | |
56 | <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// alpha</span> | |
57 | <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_bound_on_p</span><span class="special">(</span> | |
58 | <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> | |
59 | <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span> | |
60 | <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// alpha</span> | |
61 | ||
62 | <span class="comment">// Estimate min/max number of trials:</span> | |
63 | <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span> | |
64 | <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// Number of failures.</span> | |
65 | <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// Success fraction.</span> | |
66 | <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// Probability threshold alpha.</span> | |
67 | <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span> | |
68 | <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// Number of failures.</span> | |
69 | <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// Success fraction.</span> | |
70 | <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// Probability threshold alpha.</span> | |
71 | <span class="special">};</span> | |
72 | ||
73 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
74 | </pre> | |
75 | <p> | |
76 | The class type <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span></code> | |
77 | represents a <a href="http://en.wikipedia.org/wiki/Negative_binomial_distribution" target="_top">negative_binomial | |
78 | distribution</a>: it is used when there are exactly two mutually exclusive | |
79 | outcomes of a <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli | |
80 | trial</a>: these outcomes are labelled "success" and "failure". | |
81 | </p> | |
82 | <p> | |
83 | For k + r Bernoulli trials each with success fraction p, the negative_binomial | |
84 | distribution gives the probability of observing k failures and r successes | |
85 | with success on the last trial. The negative_binomial distribution assumes | |
86 | that success_fraction p is fixed for all (k + r) trials. | |
87 | </p> | |
88 | <div class="note"><table border="0" summary="Note"> | |
89 | <tr> | |
90 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> | |
91 | <th align="left">Note</th> | |
92 | </tr> | |
93 | <tr><td align="left" valign="top"><p> | |
94 | The random variable for the negative binomial distribution is the number | |
95 | of trials, (the number of successes is a fixed property of the distribution) | |
96 | whereas for the binomial, the random variable is the number of successes, | |
97 | for a fixed number of trials. | |
98 | </p></td></tr> | |
99 | </table></div> | |
100 | <p> | |
101 | It has the PDF: | |
102 | </p> | |
103 | <p> | |
104 | <span class="inlinemediaobject"><img src="../../../../equations/neg_binomial_ref.svg"></span> | |
105 | </p> | |
106 | <p> | |
107 | The following graph illustrate how the PDF varies as the success fraction | |
108 | <span class="emphasis"><em>p</em></span> changes: | |
109 | </p> | |
110 | <p> | |
111 | <span class="inlinemediaobject"><img src="../../../../graphs/negative_binomial_pdf_1.svg" align="middle"></span> | |
112 | </p> | |
113 | <p> | |
114 | Alternatively, this graph shows how the shape of the PDF varies as the | |
115 | number of successes changes: | |
116 | </p> | |
117 | <p> | |
118 | <span class="inlinemediaobject"><img src="../../../../graphs/negative_binomial_pdf_2.svg" align="middle"></span> | |
119 | </p> | |
120 | <h5> | |
121 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h0"></a> | |
122 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.related_distributions"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.related_distributions">Related | |
123 | Distributions</a> | |
124 | </h5> | |
125 | <p> | |
126 | The name negative binomial distribution is reserved by some to the case | |
127 | where the successes parameter r is an integer. This integer version is | |
128 | also called the <a href="http://mathworld.wolfram.com/PascalDistribution.html" target="_top">Pascal | |
129 | distribution</a>. | |
130 | </p> | |
131 | <p> | |
132 | This implementation uses real numbers for the computation throughout (because | |
133 | it uses the <span class="bold"><strong>real-valued</strong></span> incomplete beta | |
134 | function family of functions). This real-valued version is also called | |
135 | the Polya Distribution. | |
136 | </p> | |
137 | <p> | |
