]>
Commit | Line | Data |
---|---|---|
7c673cae FG |
1 | <html> |
2 | <head> | |
3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
4 | <title>Laplace Distribution</title> | |
5 | <link rel="stylesheet" href="../../../math.css" type="text/css"> | |
6 | <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> | |
7 | <link rel="home" href="../../../index.html" title="Math Toolkit 2.5.1"> | |
8 | <link rel="up" href="../dists.html" title="Distributions"> | |
9 | <link rel="prev" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution"> | |
10 | <link rel="next" href="logistic_dist.html" title="Logistic Distribution"> | |
11 | </head> | |
12 | <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> | |
13 | <table cellpadding="2" width="100%"><tr> | |
14 | <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> | |
15 | <td align="center"><a href="../../../../../../../index.html">Home</a></td> | |
16 | <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> | |
17 | <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> | |
18 | <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> | |
19 | <td align="center"><a href="../../../../../../../more/index.htm">More</a></td> | |
20 | </tr></table> | |
21 | <hr> | |
22 | <div class="spirit-nav"> | |
23 | <a accesskey="p" href="inverse_gaussian_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="logistic_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> | |
24 | </div> | |
25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h4 class="title"> | |
27 | <a name="math_toolkit.dist_ref.dists.laplace_dist"></a><a class="link" href="laplace_dist.html" title="Laplace Distribution">Laplace Distribution</a> | |
28 | </h4></div></div></div> | |
29 | <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">laplace</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> | |
30 | <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> | |
31 | ||
32 | <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> | |
33 | <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> | |
34 | <span class="keyword">class</span> <span class="identifier">laplace_distribution</span><span class="special">;</span> | |
35 | ||
36 | <span class="keyword">typedef</span> <span class="identifier">laplace_distribution</span><span class="special"><></span> <span class="identifier">laplace</span><span class="special">;</span> | |
37 | ||
38 | <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> | |
39 | <span class="keyword">class</span> <span class="identifier">laplace_distribution</span> | |
40 | <span class="special">{</span> | |
41 | <span class="keyword">public</span><span class="special">:</span> | |
42 | <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> | |
43 | <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> | |
44 | <span class="comment">// Construct:</span> | |
45 | <span class="identifier">laplace_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</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> | |
46 | <span class="comment">// Accessors:</span> | |
47 | <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
48 | <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
49 | <span class="special">};</span> | |
50 | ||
51 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
52 | </pre> | |
53 | <p> | |
54 | Laplace distribution is the distribution of differences between two independent | |
55 | variates with identical exponential distributions (Abramowitz and Stegun | |
56 | 1972, p. 930). It is also called the double exponential distribution. | |
57 | </p> | |
58 | <p> | |
59 | For location parameter μ   and scale parameter σ  , it is defined by the probability | |
60 | density function: | |
61 | </p> | |
62 | <p> | |
63 | <span class="inlinemediaobject"><img src="../../../../equations/laplace_pdf.svg"></span> | |
64 | </p> | |
65 | <p> | |
66 | The location and scale parameters are equivalent to the mean and standard | |
67 | deviation of the normal or Gaussian distribution. | |
68 | </p> | |
69 | <p> | |
70 | The following graph illustrates the effect of the parameters μ   and σ   on the | |
71 | PDF. Note that the domain of the random variable remains [-∞,+∞] irrespective | |
72 | of the value of the location parameter: | |
