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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.inverse_gaussian_dist"></a><a class="link" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution">Inverse | |
28 | Gaussian (or Inverse Normal) 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">inverse_gaussian</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">inverse_gaussian_distribution</span> | |
36 | <span class="special">{</span> | |
37 | <span class="keyword">public</span><span class="special">:</span> | |
38 | <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> | |
39 | <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> | |
40 | ||
41 | <span class="identifier">inverse_gaussian_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> | |
42 | ||
43 | <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// mean default 1.</span> | |
44 | <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Optional scale, default 1 (unscaled).</span> | |
45 | <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Shape = scale/mean.</span> | |
46 | <span class="special">};</span> | |
47 | <span class="keyword">typedef</span> <span class="identifier">inverse_gaussian_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">inverse_gaussian</span><span class="special">;</span> | |
48 | ||
49 | <span class="special">}}</span> <span class="comment">// namespace boost // namespace math</span> | |
50 | </pre> | |
51 | <p> | |
52 | The Inverse Gaussian distribution distribution is a continuous probability | |
53 | distribution. | |
54 | </p> | |
55 | <p> | |
56 | The distribution is also called 'normal-inverse Gaussian distribution', | |
57 | and 'normal Inverse' distribution. | |
58 | </p> | |
59 | <p> | |
60 | It is also convenient to provide unity as default for both mean and scale. | |
61 | This is the Standard form for all distributions. The Inverse Gaussian distribution | |
62 | was first studied in relation to Brownian motion. In 1956 M.C.K. Tweedie | |
63 | used the name Inverse Gaussian because there is an inverse relationship | |
64 | between the time to cover a unit distance and distance covered in unit | |
65 | time. The inverse Gaussian is one of family of distributions that have | |
66 | been called the <a href="http://en.wikipedia.org/wiki/Tweedie_distributions" target="_top">Tweedie | |
67 | distributions</a>. | |
68 | </p> | |
69 | <p> | |
70 | (So <span class="emphasis"><em>inverse</em></span> in the name may mislead: it does <span class="bold"><strong>not</strong></span> relate to the inverse of a distribution). | |
71 | </p> | |
72 | <p> | |
73 | The tails of the distribution decrease more slowly than the normal distribution. | |
74 | It is therefore suitable to model phenomena where numerically large values | |
75 | are more probable than is the case for the normal distribution. For stock | |
76 | market returns and prices, a key characteristic is that it models that | |
77 | extremely large variations from typical (crashes) can occur even when almost | |
78 | all (normal) variations are small. | |
79 | </p> | |
80 | <p> | |
81 | Examples are returns from financial assets and turbulent wind speeds. | |
82 | </p> | |
83 | <p> | |
84 | The normal-inverse Gaussian distributions form a subclass of the generalised | |
85 | hyperbolic distributions. | |
86 | </p> | |
87 | <p> | |
88 | See <a href="http://en.wikipedia.org/wiki/Normal-inverse_Gaussian_distribution" target="_top">distribution</a>. | |
89 | <a href="http://mathworld.wolfram.com/InverseGaussianDistribution.html" target="_top">Weisstein, | |
90 | Eric W. "Inverse Gaussian Distribution." From MathWorld--A Wolfram | |
91 | Web Resource.</a> | |
92 | </p> | |
93 | <p> | |
94 | If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision | |
95 | inverse_gaussian distribution you can use | |
96 | </p> | |
97 | <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_gaussian_distribution</span><span class="special"><></span></pre> | |
98 | <p> | |
99 | or, more conveniently, you can write | |
100 | </p> | |
101 | <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_gaussian</span><span class="special">;</span> | |
102 | <span class="identifier">inverse_gaussian</span> <span class="identifier">my_ig</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span> | |
103 | </pre> | |
104 | <p> | |
105 | For mean parameters μ and scale (also called precision) parameter λ, and random | |
106 | variate x, the inverse_gaussian distribution is defined by the probability | |
107 | density function (PDF): | |
108 | </p> | |
109 | <p> | |
110 |    f(x;μ, λ) = √(λ/2πx<sup>3</sup>) e<sup>-λ(x-μ)²/2μ²x</sup> | |
111 | </p> | |
112 | <p> | |
113 | and Cumulative Density Function (CDF): | |
114 | </p> | |
115 | <p> | |
116 |    F(x;μ, λ) = Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)} + e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)} | |
117 | </p> | |
118 | <p> | |
119 | where Φ is the standard normal distribution CDF. | |
120 | </p> | |
121 | <p> | |
122 | The following graphs illustrate how the PDF and CDF of the inverse_gaussian | |
123 | distribution varies for a few values of parameters μ and λ: | |
124 | </p> | |
125 | <p> | |
126 | <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_pdf.svg" align="middle"></span> | |