138 | The Poisson distribution is a generalization of the Pascal distribution, | |
139 | where the success parameter r is an integer: to obtain the Pascal distribution | |
140 | you must ensure that an integer value is provided for r, and take integer | |
141 | values (floor or ceiling) from functions that return a number of successes. | |
142 | </p> | |
143 | <p> | |
144 | For large values of r (successes), the negative binomial distribution converges | |
145 | to the Poisson distribution. | |
146 | </p> | |
147 | <p> | |
148 | The geometric distribution is a special case where the successes parameter | |
149 | r = 1, so only a first and only success is required. geometric(p) = negative_binomial(1, | |
150 | p). | |
151 | </p> | |
152 | <p> | |
153 | The Poisson distribution is a special case for large successes | |
154 | </p> | |
155 | <p> | |
156 | poisson(λ) = lim <sub>r → ∞</sub>   negative_binomial(r, r / (λ + r))) | |
157 | </p> | |
158 | <div class="caution"><table border="0" summary="Caution"> | |
159 | <tr> | |
160 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td> | |
161 | <th align="left">Caution</th> | |
162 | </tr> | |
163 | <tr><td align="left" valign="top"> | |
164 | <p> | |
165 | The Negative Binomial distribution is a discrete distribution: internally, | |
166 | functions like the <code class="computeroutput"><span class="identifier">cdf</span></code> | |
167 | and <code class="computeroutput"><span class="identifier">pdf</span></code> are treated "as | |
168 | if" they are continuous functions, but in reality the results returned | |
169 | from these functions only have meaning if an integer value is provided | |
170 | for the random variate argument. | |
171 | </p> | |
172 | <p> | |
173 | The quantile function will by default return an integer result that has | |
174 | been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles | |
175 | (where the probability is less than 0.5) are rounded downward, and upper | |
176 | quantiles (where the probability is greater than 0.5) are rounded upwards. | |
177 | This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at | |
178 | least</em></span> the requested coverage will be present in the central | |
179 | region, and <span class="emphasis"><em>no more than</em></span> the requested coverage | |
180 | will be present in the tails. | |
181 | </p> | |
182 | <p> | |
183 | This behaviour can be changed so that the quantile functions are rounded | |
184 | differently, or even return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>. | |
185 | It is strongly recommended that you read the tutorial <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding | |
186 | Quantiles of Discrete Distributions</a> before using the quantile | |
187 | function on the Negative Binomial distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference | |
188 | docs</a> describe how to change the rounding policy for these distributions. | |
189 | </p> | |
190 | </td></tr> | |
191 | </table></div> | |
192 | <h5> | |
193 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h1"></a> | |
194 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.member_functions"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.member_functions">Member | |
195 | Functions</a> | |
196 | </h5> | |
197 | <h6> | |
198 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h2"></a> | |
199 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.construct"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.construct">Construct</a> | |
200 | </h6> | |
201 | <pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span> | |
202 | </pre> | |
203 | <p> | |
204 | Constructor: <span class="emphasis"><em>r</em></span> is the total number of successes, | |
205 | <span class="emphasis"><em>p</em></span> is the probability of success of a single trial. | |
206 | </p> | |
207 | <p> | |
208 | Requires: <code class="computeroutput"><span class="identifier">r</span> <span class="special">></span> | |
209 | <span class="number">0</span></code> and <code class="computeroutput"><span class="number">0</span> | |
210 | <span class="special"><=</span> <span class="identifier">p</span> | |
211 | <span class="special"><=</span> <span class="number">1</span></code>. | |
212 | </p> | |
213 | <h6> | |
214 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h3"></a> | |
215 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.accessors"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.accessors">Accessors</a> | |
216 | </h6> | |
