mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-11-03 21:40:52 -05:00
281 lines
10 KiB
Plaintext
281 lines
10 KiB
Plaintext
[section:beta_dist Beta Distribution]
|
|
|
|
``#include <boost/math/distributions/beta.hpp>``
|
|
|
|
namespace boost{ namespace math{
|
|
|
|
template <class RealType = double,
|
|
class ``__Policy`` = ``__policy_class`` >
|
|
class beta_distribution;
|
|
|
|
// typedef beta_distribution<double> beta;
|
|
// Note that this is deliberately NOT provided,
|
|
// to avoid a clash with the function name beta.
|
|
|
|
template <class RealType, class ``__Policy``>
|
|
class beta_distribution
|
|
{
|
|
public:
|
|
typedef RealType value_type;
|
|
typedef Policy policy_type;
|
|
// Constructor from two shape parameters, alpha & beta:
|
|
beta_distribution(RealType a, RealType b);
|
|
|
|
// Parameter accessors:
|
|
RealType alpha() const;
|
|
RealType beta() const;
|
|
|
|
// Parameter estimators of alpha or beta from mean and variance.
|
|
static RealType find_alpha(
|
|
RealType mean, // Expected value of mean.
|
|
RealType variance); // Expected value of variance.
|
|
|
|
static RealType find_beta(
|
|
RealType mean, // Expected value of mean.
|
|
RealType variance); // Expected value of variance.
|
|
|
|
// Parameter estimators from
|
|
// either alpha or beta, and x and probability.
|
|
|
|
static RealType find_alpha(
|
|
RealType beta, // from beta.
|
|
RealType x, // x.
|
|
RealType probability); // cdf
|
|
|
|
static RealType find_beta(
|
|
RealType alpha, // alpha.
|
|
RealType x, // probability x.
|
|
RealType probability); // probability cdf.
|
|
};
|
|
|
|
}} // namespaces
|
|
|
|
The class type `beta_distribution` represents a
|
|
[@http://en.wikipedia.org/wiki/Beta_distribution beta ]
|
|
[@http://en.wikipedia.org/wiki/Probability_distribution probability distribution function].
|
|
|
|
The [@http://mathworld.wolfram.com/BetaDistribution.htm beta distribution ]
|
|
is used as a [@http://en.wikipedia.org/wiki/Prior_distribution prior distribution]
|
|
for binomial proportions in
|
|
[@http://mathworld.wolfram.com/BayesianAnalysis.html Bayesian analysis].
|
|
|
|
See also:
|
|
[@http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/BetaDistribution.html beta distribution]
|
|
and [@http://en.wikipedia.org/wiki/Bayesian_statistics Bayesian statistics].
|
|
|
|
How the beta distribution is used for
|
|
[@http://home.uchicago.edu/~grynav/bayes/ABSLec5.ppt
|
|
Bayesian analysis of one parameter models]
|
|
is discussed by Jeff Grynaviski.
|
|
|
|
The [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF]
|
|
for the [@http://en.wikipedia.org/wiki/Beta_distribution beta distribution]
|
|
defined on the interval \[0,1\] is given by:
|
|
|
|
f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta])
|
|
|
|
where B([alpha], [beta]) is the
|
|
[@http://en.wikipedia.org/wiki/Beta_function beta function],
|
|
implemented in this library as __beta. Division by the beta function
|
|
ensures that the pdf is normalized to the range zero to unity.
|
|
|
|
The following graph illustrates examples of the pdf for various values
|
|
of the shape parameters. Note the [alpha] = [beta] = 2 (blue line)
|
|
is dome-shaped, and might be approximated by a symmetrical triangular
|
|
distribution.
|
|
|
|
[graph beta_pdf]
|
|
|
|
If [alpha] = [beta] = 1, then it is a __space
|
|
[@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution],
|
|
equal to unity in the entire interval x = 0 to 1.
|
|
If [alpha] __space and [beta] __space are < 1, then the pdf is U-shaped.