73 | </p> | |
74 | <p> | |
75 | <span class="inlinemediaobject"><img src="../../../../graphs/laplace_pdf.svg" align="middle"></span> | |
76 | </p> | |
77 | <h5> | |
78 | <a name="math_toolkit.dist_ref.dists.laplace_dist.h0"></a> | |
79 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.member_functions"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.member_functions">Member | |
80 | Functions</a> | |
81 | </h5> | |
82 | <pre class="programlisting"><span class="identifier">laplace_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</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> | |
83 | </pre> | |
84 | <p> | |
85 | Constructs a laplace distribution with location <span class="emphasis"><em>location</em></span> | |
86 | and scale <span class="emphasis"><em>scale</em></span>. | |
87 | </p> | |
88 | <p> | |
89 | The location parameter is the same as the mean of the random variate. | |
90 | </p> | |
91 | <p> | |
92 | The scale parameter is proportional to the standard deviation of the random | |
93 | variate. | |
94 | </p> | |
95 | <p> | |
96 | Requires that the scale parameter is greater than zero, otherwise calls | |
97 | <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. | |
98 | </p> | |
99 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
100 | </pre> | |
101 | <p> | |
102 | Returns the <span class="emphasis"><em>location</em></span> parameter of this distribution. | |
103 | </p> | |
104 | <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> | |
105 | </pre> | |
106 | <p> | |
107 | Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution. | |
108 | </p> | |
109 | <h5> | |
110 | <a name="math_toolkit.dist_ref.dists.laplace_dist.h1"></a> | |
111 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.non_member_accessors"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.non_member_accessors">Non-member | |
112 | Accessors</a> | |
113 | </h5> | |
114 | <p> | |
115 | All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor | |
116 | functions</a> that are generic to all distributions are supported: | |
117 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, | |
118 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, | |
119 | <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>, | |
120 | <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>, | |
121 | <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>, | |
122 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, | |
123 | <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>, | |
124 | <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>. | |
125 | </p> | |
126 | <p> | |
127 | The domain of the random variable is [-∞,+∞]. | |
128 | </p> | |
129 | <h5> | |
130 | <a name="math_toolkit.dist_ref.dists.laplace_dist.h2"></a> | |
131 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.accuracy"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.accuracy">Accuracy</a> | |
132 | </h5> | |
133 | <p> | |
134 | The laplace distribution is implemented in terms of the standard library | |
135 | log and exp functions and as such should have very small errors. | |
136 | </p> | |
137 | <h5> | |
138 | <a name="math_toolkit.dist_ref.dists.laplace_dist.h3"></a> | |
139 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.implementation"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.implementation">Implementation</a> | |
140 | </h5> | |
141 | <p> | |
142 | In the following table μ is the location parameter of the distribution, σ is | |
143 | its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> | |
144 | is the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>. | |
145 | </p> | |
146 | <div class="informaltable"><table class="table"> | |
147 | <colgroup> | |
148 | <col> | |
149 | <col> | |
150 | </colgroup> | |
151 | <thead><tr> | |
152 | <th> | |
153 | <p> | |
154 | Function | |
155 | </p> | |
156 | </th> | |
157 | <th> | |
158 | <p> | |
159 | Implementation Notes | |
160 | </p> | |
161 | </th> | |
162 | </tr></thead> | |
163 | <tbody> | |
164 | <tr> | |
165 | <td> | |
166 | <p> | |
167 | ||
168 | </p> | |
169 | </td> | |
170 | <td> | |
171 | <p> | |
172 | Using the relation: pdf = e<sup>-abs(x-μ) / σ</sup> / (2 * σ) | |
173 | </p> | |
174 | </td> | |
175 | </tr> | |
176 | <tr> | |