217 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> <span class="comment">// successes / trials (0 <= p <= 1)</span> | |
218 | </pre> | |
219 | <p> | |
220 | Returns the parameter <span class="emphasis"><em>p</em></span> from which this distribution | |
221 | was constructed. | |
222 | </p> | |
223 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> <span class="comment">// required successes (r > 0)</span> | |
224 | </pre> | |
225 | <p> | |
226 | Returns the parameter <span class="emphasis"><em>r</em></span> from which this distribution | |
227 | was constructed. | |
228 | </p> | |
229 | <p> | |
230 | The best method of calculation for the following functions is disputed: | |
231 | see <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial | |
232 | Distribution</a> for more discussion. | |
233 | </p> | |
234 | <h6> | |
235 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h4"></a> | |
236 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.lower_bound_on_parameter_p"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.lower_bound_on_parameter_p">Lower | |
237 | Bound on Parameter p</a> | |
238 | </h6> | |
239 | <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span> | |
240 | <span class="identifier">RealType</span> <span class="identifier">failures</span><span class="special">,</span> | |
241 | <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span> | |
242 | <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">)</span> <span class="comment">// (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.</span> | |
243 | </pre> | |
244 | <p> | |
245 | Returns a <span class="bold"><strong>lower bound</strong></span> on the success fraction: | |
246 | </p> | |
247 | <div class="variablelist"> | |
248 | <p class="title"><b></b></p> | |
249 | <dl class="variablelist"> | |
250 | <dt><span class="term">failures</span></dt> | |
251 | <dd><p> | |
252 | The total number of failures before the <span class="emphasis"><em>r</em></span>th | |
253 | success. | |
254 | </p></dd> | |
255 | <dt><span class="term">successes</span></dt> | |
256 | <dd><p> | |
257 | The number of successes required. | |
258 | </p></dd> | |
259 | <dt><span class="term">alpha</span></dt> | |
260 | <dd><p> | |
261 | The largest acceptable probability that the true value of the success | |
262 | fraction is <span class="bold"><strong>less than</strong></span> the value | |
263 | returned. | |
264 | </p></dd> | |
265 | </dl> | |
266 | </div> | |
267 | <p> | |
268 | For example, if you observe <span class="emphasis"><em>k</em></span> failures and <span class="emphasis"><em>r</em></span> | |
269 | successes from <span class="emphasis"><em>n</em></span> = k + r trials the best estimate | |
270 | for the success fraction is simply <span class="emphasis"><em>r/n</em></span>, but if you | |
271 | want to be 95% sure that the true value is <span class="bold"><strong>greater | |
272 | than</strong></span> some value, <span class="emphasis"><em>p<sub>min</sub></em></span>, then: | |
273 | </p> | |
274 | <pre class="programlisting"><span class="identifier">p</span><sub>min</sub> <span class="special">=</span> <span class="identifier">negative_binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span> | |
275 | <span class="identifier">failures</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> | |
276 | </pre> | |
277 | <p> | |
278 | <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See | |
279 | negative binomial confidence interval example.</a> | |
280 | </p> | |
281 | <p> | |
282 | This function uses the Clopper-Pearson method of computing the lower bound | |
283 | on the success fraction, whilst many texts refer to this method as giving | |
284 | an "exact" result in practice it produces an interval that guarantees | |
285 | <span class="emphasis"><em>at least</em></span> the coverage required, and may produce pessimistic | |
286 | estimates for some combinations of <span class="emphasis"><em>failures</em></span> and <span class="emphasis"><em>successes</em></span>. | |
287 | See: | |
288 | </p> | |
289 | <p> | |
290 | <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong | |
291 | Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some | |
292 | Discrete Distributions. Computational statistics and data analysis, 2005, | |
293 | vol. 48, no3, 605-621</a>. | |
294 | </p> | |
295 | <h6> | |
296 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h5"></a> | |
297 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.upper_bound_on_parameter_p"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.upper_bound_on_parameter_p">Upper | |