|
|
If [alpha] != [beta], then the shape is asymmetric
|
|
and could be approximated by a triangle
|
|
whose apex is away from the centre (where x = half).
|
|
|
|
[h4 Member Functions]
|
|
|
|
[h5 Constructor]
|
|
|
|
beta_distribution(RealType alpha, RealType beta);
|
|
|
|
Constructs a beta distribution with shape parameters /alpha/ and /beta/.
|
|
|
|
Requires alpha,beta > 0,otherwise __domain_error is called. Note that
|
|
technically the beta distribution is defined for alpha,beta >= 0, but
|
|
it's not clear whether any program can actually make use of that latitude
|
|
or how many of the non-member functions can be usefully defined in that case.
|
|
Therefore for now, we regard it as an error if alpha or beta is zero.
|
|
|
|
For example:
|
|
|
|
beta_distribution<> mybeta(2, 5);
|
|
|
|
Constructs a the beta distribution with alpha=2 and beta=5 (shown in yellow
|
|
in the graph above).
|
|
|
|
[h5 Parameter Accessors]
|
|
|
|
RealType alpha() const;
|
|
|
|
Returns the parameter /alpha/ from which this distribution was constructed.
|
|
|
|
RealType beta() const;
|
|
|
|
Returns the parameter /beta/ from which this distribution was constructed.
|
|
|
|
So for example:
|
|
|
|
beta_distribution<> mybeta(2, 5);
|
|
assert(mybeta.alpha() == 2.); // mybeta.alpha() returns 2
|
|
assert(mybeta.beta() == 5.); // mybeta.beta() returns 5
|
|
|
|
[h4 Parameter Estimators]
|
|
|
|
Two pairs of parameter estimators are provided.
|
|
|
|
One estimates either [alpha] __space or [beta] __space
|
|
from presumed-known mean and variance.
|
|
|
|
The other pair estimates either [alpha] __space or [beta] __space from
|
|
the cdf and x.
|
|
|
|
It is also possible to estimate [alpha] __space and [beta] __space from
|
|
'known' mode & quantile. For example, calculators are provided by the
|
|
[@http://www.ausvet.com.au/pprev/content.php?page=PPscript
|
|
Pooled Prevalence Calculator] and
|
|
[@http://www.epi.ucdavis.edu/diagnostictests/betabuster.html Beta Buster]
|
|
but this is not yet implemented here.
|
|
|
|
static RealType find_alpha(
|
|
RealType mean, // Expected value of mean.
|
|
RealType variance); // Expected value of variance.
|
|
|
|
Returns the unique value of [alpha][space] that corresponds to a
|
|
beta distribution with mean /mean/ and variance /variance/.
|
|
|
|
static RealType find_beta(
|
|
RealType mean, // Expected value of mean.
|
|
RealType variance); // Expected value of variance.
|
|
|
|
Returns the unique value of [beta][space] that corresponds to a
|
|
beta distribution with mean /mean/ and variance /variance/.
|
|
|
|
static RealType find_alpha(
|
|
RealType beta, // from beta.
|
|
RealType x, // x.
|
|
RealType probability); // probability cdf
|
|
|
|
Returns the value of [alpha][space] that gives:
|
|
`cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.
|
|
|
|
static RealType find_beta(
|
|
RealType alpha, // alpha.
|
|
RealType x, // probability x.
|
|
RealType probability); // probability cdf.
|
|
|
|
Returns the value of [beta][space] that gives:
|
|
`cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.
|
|
|
|
[h4 Non-member Accessor Functions]
|
|
|
|
All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
|
|
that are generic to all distributions are supported: __usual_accessors.
|
|
|
|
The formulae for calculating these are shown in the table below, and at
|
|
[@http://mathworld.wolfram.com/BetaDistribution.html Wolfram Mathworld].
|
|
|
|
[h4 Applications]
|
|
|
|
The beta distribution can be used to model events constrained
|
|
to take place within an interval defined by a minimum and maximum value:
|
|
so it is used in project management systems.
