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140 lines
4.5 KiB
Plaintext
140 lines
4.5 KiB
Plaintext
[section:gamma_dist Gamma (and Erlang) Distribution]
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``#include <boost/math/distributions/gamma.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 gamma_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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gamma_distribution(RealType shape, RealType scale = 1)
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RealType shape()const;
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RealType scale()const;
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};
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}} // namespaces
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The gamma distribution is a continuous probability distribution.
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When the shape parameter is an integer then it is known as the
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Erlang Distribution. It is also closely related to the Poisson
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and Chi Squared Distributions.
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When the shape parameter has an integer value, the distribution is the
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[@http://en.wikipedia.org/wiki/Erlang_distribution Erlang distribution].
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Since this can be produced by ensuring that the shape parameter has an
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integer value > 0, the Erlang distribution is not separately implemented.
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[note
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To avoid potential confusion with the gamma functions, this
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distribution does not provide the typedef:
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``typedef gamma_distribution<double> gamma;``
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Instead if you want a double precision gamma distribution you can write
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``boost::math::gamma_distribution<> my_gamma(1, 1);``
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]
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For shape parameter /k/ and scale parameter [theta][space] it is defined by the
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probability density function:
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[equation gamma_dist_ref1]
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Sometimes an alternative formulation is used: given parameters
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[alpha][space]= k and [beta][space]= 1 / [theta], then the
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distribution can be defined by the PDF:
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[equation gamma_dist_ref2]
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In this form the inverse scale parameter is called a /rate parameter/.
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Both forms are in common usage: this library uses the first definition
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throughout. Therefore to construct a Gamma Distribution from a ['rate
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parameter], you should pass the reciprocal of the rate as the scale parameter.
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The following two graphs illustrate how the PDF of the gamma distribution
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varies as the parameters vary:
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[graph gamma1_pdf]
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[graph gamma2_pdf]
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The [*Erlang Distribution] is the same as the Gamma, but with the shape parameter
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an integer. It is often expressed using a /rate/ rather than a /scale/ as the
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second parameter (remember that the rate is the reciprocal of the scale).
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Internally the functions used to implement the Gamma Distribution are
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already optimised for small-integer arguments, so in general there should
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be no great loss of performance from using a Gamma Distribution rather than
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a dedicated Erlang Distribution.
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[h4 Member Functions]
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gamma_distribution(RealType shape, RealType scale = 1);
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Constructs a gamma distribution with shape /shape/ and
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scale /scale/.
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Requires that the shape and scale parameters are greater than zero, otherwise calls
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__domain_error.
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RealType shape()const;
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Returns the /shape/ parameter of this distribution.
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RealType scale()const;
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Returns the /scale/ 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 \[0,+[infin]\].
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[h4 Accuracy]
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The lognormal distribution is implemented in terms of the
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incomplete gamma functions __gamma_p and __gamma_q and their
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inverses __gamma_p_inv and __gamma_q_inv: refer to the accuracy
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data for those functions for more information.
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[h4 Implementation]
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In the following table /k/ is the shape parameter of the distribution,
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[theta][space] is its scale parameter, /x/ is the random variate, /p/ is the probability
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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 = __gamma_p_derivative(k, x / [theta]) / [theta] ]]
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[[cdf][Using the relation: p = __gamma_p(k, x / [theta]) ]]
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[[cdf complement][Using the relation: q = __gamma_q(k, x / [theta]) ]]
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[[quantile][Using the relation: x = [theta][space]* __gamma_p_inv(k, p) ]]
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[[quantile from the complement][Using the relation: x = [theta][space]* __gamma_q_inv(k, p) ]]
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[[mean][k[theta] ]]
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[[variance][k[theta][super 2] ]]
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[[mode][(k-1)[theta][space] for ['k>1] otherwise a __domain_error ]]
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[[skewness][2 / sqrt(k) ]]
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[[kurtosis][3 + 6 / k]]
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[[kurtosis excess][6 / k ]]
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]
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[endsect][/section:gamma_dist Gamma (and Erlang) Distribution]
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[/
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Copyright 2006, 2010 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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