]> git.proxmox.com Git - ceph.git/blame - ceph/src/boost/libs/math/doc/html/math_toolkit/dist_ref/dists/rayleigh.html
projects / ceph.git / blame
? search:
summary | shortlog | log | commit | commitdiff | tree
blob | blame (incremental) | history | HEAD
bump version to 12.2.2-pve1
[ceph.git] / ceph / src / boost / libs / math / doc / html / math_toolkit / dist_ref / dists / rayleigh.html
CommitLineData
7c673cae
FG		 1<html>
2<head>
3<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
4<title>Rayleigh 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="poisson_dist.html" title="Poisson Distribution">
10<link rel="next" href="skew_normal_dist.html" title="Skew Normal 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="poisson_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="skew_normal_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.rayleigh"></a><a class="link" href="rayleigh.html" title="Rayleigh Distribution">Rayleigh Distribution</a>
28</h4></div></div></div>
29<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">rayleigh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">&lt;</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&#160;15.&#160;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&lt;&gt;</a> <span class="special">&gt;</span>
34<span class="keyword">class</span> <span class="identifier">rayleigh_distribution</span><span class="special">;</span>
35
36<span class="keyword">typedef</span> <span class="identifier">rayleigh_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">rayleigh</span><span class="special">;</span>
37
38<span class="keyword">template</span> <span class="special">&lt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
39<span class="keyword">class</span> <span class="identifier">rayleigh_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">rayleigh_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">sigma</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">sigma</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
48<span class="special">};</span>
49
50<span class="special">}}</span> <span class="comment">// namespaces</span>
51</pre>
52<p>
53 The <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
54 distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
55 density function</a>:
56 </p>
57<p>
58 f(x; sigma) = x * exp(-x<sup>2</sup>/2 &#963;<sup>2</sup>) / &#963;<sup>2</sup>
59 </p>
60<p>
61 For sigma parameter &#963; &#160; &gt; 0, and x &gt; 0.
62 </p>
63<p>
64 The Rayleigh distribution is often used where two orthogonal components
65 have an absolute value, for example, wind velocity and direction may be
66 combined to yield a wind speed, or real and imaginary components may have
67 absolute values that are Rayleigh distributed.
68 </p>
69<p>
70 The following graph illustrates how the Probability density Function(pdf)
71 varies with the shape parameter &#963;:
72 </p>
73<p>
74 <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_pdf.svg" align="middle"></span>
75 </p>
76<p>
77 and the Cumulative Distribution Function (cdf)
78 </p>
79<p>
80 <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_cdf.svg" align="middle"></span>
81 </p>
82<h5>
83<a name="math_toolkit.dist_ref.dists.rayleigh.h0"></a>
84 <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.related_distributions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.related_distributions">Related
85 distributions</a>
86 </h5>
87<p>
88 The absolute value of two independent normal distributions X and Y, &#8730; (X<sup>2</sup> +
89 Y<sup>2</sup>) is a Rayleigh distribution.
90 </p>
91<p>
92 The <a href="http://en.wikipedia.org/wiki/Chi_distribution" target="_top">Chi</a>,
93 <a href="http://en.wikipedia.org/wiki/Rice_distribution" target="_top">Rice</a>
94 and <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull</a>
95 distributions are generalizations of the <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
96 distribution</a>.
97 </p>
98<h5>
99<a name="math_toolkit.dist_ref.dists.rayleigh.h1"></a>
100 <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.member_functions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.member_functions">Member
101 Functions</a>
102 </h5>
103<pre class="programlisting"><span class="identifier">rayleigh_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
104</pre>
105<p>
106 Constructs a <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
107 distribution</a> with &#963; <span class="emphasis"><em>sigma</em></span>.
108 </p>
109<p>
110 Requires that the &#963; parameter is greater than zero, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
111 </p>
112<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">sigma</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
113</pre>
114<p>
115 Returns the <span class="emphasis"><em>sigma</em></span> parameter of this distribution.
116 </p>
117<h5>
118<a name="math_toolkit.dist_ref.dists.rayleigh.h2"></a>
119 <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.non_member_accessors"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.non_member_accessors">Non-member
120 Accessors</a>
121 </h5>
122<p>
123 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
124 functions</a> that are generic to all distributions are supported:
125 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
126 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
127 <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>,
128 <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>,
129 <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>,
130 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
131 <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>,
132 <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>.
133 </p>
134<p>
135 The domain of the random variable is [0, max_value].
136 </p>
137<h5>
138<a name="math_toolkit.dist_ref.dists.rayleigh.h3"></a>
139 <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.accuracy"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.accuracy">Accuracy</a>
140 </h5>
141<p>
142 The Rayleigh distribution is implemented in terms of the standard library
143 <code class="computeroutput"><span class="identifier">sqrt</span></code> and <code class="computeroutput"><span class="identifier">exp</span></code> and as such should have very low
144 error rates. Some constants such as skewness and kurtosis were calculated
145 using NTL RR type with 150-bit accuracy, about 50 decimal digits.