127 | </p> | |
128 | <p> | |
129 | <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_cdf.svg" align="middle"></span> | |
130 | </p> | |
131 | <p> | |
132 | Tweedie also provided 3 other parameterisations where (μ and λ) are replaced | |
133 | by their ratio φ = λ/μ and by 1/μ: these forms may be more suitable for Bayesian | |
134 | applications. These can be found on Seshadri, page 2 and are also discussed | |
135 | by Chhikara and Folks on page 105. Another related parameterisation, the | |
136 | __wald_distrib (where mean μ is unity) is also provided. | |
137 | </p> | |
138 | <h5> | |
139 | <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h0"></a> | |
140 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.member_functions"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.member_functions">Member | |
141 | Functions</a> | |
142 | </h5> | |
143 | <pre class="programlisting"><span class="identifier">inverse_gaussian_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// optionally scaled.</span> | |
144 | </pre> | |
145 | <p> | |
146 | Constructs an inverse_gaussian distribution with μ mean, and scale λ, with | |
147 | both default values 1. | |
148 | </p> | |
149 | <p> | |
150 | Requires that both the mean μ parameter and scale λ are greater than zero, | |
151 | otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. | |
152 | </p> | |
153 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
154 | </pre> | |
155 | <p> | |
156 | Returns the mean μ parameter of this distribution. | |
157 | </p> | |
158 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
159 | </pre> | |
160 | <p> | |
161 | Returns the scale λ parameter of this distribution. | |
162 | </p> | |
163 | <h5> | |
164 | <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h1"></a> | |
165 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.non_member_accessors"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.non_member_accessors">Non-member | |
166 | Accessors</a> | |
167 | </h5> | |
168 | <p> | |
169 | All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor | |
170 | functions</a> that are generic to all distributions are supported: | |
171 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, | |
172 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, | |
173 | <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>, | |
174 | <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>, | |
175 | <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>, | |
176 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, | |
177 | <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>, | |
178 | <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>. | |
179 | </p> | |
180 | <p> | |
181 | The domain of the random variate is [0,+∞). | |
182 | </p> | |
183 | <div class="note"><table border="0" summary="Note"> | |
184 | <tr> | |
185 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> | |
186 | <th align="left">Note</th> | |
187 | </tr> | |
188 | <tr><td align="left" valign="top"><p> | |
189 | Unlike some definitions, this implementation supports a random variate | |
190 | equal to zero as a special case, returning zero for both pdf and cdf. | |
191 | </p></td></tr> | |
192 | </table></div> | |
193 | <h5> | |
194 | <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h2"></a> | |
195 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.accuracy"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.accuracy">Accuracy</a> | |
196 | </h5> | |
197 | <p> | |
198 | The inverse_gaussian distribution is implemented in terms of the exponential | |
199 | function and standard normal distribution <span class="emphasis"><em>N</em></span>0,1 Φ : refer | |
200 | to the accuracy data for those functions for more information. But in general, | |
201 | gamma (and thus inverse gamma) results are often accurate to a few epsilon, | |
202 | >14 decimal digits accuracy for 64-bit double. | |
203 | </p> | |
204 | <h5> | |
205 | <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h3"></a> | |
206 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.implementation"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.implementation">Implementation</a> | |
207 | </h5> | |
208 | <p> | |
209 | In the following table μ is the mean parameter and λ is the scale parameter | |
210 | of the inverse_gaussian distribution, <span class="emphasis"><em>x</em></span> is the random | |
211 | variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span> | |
212 | its complement. Parameters μ for shape and λ for scale are used for the inverse | |
213 | gaussian function. | |
214 | </p> | |
215 | <div class="informaltable"><table class="table"> | |
216 | <colgroup> | |
217 | <col> | |
218 | <col> | |
219 | </colgroup> | |
220 | <thead><tr> | |
221 | <th> | |
222 | <p> | |
223 | Function | |
224 | </p> | |
225 | </th> | |
226 | <th> | |
227 | <p> | |
228 | Implementation Notes | |
229 | </p> | |
230 | </th> | |
231 | </tr></thead> | |
232 | <tbody> | |
233 | <tr> | |
234 | <td> | |
235 | <p> | |
236 | ||
237 | </p> | |
238 | </td> | |
239 | <td> | |
240 | <p> | |
241 | √(λ/ 2πx<sup>3</sup>) e<sup>-λ(x - μ)²/ 2μ²x</sup> | |
242 | </p> | |
243 | </td> | |
244 | </tr> | |