|
|
|
|
It is also widely used in [@http://en.wikipedia.org/wiki/Bayesian_inference Bayesian statistical inference].
|
|
|
|
[h4 Related distributions]
|
|
|
|
The beta distribution with both [alpha] __space and [beta] = 1 follows a
|
|
[@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution].
|
|
|
|
The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular]
|
|
is used when less precise information is available.
|
|
|
|
The [@http://en.wikipedia.org/wiki/Binomial_distribution binomial distribution]
|
|
is closely related when [alpha] __space and [beta] __space are integers.
|
|
|
|
With integer values of [alpha] __space and [beta] __space the distribution B(i, j) is
|
|
that of the j-th highest of a sample of i + j + 1 independent random variables
|
|
uniformly distributed between 0 and 1.
|
|
The cumulative probability from 0 to x is thus
|
|
the probability that the j-th highest value is less than x.
|
|
Or it is the probability that at least i of the random variables are less than x,
|
|
a probability given by summing over the __binomial_distrib
|
|
with its p parameter set to x.
|
|
|
|
[h4 Accuracy]
|
|
|
|
This distribution is implemented using the
|
|
[link math_toolkit.sf_beta.beta_function beta functions] __beta and
|
|
[link math_toolkit.sf_beta.ibeta_function incomplete beta functions] __ibeta and __ibetac;
|
|
please refer to these functions for information on accuracy.
|
|
|
|
[h4 Implementation]
|
|
|
|
In the following table /a/ and /b/ are the parameters [alpha][space] and [beta],
|
|
/x/ is the random variable, /p/ is the probability and /q = 1-p/.
|
|
|
|
[table
|
|
[[Function][Implementation Notes]]
|
|
[[pdf]
|
|
[f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta])
|
|
|
|
Implemented using __ibeta_derivative(a, b, x).]]
|
|
|
|
[[cdf][Using the incomplete beta function __ibeta(a, b, x)]]
|
|
[[cdf complement][__ibetac(a, b, x)]]
|
|
[[quantile][Using the inverse incomplete beta function __ibeta_inv(a, b, p)]]
|
|
[[quantile from the complement][__ibetac_inv(a, b, q)]]
|
|
[[mean][`a/(a+b)`]]
|
|
[[variance][`a * b / (a+b)^2 * (a + b + 1)`]]
|
|
[[mode][`(a-1) / (a + b - 2)`]]
|
|
[[skewness][`2 (b-a) sqrt(a+b+1)/(a+b+2) * sqrt(a * b)`]]
|
|
[[kurtosis excess][ [equation beta_dist_kurtosis] ]]
|
|
[[kurtosis][`kurtosis + 3`]]
|
|
[[parameter estimation][ ]]
|
|
[[alpha
|
|
|
|
from mean and variance][`mean * (( (mean * (1 - mean)) / variance)- 1)`]]
|
|
[[beta
|
|
|
|
from mean and variance][`(1 - mean) * (((mean * (1 - mean)) /variance)-1)`]]
|
|
[[The member functions `find_alpha` and `find_beta`
|
|
|
|
from cdf and probability x
|
|
|
|
and *either* `alpha` or `beta`]
|
|
[Implemented in terms of the inverse incomplete beta functions
|
|
|
|
__ibeta_inva, and __ibeta_invb respectively.]]
|
|
[[`find_alpha`][`ibeta_inva(beta, x, probability)`]]
|
|
[[`find_beta`][`ibeta_invb(alpha, x, probability)`]]
|
|
]
|
|
|
|
[h4 References]
|
|
|
|
[@http://en.wikipedia.org/wiki/Beta_distribution Wikipedia Beta distribution]
|
|
|
|
[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm NIST Exploratory Data Analysis]
|
|
|
|
[@http://mathworld.wolfram.com/BetaDistribution.html Wolfram MathWorld]
|
|
|
|
[endsect][/section:beta_dist beta]
|
|
|
|
[/ beta.qbk
|
|
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).
|
|
]
|