mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-11-03 21:40:52 -05:00
120 lines
4.5 KiB
Plaintext
120 lines
4.5 KiB
Plaintext
|
[section:rayleigh Rayleigh Distribution]
|
||
|
|
|
||
|
|
|
||
|
|
``#include <boost/math/distributions/rayleigh.hpp>``
|
||
|
|
|
||
|
|
namespace boost{ namespace math{
|
||
|
|
|
||
|
|
template <class RealType = double,
|
||
|
|
class ``__Policy`` = ``__policy_class`` >
|
||
|
|
class rayleigh_distribution;
|
||
|
|
|
||
|
|
typedef rayleigh_distribution<> rayleigh;
|
||
|
|
|
||
|
|
template <class RealType, class ``__Policy``>
|
||
|
|
class rayleigh_distribution
|
||
|
|
{
|
||
|
|
public:
|
||
|
|
typedef RealType value_type;
|
||
|
|
typedef Policy policy_type;
|
||
|
|
// Construct:
|
||
|
|
rayleigh_distribution(RealType sigma = 1)
|
||
|
|
// Accessors:
|
||
|
|
RealType sigma()const;
|
||
|
|
};
|
||
|
|
|
||
|
|
}} // namespaces
|
||
|
|
|
||
|
|
The [@http://en.wikipedia.org/wiki/Rayleigh_distribution Rayleigh distribution]
|
||
|
|
is a continuous distribution with the
|
||
|
|
[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]:
|
||
|
|
|
||
|
|
f(x; sigma) = x * exp(-x[super 2]/2 [sigma][super 2]) / [sigma][super 2]
|
||
|
|
|
||
|
|
For sigma parameter [sigma][space] > 0, and x > 0.
|
||
|
|
|
||
|
|
The Rayleigh distribution is often used where two orthogonal components
|
||
|
|
have an absolute value,
|
||
|
|
for example, wind velocity and direction may be combined to yield a wind speed,
|
||
|
|
or real and imaginary components may have absolute values that are Rayleigh distributed.
|
||
|
|
|
||
|
|
The following graph illustrates how the Probability density Function(pdf) varies with the shape parameter [sigma]:
|
||
|
|
|
||
|
|
[graph rayleigh_pdf]
|
||
|
|
|
||
|
|
and the Cumulative Distribution Function (cdf)
|
||
|
|
|
||
|
|
[graph rayleigh_cdf]
|
||
|
|
|
||
|
|
[h4 Related distributions]
|
||
|
|
|
||
|
|
The absolute value of two independent normal distributions X and Y, [radic] (X[super 2] + Y[super 2])
|
||
|
|
is a Rayleigh distribution.
|
||
|
|
|
||
|
|
The [@http://en.wikipedia.org/wiki/Chi_distribution Chi],
|
||
|
|
[@http://en.wikipedia.org/wiki/Rice_distribution Rice]
|
||
|
|
and [@http://en.wikipedia.org/wiki/Weibull_distribution Weibull] distributions are generalizations of the
|
||
|
|
[@http://en.wikipedia.org/wiki/Rayleigh_distribution Rayleigh distribution].
|
||
|
|
|
||
|
|
[h4 Member Functions]
|
||
|
|
|
||
|
|
rayleigh_distribution(RealType sigma = 1);
|
||
|
|
|
||
|
|
Constructs a [@http://en.wikipedia.org/wiki/Rayleigh_distribution
|
||
|
|
Rayleigh distribution] with [sigma] /sigma/.
|
||
|
|
|
||
|
|
Requires that the [sigma] parameter is greater than zero,
|
||
|
|
otherwise calls __domain_error.
|
||
|
|
|
||
|
|
RealType sigma()const;
|
||
|
|
|
||
|
|
Returns the /sigma/ parameter of this distribution.
|
||
|
|
|
||
|
|
[h4 Non-member Accessors]
|
||
|
|
|
||
|
|
All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
|
||
|
|
distributions are supported: __usual_accessors.
|
||
|
|
|
||
|
|
The domain of the random variable is \[0, max_value\].
|
||
|
|
|
||
|
|
[h4 Accuracy]
|
||
|
|
|
||
|
|
The Rayleigh distribution is implemented in terms of the
|
||
|
|
standard library `sqrt` and `exp` and as such should have very low error rates.
|
||
|
|
Some constants such as skewness and kurtosis were calculated using
|
||
|
|
NTL RR type with 150-bit accuracy, about 50 decimal digits.
|
||
|
|
|
||
|
|
[h4 Implementation]
|
||
|
|
|
||
|
|
In the following table [sigma][space] is the sigma parameter of the distribution,
|
||
|
|
/x/ is the random variate, /p/ is the probability and /q = 1-p/.
|
||
|
|
|
||
|
|
[table
|
||
|
|
[[Function][Implementation Notes]]
|
||
|
|
[[pdf][Using the relation: pdf = x * exp(-x[super 2])/2 [sigma][super 2] ]]
|
||
|
|
[[cdf][Using the relation: p = 1 - exp(-x[super 2]/2) [sigma][super 2][space] = -__expm1(-x[super 2]/2) [sigma][super 2]]]
|
||
|
|
[[cdf complement][Using the relation: q = exp(-x[super 2]/ 2) * [sigma][super 2] ]]
|
||
|
|
[[quantile][Using the relation: x = sqrt(-2 * [sigma] [super 2]) * log(1 - p)) = sqrt(-2 * [sigma] [super 2]) * __log1p(-p))]]
|
||
|
|
[[quantile from the complement][Using the relation: x = sqrt(-2 * [sigma] [super 2]) * log(q)) ]]
|
||
|
|
[[mean][[sigma] * sqrt([pi]/2) ]]
|
||
|
|
[[variance][[sigma][super 2] * (4 - [pi]/2) ]]
|
||
|
|
[[mode][[sigma] ]]
|
||
|
|
[[skewness][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
|
||
|
|
[[kurtosis][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
|
||
|
|
[[kurtosis excess][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
|
||
|
|
]
|
||
|
|
|
||
|
|
[h4 References]
|
||
|
|
* [@http://en.wikipedia.org/wiki/Rayleigh_distribution ]
|
||
|
|
* [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram Web Resource.]
|
||
|
|
|
||
|
|
[endsect] [/section:Rayleigh Rayleigh]
|
||
|
|
|
||
|
|
[/
|
||
|
|
Copyright 2006 John Maddock and Paul A. Bristow.
|
||
|
|
Distributed under the Boost Software License, Version 1.0.
|
||
|
|
(See accompanying file LICENSE_1_0.txt or copy at
|
||
|
|
http://www.boost.org/LICENSE_1_0.txt).
|
||
|
|
]
|
||
|
|
|