mirror of
https://github.com/boostorg/math.git
synced 2026-01-19 04:22:09 +00:00
e9cd6c96fd
Add SYCL testing of normal dist Add CUDA testing of normal dist Add NVRTC testing of normal dist NVRTC fixes Move headers for NVRTC support Add GPU support to inverse gaussian dist Add NVRTC testing of inverse Gaussian dist Add CUDA testing of inverse gaussian dist Add SYCL testing of inverse gaussian dist Add GPU support to lognormal dist Add SYCL testing of lognormal dist Add CUDA testing of lognormal dist Add nvrtc testing of lognormal dist Add GPU support to negative binomial dist Avoid float_prior on GPU platform Add NVRTC testing of negative binomial dist Fix ambiguous use of nextafter Add CUDA testing of negative binomial dist Fix float_prior workaround Add SYCL testing of negative binomial dist Add GPU support to non_central_beta dist Add SYCL testing of nc beta dist Add CUDA testing of nc beta dist Enable generic dist handling on GPU Add GPU support to brent_find_minima Add NVRTC testing of nc beta dist Add utility header Replace non-functional macro with new function Add GPU support to non central chi squared dist Add SYCL testing of non central chi squared dist Add missing macro definition Markup generic quantile finder Add CUDA testing of non central chi squared dist Add NVRTC testing of non central chi squared dist Add GPU support to the non-central f dist Add SYCL testing of ncf Add CUDA testing of ncf dist Add NVRTC testing of ncf dist Add GPU support to students_t dist Add SYCL testing of students_t dist Add CUDA testing of students_t Add NVRTC testing of students_t dist Workaround for header cycle Add GPU support to pareto dist Add SYCL testing of pareto dist Add CUDA testing of pareto dist Add NVRTC testing of pareto dist Add missing header Add GPU support to poisson dist Add SYCL testing of poisson dist Add CUDA testing of poisson dist Add NVRTC testing of poisson dist Add forward decl for NVRTC platform Add GPU support to rayleigh dist Add CUDA testing of rayleigh dist Add SYCL testing of rayleigh dist Add NVRTC testing of rayleigh dist Add GPU support to triangular dist Add SYCL testing of triangular dist Add NVRTC testing of triangular dist Add CUDA testing of triangular dist Add GPU support to the uniform dist Add CUDA testing of uniform dist Add SYCL testing of uniform dist Add NVRTC testing of uniform dist Fix missing header Add markers to docs
127 lines
4.9 KiB
Plaintext
127 lines
4.9 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:
|
|
BOOST_MATH_GPU_ENABLED rayleigh_distribution(RealType sigma = 1)
|
|
// Accessors:
|
|
BOOST_MATH_GPU_ENABLED 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]:
|
|
|
|
[expression f(x; sigma) = x * exp(-x[super 2]/2 [sigma][super 2]) / [sigma][super 2]]
|
|
|
|
For sigma parameter ['[sigma]] > 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]
|
|
|
|
BOOST_MATH_GPU_ENABLED 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.
|
|
|
|
BOOST_MATH_GPU_ENABLED 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.
|
|
For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
|
|
be run on both host and device.
|
|
|
|
The domain of the random variable is \[0, max_value\].
|
|
|
|
In this distribution the implementation of both `logcdf`, and `logpdf` are specialized
|
|
to improve numerical accuracy.
|
|
|
|
[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] 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] ]]
|
|
[[logpdf][log(pdf) = -(x[super 2])/(2*[sigma][super 2]) - 2*log([sigma]) + log(x) ]]
|
|
[[cdf][Using the relation: p = 1 - exp(-x[super 2]/2) [sigma][super 2]= -__expm1(-x[super 2]/2) [sigma][super 2]]]
|
|
[[logcdf][log(cdf) = log1p(-exp(-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).
|
|
]
|
|
|