298 | Bound on Parameter p</a> | |
299 | </h6> | |
300 | <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_bound_on_p</span><span class="special">(</span> | |
301 | <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> | |
302 | <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span> | |
303 | <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.</span> | |
304 | </pre> | |
305 | <p> | |
306 | Returns an <span class="bold"><strong>upper bound</strong></span> on the success | |
307 | fraction: | |
308 | </p> | |
309 | <div class="variablelist"> | |
310 | <p class="title"><b></b></p> | |
311 | <dl class="variablelist"> | |
312 | <dt><span class="term">trials</span></dt> | |
313 | <dd><p> | |
314 | The total number of trials conducted. | |
315 | </p></dd> | |
316 | <dt><span class="term">successes</span></dt> | |
317 | <dd><p> | |
318 | The number of successes that occurred. | |
319 | </p></dd> | |
320 | <dt><span class="term">alpha</span></dt> | |
321 | <dd><p> | |
322 | The largest acceptable probability that the true value of the success | |
323 | fraction is <span class="bold"><strong>greater than</strong></span> the value | |
324 | returned. | |
325 | </p></dd> | |
326 | </dl> | |
327 | </div> | |
328 | <p> | |
329 | For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span> | |
330 | trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>, | |
331 | but if you want to be 95% sure that the true value is <span class="bold"><strong>less | |
332 | than</strong></span> some value, <span class="emphasis"><em>p<sub>max</sub></em></span>, then: | |
333 | </p> | |
334 | <pre class="programlisting"><span class="identifier">p</span><sub>max</sub> <span class="special">=</span> <span class="identifier">negative_binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span> | |
335 | <span class="identifier">r</span><span class="special">,</span> <span class="identifier">k</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> | |
336 | </pre> | |
337 | <p> | |
338 | <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See | |
339 | negative binomial confidence interval example.</a> | |
340 | </p> | |
341 | <p> | |
342 | This function uses the Clopper-Pearson method of computing the lower bound | |
343 | on the success fraction, whilst many texts refer to this method as giving | |
344 | an "exact" result in practice it produces an interval that guarantees | |
345 | <span class="emphasis"><em>at least</em></span> the coverage required, and may produce pessimistic | |
346 | estimates for some combinations of <span class="emphasis"><em>failures</em></span> and <span class="emphasis"><em>successes</em></span>. | |
347 | See: | |
348 | </p> | |
349 | <p> | |
350 | <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong | |
351 | Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some | |
352 | Discrete Distributions. Computational statistics and data analysis, 2005, | |
353 | vol. 48, no3, 605-621</a>. | |
354 | </p> | |
355 | <h6> | |
356 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h6"></a> | |
357 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.estimating_number_of_trials_to_e"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.estimating_number_of_trials_to_e">Estimating | |
358 | Number of Trials to Ensure at Least a Certain Number of Failures</a> | |
359 | </h6> | |
360 | <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span> | |
361 | <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of failures.</span> | |
362 | <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction.</span> | |
363 | <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold (0.05 equivalent to 95%).</span> | |
364 | </pre> | |
365 | <p> | |
366 | This functions estimates the number of trials required to achieve a certain | |
367 | probability that <span class="bold"><strong>more than k failures will be observed</strong></span>. | |
368 | </p> | |
369 | <div class="variablelist"> | |
370 | <p class="title"><b></b></p> | |
371 | <dl class="variablelist"> | |
372 | <dt><span class="term">k</span></dt> | |
373 | <dd><p> | |
374 | The target number of failures to be observed. | |
375 | </p></dd> | |
376 | <dt><span class="term">p</span></dt> | |
377 | <dd><p> | |
378 | The probability of <span class="emphasis"><em>success</em></span> for each trial. | |
379 | </p></dd> | |
380 | <dt><span class="term">alpha</span></dt> | |
381 | <dd><p> | |
382 | The maximum acceptable risk that only k failures or fewer will be | |