146 </p>
147<h5>
148<a name="math_toolkit.dist_ref.dists.rayleigh.h4"></a>
149 <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.implementation"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.implementation">Implementation</a>
150 </h5>
151<p>
152 In the following table &#963; &#160; is the sigma parameter of the distribution, <span class="emphasis"><em>x</em></span>
153 is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
154 = 1-p</em></span>.
155 </p>
156<div class="informaltable"><table class="table">
157<colgroup>
158<col>
159<col>
160</colgroup>
161<thead><tr>
162<th>
163 <p>
164 Function
165 </p>
166 </th>
167<th>
168 <p>
169 Implementation Notes
170 </p>
171 </th>
172</tr></thead>
173<tbody>
174<tr>
175<td>
176 <p>
177 pdf
178 </p>
179 </td>
180<td>
181 <p>
182 Using the relation: pdf = x * exp(-x<sup>2</sup>)/2 &#963;<sup>2</sup>
183 </p>
184 </td>
185</tr>
186<tr>
187<td>
188 <p>
189 cdf
190 </p>
191 </td>
192<td>
193 <p>
194 Using the relation: p = 1 - exp(-x<sup>2</sup>/2) &#963;<sup>2</sup> &#160; = -<a class="link" href="../../powers/expm1.html" title="expm1">expm1</a>(-x<sup>2</sup>/2)
195 &#963;<sup>2</sup>
196 </p>
197 </td>
198</tr>
199<tr>
200<td>
201 <p>
202 cdf complement
203 </p>
204 </td>
205<td>
206 <p>
207 Using the relation: q = exp(-x<sup>2</sup>/ 2) * &#963;<sup>2</sup>
208 </p>
209 </td>
210</tr>
211<tr>
212<td>
213 <p>
214 quantile
215 </p>
216 </td>
217<td>
218 <p>
219 Using the relation: x = sqrt(-2 * &#963; <sup>2</sup>) * log(1 - p)) = sqrt(-2
220 * &#963; <sup>2</sup>) * <a class="link" href="../../powers/log1p.html" title="log1p">log1p</a>(-p))
221 </p>
222 </td>
223</tr>
224<tr>
225<td>
226 <p>
227 quantile from the complement
228 </p>
229 </td>
230<td>
231 <p>
232 Using the relation: x = sqrt(-2 * &#963; <sup>2</sup>) * log(q))
233 </p>
234 </td>
235</tr>
236<tr>
237<td>
238 <p>
239 mean
240 </p>
241 </td>
242<td>
243 <p>
244 &#963; * sqrt(&#960;/2)
245 </p>
246 </td>
247</tr>
248<tr>
249<td>
250 <p>
251 variance
252 </p>
253 </td>
254<td>
255 <p>
256 &#963;<sup>2</sup> * (4 - &#960;/2)
257 </p>
258 </td>
259</tr>
260<tr>
261<td>
262 <p>
263 mode
264 </p>
265 </td>
266<td>
267 <p>
268 &#963;
269 </p>
270 </td>
271</tr>
272<tr>
273<td>
274 <p>
275 skewness
276 </p>
277 </td>
278<td>
279 <p>
280 Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
281 Eric W. "Weibull Distribution." From MathWorld--A Wolfram
282 Web Resource.</a>
283 </p>
284 </td>
285</tr>
286<tr>
287<td>
288 <p>
289 kurtosis
290 </p>
291 </td>
292<td>
293 <p>
294 Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
295 Eric W. "Weibull Distribution." From MathWorld--A Wolfram
296 Web Resource.</a>
297 </p>
298 </td>
299</tr>
300<tr>
301<td>
302 <p>
303 kurtosis excess
304 </p>
305 </td>
306<td>
307 <p>
308 Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
309 Eric W. "Weibull Distribution." From MathWorld--A Wolfram
310 Web Resource.</a>
311 </p>
312 </td>
313</tr>
314</tbody>
315</table></div>
316<h5>
317<a name="math_toolkit.dist_ref.dists.rayleigh.h5"></a>
318 <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.references"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.references">References</a>
319 </h5>
320<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
321<li class="listitem">
322 <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">http://en.wikipedia.org/wiki/Rayleigh_distribution</a>
323 </li>
324<li class="listitem">
325 <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
326 Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram
327 Web Resource.</a>
328 </li>
329</ul></div>
330</div>
331<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
332<td align="left"></td>
333<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
334 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
335 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
336 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
337 Distributed under the Boost Software License, Version 1.0. (See accompanying
338 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>)
339 </p>
340</div></td>
341</tr></table>
342<hr>
343<div class="spirit-nav">
344<a accesskey="p" href="poisson_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="skew_normal_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
345</div>
346</body>
347</html>
Ceph