245 | <tr> | |
246 | <td> | |
247 | <p> | |
248 | cdf | |
249 | </p> | |
250 | </td> | |
251 | <td> | |
252 | <p> | |
253 | Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)} + e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)} | |
254 | </p> | |
255 | </td> | |
256 | </tr> | |
257 | <tr> | |
258 | <td> | |
259 | <p> | |
260 | cdf complement | |
261 | </p> | |
262 | </td> | |
263 | <td> | |
264 | <p> | |
265 | using complement of Φ above. | |
266 | </p> | |
267 | </td> | |
268 | </tr> | |
269 | <tr> | |
270 | <td> | |
271 | <p> | |
272 | quantile | |
273 | </p> | |
274 | </td> | |
275 | <td> | |
276 | <p> | |
277 | No closed form known. Estimated using a guess refined by Newton-Raphson | |
278 | iteration. | |
279 | </p> | |
280 | </td> | |
281 | </tr> | |
282 | <tr> | |
283 | <td> | |
284 | <p> | |
285 | quantile from the complement | |
286 | </p> | |
287 | </td> | |
288 | <td> | |
289 | <p> | |
290 | No closed form known. Estimated using a guess refined by Newton-Raphson | |
291 | iteration. | |
292 | </p> | |
293 | </td> | |
294 | </tr> | |
295 | <tr> | |
296 | <td> | |
297 | <p> | |
298 | mode | |
299 | </p> | |
300 | </td> | |
301 | <td> | |
302 | <p> | |
303 | μ {√(1+9μ²/4λ²)² - 3μ/2λ} | |
304 | </p> | |
305 | </td> | |
306 | </tr> | |
307 | <tr> | |
308 | <td> | |
309 | <p> | |
310 | median | |
311 | </p> | |
312 | </td> | |
313 | <td> | |
314 | <p> | |
315 | No closed form analytic equation is known, but is evaluated as | |
316 | quantile(0.5) | |
317 | </p> | |
318 | </td> | |
319 | </tr> | |
320 | <tr> | |
321 | <td> | |
322 | <p> | |
323 | mean | |
324 | </p> | |
325 | </td> | |
326 | <td> | |
327 | <p> | |
328 | μ | |
329 | </p> | |
330 | </td> | |
331 | </tr> | |
332 | <tr> | |
333 | <td> | |
334 | <p> | |
335 | variance | |
336 | </p> | |
337 | </td> | |
338 | <td> | |
339 | <p> | |
340 | μ³/λ | |
341 | </p> | |
342 | </td> | |
343 | </tr> | |
344 | <tr> | |
345 | <td> | |
346 | <p> | |
347 | skewness | |
348 | </p> | |
349 | </td> | |
350 | <td> | |
351 | <p> | |
352 | 3 √ (μ/λ) | |
353 | </p> | |
354 | </td> | |
355 | </tr> | |
356 | <tr> | |
357 | <td> | |
358 | <p> | |
359 | kurtosis_excess | |
360 | </p> | |
361 | </td> | |
362 | <td> | |
363 | <p> | |
364 | 15μ/λ | |
365 | </p> | |
366 | </td> | |
367 | </tr> | |
368 | <tr> | |
369 | <td> | |
370 | <p> | |
371 | kurtosis | |
372 | </p> | |
373 | </td> | |
374 | <td> | |
375 | <p> | |
376 | 12μ/λ | |
377 | </p> | |
378 | </td> | |
379 | </tr> | |
380 | </tbody> | |
381 | </table></div> | |
382 | <h5> | |
383 | <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h4"></a> | |
384 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.references"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.references">References</a> | |
385 | </h5> | |
386 | <div class="orderedlist"><ol class="orderedlist" type="1"> | |
387 | <li class="listitem"> | |
388 | Wald, A. (1947). Sequential analysis. Wiley, NY. | |
389 | </li> | |
390 | <li class="listitem"> | |
391 | The Inverse Gaussian distribution : theory, methodology, and applications, | |
392 | Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989). | |
393 | </li> | |
394 | <li class="listitem"> | |
395 | The Inverse Gaussian distribution : statistical theory and applications, | |
396 | Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998). | |
397 | </li> | |
398 | <li class="listitem"> | |
399 | <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.wald.html" target="_top">Numpy | |
400 | and Scipy Documentation</a>. | |
401 | </li> | |
402 | <li class="listitem"> | |
403 | <a href="http://bm2.genes.nig.ac.jp/RGM2/R_current/library/statmod/man/invgauss.html" target="_top">R | |
404 | statmod invgauss functions</a>. | |
405 | </li> | |
406 | <li class="listitem"> | |
407 | <a href="http://cran.r-project.org/web/packages/SuppDists/index.html" target="_top">R | |
408 | SuppDists invGauss functions</a>. (Note that these R implementations | |
409 | names differ in case). | |
410 | </li> | |
411 | <li class="listitem"> | |
412 | <a href="http://www.statsci.org/s/invgauss.html" target="_top">StatSci.org invgauss | |
413 | help</a>. | |
414 | </li> | |
415 | <li class="listitem"> | |
416 | <a href="http://www.statsci.org/s/invgauss.statSci.org" target="_top">invgauss | |
417 | R source</a>. | |
418 | </li> | |
419 | <li class="listitem"> | |
420 | <a href="http://www.biostat.wustl.edu/archives/html/s-news/2001-12/msg00144.html" target="_top">pwald, | |
421 | qwald</a>. | |
422 | </li> | |
423 | <li class="listitem"> | |
424 | <a href="http://www.brighton-webs.co.uk/distributions/wald.asp" target="_top">Brighton | |
425 | Webs wald</a>. | |
426 | </li> | |
427 | </ol></div> | |
428 | </div> | |
429 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
430 | <td align="left"></td> | |
431 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
432 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
433 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
434 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
435 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
436 | 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>) | |
437 | </p> | |
438 | </div></td> | |
439 | </tr></table> | |
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