383 | observed. | |
384 | </p></dd> | |
385 | </dl> | |
386 | </div> | |
387 | <p> | |
388 | For example: | |
389 | </p> | |
390 | <pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span><span class="number">10</span><span class="special">,</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> | |
391 | </pre> | |
392 | <p> | |
393 | Returns the smallest number of trials we must conduct to be 95% sure of | |
394 | seeing 10 failures that occur with frequency one half. | |
395 | </p> | |
396 | <p> | |
397 | <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html" title="Estimating Sample Sizes for the Negative Binomial.">Worked | |
398 | Example.</a> | |
399 | </p> | |
400 | <p> | |
401 | This function uses numeric inversion of the negative binomial distribution | |
402 | to obtain the result: another interpretation of the result, is that it | |
403 | finds the number of trials (success+failures) that will lead to an <span class="emphasis"><em>alpha</em></span> | |
404 | probability of observing k failures or fewer. | |
405 | </p> | |
406 | <h6> | |
407 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h7"></a> | |
408 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.estimating_number_of_trials_to_0"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.estimating_number_of_trials_to_0">Estimating | |
409 | Number of Trials to Ensure a Maximum Number of Failures or Less</a> | |
410 | </h6> | |
411 | <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span> | |
412 | <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of failures.</span> | |
413 | <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction.</span> | |
414 | <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold (0.05 equivalent to 95%).</span> | |
415 | </pre> | |
416 | <p> | |
417 | This functions estimates the maximum number of trials we can conduct and | |
418 | achieve a certain probability that <span class="bold"><strong>k failures or | |
419 | fewer will be observed</strong></span>. | |
420 | </p> | |
421 | <div class="variablelist"> | |
422 | <p class="title"><b></b></p> | |
423 | <dl class="variablelist"> | |
424 | <dt><span class="term">k</span></dt> | |
425 | <dd><p> | |
426 | The maximum number of failures to be observed. | |
427 | </p></dd> | |
428 | <dt><span class="term">p</span></dt> | |
429 | <dd><p> | |
430 | The probability of <span class="emphasis"><em>success</em></span> for each trial. | |
431 | </p></dd> | |
432 | <dt><span class="term">alpha</span></dt> | |
433 | <dd><p> | |
434 | The maximum acceptable <span class="emphasis"><em>risk</em></span> that more than k | |
435 | failures will be observed. | |
436 | </p></dd> | |
437 | </dl> | |
438 | </div> | |
439 | <p> | |
440 | For example: | |
441 | </p> | |
442 | <pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1.0</span><span class="special">-</span><span class="number">1.0</span><span class="special">/</span><span class="number">1000000</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> | |
443 | </pre> | |
444 | <p> | |
445 | Returns the largest number of trials we can conduct and still be 95% sure | |
446 | of seeing no failures that occur with frequency one in one million. | |
447 | </p> | |
448 | <p> | |
449 | This function uses numeric inversion of the negative binomial distribution | |
450 | to obtain the result: another interpretation of the result, is that it | |
451 | finds the number of trials (success+failures) that will lead to an <span class="emphasis"><em>alpha</em></span> | |
452 | probability of observing more than k failures. | |
453 | </p> | |
454 | <h5> | |
455 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h8"></a> | |
456 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.non_member_accessors"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.non_member_accessors">Non-member | |
457 | Accessors</a> | |
458 | </h5> | |
459 | <p> | |
460 | All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor | |
461 | functions</a> that are generic to all distributions are supported: | |
462 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, | |
463 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, | |
464 | <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>, | |
465 | <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>, | |
466 | <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>, | |
467 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, | |
468 | <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>, | |
