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120 lines
4.5 KiB
Plaintext
120 lines
4.5 KiB
Plaintext
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[section:rayleigh Rayleigh Distribution]
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``#include <boost/math/distributions/rayleigh.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class rayleigh_distribution;
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typedef rayleigh_distribution<> rayleigh;
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template <class RealType, class ``__Policy``>
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class rayleigh_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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// Construct:
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rayleigh_distribution(RealType sigma = 1)
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// Accessors:
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RealType sigma()const;
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};
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}} // namespaces
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The [@http://en.wikipedia.org/wiki/Rayleigh_distribution Rayleigh distribution]
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is a continuous distribution with the
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[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]:
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f(x; sigma) = x * exp(-x[super 2]/2 [sigma][super 2]) / [sigma][super 2]
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For sigma parameter [sigma][space] > 0, and x > 0.
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The Rayleigh distribution is often used where two orthogonal components
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have an absolute value,
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for example, wind velocity and direction may be combined to yield a wind speed,
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or real and imaginary components may have absolute values that are Rayleigh distributed.
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The following graph illustrates how the Probability density Function(pdf) varies with the shape parameter [sigma]:
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[graph rayleigh_pdf]
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and the Cumulative Distribution Function (cdf)
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[graph rayleigh_cdf]
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[h4 Related distributions]
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The absolute value of two independent normal distributions X and Y, [radic] (X[super 2] + Y[super 2])
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is a Rayleigh distribution.
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The [@http://en.wikipedia.org/wiki/Chi_distribution Chi],
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[@http://en.wikipedia.org/wiki/Rice_distribution Rice]
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and [@http://en.wikipedia.org/wiki/Weibull_distribution Weibull] distributions are generalizations of the
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[@http://en.wikipedia.org/wiki/Rayleigh_distribution Rayleigh distribution].
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[h4 Member Functions]
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rayleigh_distribution(RealType sigma = 1);
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Constructs a [@http://en.wikipedia.org/wiki/Rayleigh_distribution
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Rayleigh distribution] with [sigma] /sigma/.
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Requires that the [sigma] parameter is greater than zero,
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otherwise calls __domain_error.
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RealType sigma()const;
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Returns the /sigma/ parameter of this distribution.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
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distributions are supported: __usual_accessors.
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The domain of the random variable is \[0, max_value\].
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[h4 Accuracy]
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The Rayleigh distribution is implemented in terms of the
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standard library `sqrt` and `exp` and as such should have very low error rates.
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Some constants such as skewness and kurtosis were calculated using
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NTL RR type with 150-bit accuracy, about 50 decimal digits.
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[h4 Implementation]
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In the following table [sigma][space] is the sigma parameter of the distribution,
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/x/ is the random variate, /p/ is the probability and /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Using the relation: pdf = x * exp(-x[super 2])/2 [sigma][super 2] ]]
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[[cdf][Using the relation: p = 1 - exp(-x[super 2]/2) [sigma][super 2][space] = -__expm1(-x[super 2]/2) [sigma][super 2]]]
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[[cdf complement][Using the relation: q = exp(-x[super 2]/ 2) * [sigma][super 2] ]]
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[[quantile][Using the relation: x = sqrt(-2 * [sigma] [super 2]) * log(1 - p)) = sqrt(-2 * [sigma] [super 2]) * __log1p(-p))]]
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[[quantile from the complement][Using the relation: x = sqrt(-2 * [sigma] [super 2]) * log(q)) ]]
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[[mean][[sigma] * sqrt([pi]/2) ]]
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[[variance][[sigma][super 2] * (4 - [pi]/2) ]]
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[[mode][[sigma] ]]
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[[skewness][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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[[kurtosis][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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[[kurtosis excess][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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]
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[h4 References]
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* [@http://en.wikipedia.org/wiki/Rayleigh_distribution ]
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* [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram Web Resource.]
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[endsect] [/section:Rayleigh Rayleigh]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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