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