469 | <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>. | |
470 | </p> | |
471 | <p> | |
472 | However it's worth taking a moment to define what these actually mean in | |
473 | the context of this distribution: | |
474 | </p> | |
475 | <div class="table"> | |
476 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table 5.3. Meaning of the non-member accessors.</b></p> | |
477 | <div class="table-contents"><table class="table" summary="Meaning of the non-member accessors."> | |
478 | <colgroup> | |
479 | <col> | |
480 | <col> | |
481 | </colgroup> | |
482 | <thead><tr> | |
483 | <th> | |
484 | <p> | |
485 | Function | |
486 | </p> | |
487 | </th> | |
488 | <th> | |
489 | <p> | |
490 | Meaning | |
491 | </p> | |
492 | </th> | |
493 | </tr></thead> | |
494 | <tbody> | |
495 | <tr> | |
496 | <td> | |
497 | <p> | |
498 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density | |
499 | Function</a> | |
500 | </p> | |
501 | </td> | |
502 | <td> | |
503 | <p> | |
504 | The probability of obtaining <span class="bold"><strong>exactly k | |
505 | failures</strong></span> from k+r trials with success fraction p. | |
506 | For example: | |
507 | </p> | |
508 | <pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">)</span></pre> | |
509 | </td> | |
510 | </tr> | |
511 | <tr> | |
512 | <td> | |
513 | <p> | |
514 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution | |
515 | Function</a> | |
516 | </p> | |
517 | </td> | |
518 | <td> | |
519 | <p> | |
520 | The probability of obtaining <span class="bold"><strong>k failures | |
521 | or fewer</strong></span> from k+r trials with success fraction p and | |
522 | success on the last trial. For example: | |
523 | </p> | |
524 | <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">)</span></pre> | |
525 | </td> | |
526 | </tr> | |
527 | <tr> | |
528 | <td> | |
529 | <p> | |
530 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of | |
531 | the Cumulative Distribution Function</a> | |
532 | </p> | |
533 | </td> | |
534 | <td> | |
535 | <p> | |
536 | The probability of obtaining <span class="bold"><strong>more than | |
537 | k failures</strong></span> from k+r trials with success fraction p | |
538 | and success on the last trial. For example: | |
539 | </p> | |
540 | <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">))</span></pre> | |
541 | </td> | |
542 | </tr> | |
543 | <tr> | |
544 | <td> | |
545 | <p> | |
546 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a> | |
547 | </p> | |
548 | </td> | |
549 | <td> | |
550 | <p> | |
551 | The <span class="bold"><strong>greatest</strong></span> number of failures | |
552 | k expected to be observed from k+r trials with success fraction | |
553 | p, at probability P. Note that the value returned is a real-number, | |
554 | and not an integer. Depending on the use case you may want to | |
555 | take either the floor or ceiling of the real result. For example: | |
556 | </p> | |
557 | <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">P</span><span class="special">)</span></pre> | |
558 | </td> | |
559 | </tr> | |
560 | <tr> | |
561 | <td> | |
562 | <p> | |
563 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile | |
564 | from the complement of the probability</a> | |
565 | </p> | |
566 | </td> | |
567 | <td> | |
568 | <p> | |
569 | The <span class="bold"><strong>smallest</strong></span> number of failures | |
570 | k expected to be observed from k+r trials with success fraction | |
571 | p, at probability P. Note that the value returned is a real-number, | |
572 | and not an integer. Depending on the use case you may want to | |
573 | take either the floor or ceiling of the real result. For example: | |
574 | </p> | |
575 | <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">P</span><span class="special">))</span></pre> | |
576 | </td> | |
577 | </tr> | |
578 | </tbody> | |
579 | </table></div> | |
580 | </div> | |
581 | <br class="table-break"><h5> | |
582 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h9"></a> | |
583 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.accuracy"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.accuracy">Accuracy</a> | |
584 | </h5> | |
585 | <p> | |
586 | This distribution is implemented using the incomplete beta functions <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>: | |
587 | please refer to these functions for information on accuracy. | |
588 | </p> | |
589 | <h5> | |
590 | <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h10"></a> | |