177 | <td> | |
178 | <p> | |
179 | cdf | |
180 | </p> | |
181 | </td> | |
182 | <td> | |
183 | <p> | |
184 | Using the relations: | |
185 | </p> | |
186 | <p> | |
187 | x < μ : p = e<sup>(x-μ)/σ </sup> / σ | |
188 | </p> | |
189 | <p> | |
190 | x >= μ : p = 1 - e<sup>(μ-x)/σ </sup> / σ | |
191 | ||
192 | </p> | |
193 | </td> | |
194 | </tr> | |
195 | <tr> | |
196 | <td> | |
197 | <p> | |
198 | cdf complement | |
199 | </p> | |
200 | </td> | |
201 | <td> | |
202 | <p> | |
203 | Using the relation: | |
204 | </p> | |
205 | <p> | |
206 | -x < μ : q = e<sup>(-x-μ)/σ </sup> / σ | |
207 | </p> | |
208 | <p> | |
209 | -x >= μ : q = 1 - e<sup>(μ+x)/σ </sup> / σ | |
210 | ||
211 | </p> | |
212 | </td> | |
213 | </tr> | |
214 | <tr> | |
215 | <td> | |
216 | <p> | |
217 | quantile | |
218 | </p> | |
219 | </td> | |
220 | <td> | |
221 | <p> | |
222 | Using the relations: | |
223 | </p> | |
224 | <p> | |
225 | p < 0.5 : x = μ + σ * log(2*p) | |
226 | </p> | |
227 | <p> | |
228 | p >= 0.5 : x = μ - σ * log(2-2*p) | |
229 | </p> | |
230 | </td> | |
231 | </tr> | |
232 | <tr> | |
233 | <td> | |
234 | <p> | |
235 | quantile from the complement | |
236 | </p> | |
237 | </td> | |
238 | <td> | |
239 | <p> | |
240 | Using the relation: | |
241 | </p> | |
242 | <p> | |
243 | q > 0.5: x = μ + σ*log(2-2*q) | |
244 | </p> | |
245 | <p> | |
246 | q <=0.5: x = μ - σ*log( 2*q ) | |
247 | </p> | |
248 | </td> | |
249 | </tr> | |
250 | <tr> | |
251 | <td> | |
252 | <p> | |
253 | mean | |
254 | </p> | |
255 | </td> | |
256 | <td> | |
257 | <p> | |
258 | μ | |
259 | </p> | |
260 | </td> | |
261 | </tr> | |
262 | <tr> | |
263 | <td> | |
264 | <p> | |
265 | variance | |
266 | </p> | |
267 | </td> | |
268 | <td> | |
269 | <p> | |
270 | 2 * σ<sup>2</sup> | |
271 | </p> | |
272 | </td> | |
273 | </tr> | |
274 | <tr> | |
275 | <td> | |
276 | <p> | |
277 | mode | |
278 | </p> | |
279 | </td> | |
280 | <td> | |
281 | <p> | |
282 | μ | |
283 | </p> | |
284 | </td> | |
285 | </tr> | |
286 | <tr> | |
287 | <td> | |
288 | <p> | |
289 | skewness | |
290 | </p> | |
291 | </td> | |
292 | <td> | |
293 | <p> | |
294 | 0 | |
295 | </p> | |
296 | </td> | |
297 | </tr> | |
298 | <tr> | |
299 | <td> | |
300 | <p> | |
301 | kurtosis | |
302 | </p> | |
303 | </td> | |
304 | <td> | |
305 | <p> | |
306 | 6 | |
307 | </p> | |
308 | </td> | |
309 | </tr> | |
310 | <tr> | |
311 | <td> | |
312 | <p> | |
313 | kurtosis excess | |
314 | </p> | |
315 | </td> | |
316 | <td> | |
317 | <p> | |
318 | 3 | |
319 | </p> | |
320 | </td> | |
321 | </tr> | |
322 | </tbody> | |
323 | </table></div> | |
324 | <h5> | |
325 | <a name="math_toolkit.dist_ref.dists.laplace_dist.h4"></a> | |
326 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.references"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.references">References</a> | |
327 | </h5> | |
328 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
329 | <li class="listitem"> | |
330 | <a href="http://mathworld.wolfram.com/LaplaceDistribution.html" target="_top">Weisstein, | |
331 | Eric W. "Laplace Distribution."</a> From MathWorld--A | |
332 | Wolfram Web Resource. | |
333 | </li> | |
334 | <li class="listitem"> | |
335 | <a href="http://en.wikipedia.org/wiki/Laplace_distribution" target="_top">Laplace | |
336 | Distribution</a> | |
337 | </li> | |
338 | <li class="listitem"> | |
339 | M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, | |
340 | 1972, p. 930. | |
341 | </li> | |
342 | </ul></div> | |
343 | </div> | |
344 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
345 | <td align="left"></td> | |
346 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
347 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
348 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
349 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
350 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
351 | 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>) | |
352 | </p> | |
353 | </div></td> | |
354 | </tr></table> | |
355 | <hr> | |
356 | <div class="spirit-nav"> | |
357 | <a accesskey="p" href="inverse_gaussian_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="logistic_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> | |
358 | </div> | |
359 | </body> | |
360 | </html> |