591 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.implementation"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.implementation">Implementation</a> | |
592 | </h5> | |
593 | <p> | |
594 | In the following table, <span class="emphasis"><em>p</em></span> is the probability that | |
595 | any one trial will be successful (the success fraction), <span class="emphasis"><em>r</em></span> | |
596 | is the number of successes, <span class="emphasis"><em>k</em></span> is the number of failures, | |
597 | <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>. | |
598 | </p> | |
599 | <div class="informaltable"><table class="table"> | |
600 | <colgroup> | |
601 | <col> | |
602 | <col> | |
603 | </colgroup> | |
604 | <thead><tr> | |
605 | <th> | |
606 | <p> | |
607 | Function | |
608 | </p> | |
609 | </th> | |
610 | <th> | |
611 | <p> | |
612 | Implementation Notes | |
613 | </p> | |
614 | </th> | |
615 | </tr></thead> | |
616 | <tbody> | |
617 | <tr> | |
618 | <td> | |
619 | <p> | |
620 | ||
621 | </p> | |
622 | </td> | |
623 | <td> | |
624 | <p> | |
625 | pdf = exp(lgamma(r + k) - lgamma(r) - lgamma(k+1)) * pow(p, r) | |
626 | * pow((1-p), k) | |
627 | </p> | |
628 | <p> | |
629 | Implementation is in terms of <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>: | |
630 | </p> | |
631 | <p> | |
632 | (p/(r + k)) * ibeta_derivative(r, static_cast<RealType>(k+1), | |
633 | p) The function <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a> | |
634 | is used here, since it has already been optimised for the lowest | |
635 | possible error - indeed this is really just a thin wrapper around | |
636 | part of the internals of the incomplete beta function. | |
637 | </p> | |
638 | </td> | |
639 | </tr> | |
640 | <tr> | |
641 | <td> | |
642 | <p> | |
643 | cdf | |
644 | </p> | |
645 | </td> | |
646 | <td> | |
647 | <p> | |
648 | Using the relation: | |
649 | </p> | |
650 | <p> | |
651 | cdf = I<sub>p</sub>(r, k+1) = ibeta(r, k+1, p) | |
652 | </p> | |
653 | <p> | |
654 | = ibeta(r, static_cast<RealType>(k+1), p) | |
655 | </p> | |
656 | </td> | |
657 | </tr> | |
658 | <tr> | |
659 | <td> | |
660 | <p> | |
661 | cdf complement | |
662 | </p> | |
663 | </td> | |
664 | <td> | |
665 | <p> | |
666 | Using the relation: | |
667 | </p> | |
668 | <p> | |
669 | 1 - cdf = I<sub>p</sub>(k+1, r) | |
670 | </p> | |
671 | <p> | |
672 | = ibetac(r, static_cast<RealType>(k+1), p) | |
673 | </p> | |
674 | </td> | |
675 | </tr> | |
676 | <tr> | |
677 | <td> | |
678 | <p> | |
679 | quantile | |
680 | </p> | |
681 | </td> | |
682 | <td> | |
683 | <p> | |
684 | ibeta_invb(r, p, P) - 1 | |
685 | </p> | |
686 | </td> | |
687 | </tr> | |
688 | <tr> | |
689 | <td> | |
690 | <p> | |
691 | quantile from the complement | |
692 | </p> | |
693 | </td> | |
694 | <td> | |
695 | <p> | |
696 | ibetac_invb(r, p, Q) -1) | |
697 | </p> | |
698 | </td> | |
699 | </tr> | |
700 | <tr> | |
701 | <td> | |
702 | <p> | |
703 | mean | |
704 | </p> | |
705 | </td> | |
706 | <td> | |
707 | <p> | |
708 | <code class="computeroutput"><span class="identifier">r</span><span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">)/</span><span class="identifier">p</span></code> | |
709 | </p> | |
710 | </td> | |
711 | </tr> | |
712 | <tr> | |
713 | <td> | |
714 | <p> | |
715 | variance | |
716 | </p> | |
717 | </td> | |
718 | <td> | |
719 | <p> | |
720 | <code class="computeroutput"><span class="identifier">r</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">)</span> | |
721 | <span class="special">/</span> <span class="identifier">p</span> | |
722 | <span class="special">*</span> <span class="identifier">p</span></code> | |
723 | </p> | |
724 | </td> | |
725 | </tr> | |
726 | <tr> | |
727 | <td> | |
728 | <p> | |
729 | mode | |
730 | </p> | |
731 | </td> | |
732 | <td> | |
733 | <p> | |
734 | <code class="computeroutput"><span class="identifier">floor</span><span class="special">((</span><span class="identifier">r</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span> <span class="identifier">p</span><span class="special">)/</span><span class="identifier">p</span><span class="special">)</span></code> | |
735 | </p> | |
736 | </td> | |
737 | </tr> | |
738 | <tr> | |
739 | <td> | |
740 | <p> | |
741 | skewness | |
742 | </p> | |
743 | </td> | |
744 | <td> | |
745 | <p> | |
746 | <code class="computeroutput"><span class="special">(</span><span class="number">2</span> | |
747 | <span class="special">-</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span> | |
748 | <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">r</span> <span class="special">*</span> | |
749 | <span class="special">(</span><span class="number">1</span> | |
750 | <span class="special">-</span> <span class="identifier">p</span><span class="special">))</span></code> | |
751 | </p> | |
752 | </td> | |
753 | </tr> | |
754 | <tr> | |
755 | <td> | |
756 | <p> | |
757 | kurtosis | |
758 | </p> | |
759 | </td> | |
760 | <td> | |
761 | <p> | |
762 | <code class="computeroutput"><span class="number">6</span> <span class="special">/</span> | |
763 | <span class="identifier">r</span> <span class="special">+</span> | |
764 | <span class="special">(</span><span class="identifier">p</span> | |
765 | <span class="special">*</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span> | |
766 | <span class="identifier">r</span> <span class="special">*</span> | |
767 | <span class="special">(</span><span class="number">1</span> | |
768 | <span class="special">-</span> <span class="identifier">p</span> | |
769 | <span class="special">)</span></code> | |
770 | </p> | |
771 | </td> | |
772 | </tr> | |
773 | <tr> | |
774 | <td> | |
775 | <p> | |
776 | kurtosis excess | |
777 | </p> | |
778 | </td> | |
779 | <td> | |
780 | <p> | |
781 | <code class="computeroutput"><span class="number">6</span> <span class="special">/</span> | |
782 | <span class="identifier">r</span> <span class="special">+</span> | |
783 | <span class="special">(</span><span class="identifier">p</span> | |
784 | <span class="special">*</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span> | |
785 | <span class="identifier">r</span> <span class="special">*</span> | |
786 | <span class="special">(</span><span class="number">1</span> | |
787 | <span class="special">-</span> <span class="identifier">p</span> | |
788 | <span class="special">)</span> <span class="special">-</span><span class="number">3</span></code> | |
789 | </p> | |
790 | </td> | |
791 | </tr> | |
792 | <tr> | |
793 | <td> | |
794 | <p> | |
795 | parameter estimation member functions | |
796 | </p> | |
797 | </td> | |
798 | <td> | |
799 | </td> | |
800 | </tr> | |
801 | <tr> | |
802 | <td> | |
803 | <p> | |
804 | <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code> | |
805 | </p> | |
806 | </td> | |
807 | <td> | |
808 | <p> | |
809 | ibeta_inv(successes, failures + 1, alpha) | |
810 | </p> | |
811 | </td> | |
812 | </tr> | |
813 | <tr> | |
814 | <td> | |
815 | <p> | |
816 | <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code> | |
817 | </p> | |
818 | </td> | |
819 | <td> | |
820 | <p> | |
821 | ibetac_inv(successes, failures, alpha) plus see comments in code. | |
822 | </p> | |
823 | </td> | |
824 | </tr> | |
825 | <tr> | |
826 | <td> | |
827 | <p> | |
828 | <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code> | |
829 | </p> | |
830 | </td> | |
831 | <td> | |
832 | <p> | |
833 | ibeta_inva(k + 1, p, alpha) | |
834 | </p> | |
835 | </td> | |
836 | </tr> | |
837 | <tr> | |
838 | <td> | |
839 | <p> | |
840 | <code class="computeroutput"><span class="identifier">find_maximum_number_of_trials</span></code> | |
841 | </p> | |
842 | </td> | |
843 | <td> | |
844 | <p> | |
845 | ibetac_inva(k + 1, p, alpha) | |
846 | </p> | |
847 | </td> | |
848 | </tr> | |
849 | </tbody> | |
850 | </table></div> | |
851 | <p> | |
852 | Implementation notes: | |
853 | </p> | |
854 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
855 | <li class="listitem"> | |
856 | The real concept type (that deliberately lacks the Lanczos approximation), | |
857 | was found to take several minutes to evaluate some extreme test values, | |
858 | so the test has been disabled for this type. | |
859 | </li> | |
860 | <li class="listitem"> | |
861 | Much greater speed, and perhaps greater accuracy, might be achieved | |
862 | for extreme values by using a normal approximation. This is NOT been | |
863 | tested or implemented. | |
864 | </li> | |
865 | </ul></div> | |
866 | </div> | |
867 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
868 | <td align="left"></td> | |
869 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
870 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
871 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
872 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
873 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
874 | 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>) | |
875 | </p> | |
876 | </div></td> | |
877 | </tr></table> | |
878 | <hr> | |
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