| // boost\math\distributions\beta.hpp
|
|
|
| // Copyright John Maddock 2006.
|
| // Copyright Paul A. Bristow 2006.
|
|
|
| // Use, modification and distribution are subject to 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)
|
|
|
| // http://en.wikipedia.org/wiki/Beta_distribution
|
| // http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm
|
| // http://mathworld.wolfram.com/BetaDistribution.html
|
|
|
| // The Beta Distribution is a continuous probability distribution.
|
| // The beta distribution is used to model events which are constrained to take place
|
| // within an interval defined by maxima and minima,
|
| // so is used extensively in PERT and other project management systems
|
| // to describe the time to completion.
|
| // The cdf of the beta distribution is used as a convenient way
|
| // of obtaining the sum over a set of binomial outcomes.
|
| // The beta distribution is also used in Bayesian statistics.
|
|
|
| #ifndef BOOST_MATH_DIST_BETA_HPP
|
| #define BOOST_MATH_DIST_BETA_HPP
|
|
|
| #include <boost/math/distributions/fwd.hpp>
|
| #include <boost/math/special_functions/beta.hpp> // for beta.
|
| #include <boost/math/distributions/complement.hpp> // complements.
|
| #include <boost/math/distributions/detail/common_error_handling.hpp> // error checks
|
| #include <boost/math/special_functions/fpclassify.hpp> // isnan.
|
| #include <boost/math/tools/roots.hpp> // for root finding.
|
|
|
| #if defined (BOOST_MSVC)
|
| # pragma warning(push)
|
| # pragma warning(disable: 4702) // unreachable code
|
| // in domain_error_imp in error_handling
|
| #endif
|
|
|
| #include <utility>
|
|
|
| namespace boost
|
| {
|
| namespace math
|
| {
|
| namespace beta_detail
|
| {
|
| // Common error checking routines for beta distribution functions:
|
| template <class RealType, class Policy>
|
| inline bool check_alpha(const char* function, const RealType& alpha, RealType* result, const Policy& pol)
|
| {
|
| if(!(boost::math::isfinite)(alpha) || (alpha <= 0))
|
| {
|
| *result = policies::raise_domain_error<RealType>(
|
| function,
|
| "Alpha argument is %1%, but must be > 0 !", alpha, pol);
|
| return false;
|
| }
|
| return true;
|
| } // bool check_alpha
|
|
|
| template <class RealType, class Policy>
|
| inline bool check_beta(const char* function, const RealType& beta, RealType* result, const Policy& pol)
|
| {
|
| if(!(boost::math::isfinite)(beta) || (beta <= 0))
|
| {
|
| *result = policies::raise_domain_error<RealType>(
|
| function,
|
| "Beta argument is %1%, but must be > 0 !", beta, pol);
|
| return false;
|
| }
|
| return true;
|
| } // bool check_beta
|
|
|
| template <class RealType, class Policy>
|
| inline bool check_prob(const char* function, const RealType& p, RealType* result, const Policy& pol)
|
| {
|
| if((p < 0) || (p > 1) || !(boost::math::isfinite)(p))
|
| {
|
| *result = policies::raise_domain_error<RealType>(
|
| function,
|
| "Probability argument is %1%, but must be >= 0 and <= 1 !", p, pol);
|
| return false;
|
| }
|
| return true;
|
| } // bool check_prob
|
|
|
| template <class RealType, class Policy>
|
| inline bool check_x(const char* function, const RealType& x, RealType* result, const Policy& pol)
|
| {
|
| if(!(boost::math::isfinite)(x) || (x < 0) || (x > 1))
|
| {
|
| *result = policies::raise_domain_error<RealType>(
|
| function,
|
| "x argument is %1%, but must be >= 0 and <= 1 !", x, pol);
|
| return false;
|
| }
|
| return true;
|
| } // bool check_x
|
|
|
| template <class RealType, class Policy>
|
| inline bool check_dist(const char* function, const RealType& alpha, const RealType& beta, RealType* result, const Policy& pol)
|
| { // Check both alpha and beta.
|
| return check_alpha(function, alpha, result, pol)
|
| && check_beta(function, beta, result, pol);
|
| } // bool check_dist
|
|
|
| template <class RealType, class Policy>
|
| inline bool check_dist_and_x(const char* function, const RealType& alpha, const RealType& beta, RealType x, RealType* result, const Policy& pol)
|
| {
|
| return check_dist(function, alpha, beta, result, pol)
|
| && check_x(function, x, result, pol);
|
| } // bool check_dist_and_x
|
|
|
| template <class RealType, class Policy>
|
| inline bool check_dist_and_prob(const char* function, const RealType& alpha, const RealType& beta, RealType p, RealType* result, const Policy& pol)
|
| {
|
| return check_dist(function, alpha, beta, result, pol)
|
| && check_prob(function, p, result, pol);
|
| } // bool check_dist_and_prob
|
|
|
| template <class RealType, class Policy>
|
| inline bool check_mean(const char* function, const RealType& mean, RealType* result, const Policy& pol)
|
| {
|
| if(!(boost::math::isfinite)(mean) || (mean <= 0))
|
| {
|
| *result = policies::raise_domain_error<RealType>(
|
| function,
|
| "mean argument is %1%, but must be > 0 !", mean, pol);
|
| return false;
|
| }
|
| return true;
|
| } // bool check_mean
|
| template <class RealType, class Policy>
|
| inline bool check_variance(const char* function, const RealType& variance, RealType* result, const Policy& pol)
|
| {
|
| if(!(boost::math::isfinite)(variance) || (variance <= 0))
|
| {
|
| *result = policies::raise_domain_error<RealType>(
|
| function,
|
| "variance argument is %1%, but must be > 0 !", variance, pol);
|
| return false;
|
| }
|
| return true;
|
| } // bool check_variance
|
| } // namespace beta_detail
|
|
|
| // typedef beta_distribution<double> beta;
|
| // is deliberately NOT included to avoid a name clash with the beta function.
|
| // Use beta_distribution<> mybeta(...) to construct type double.
|
|
|
| template <class RealType = double, class Policy = policies::policy<> >
|
| class beta_distribution
|
| {
|
| public:
|
| typedef RealType value_type;
|
| typedef Policy policy_type;
|
|
|
| beta_distribution(RealType alpha = 1, RealType beta = 1) : m_alpha(alpha), m_beta(beta)
|
| {
|
| RealType result;
|
| beta_detail::check_dist(
|
| "boost::math::beta_distribution<%1%>::beta_distribution",
|
| m_alpha,
|
| m_beta,
|
| &result, Policy());
|
| } // beta_distribution constructor.
|
| // Accessor functions:
|
| RealType alpha() const
|
| {
|
| return m_alpha;
|
| }
|
| RealType beta() const
|
| { // .
|
| return m_beta;
|
| }
|
|
|
| // Estimation of the alpha & beta parameters.
|
| // http://en.wikipedia.org/wiki/Beta_distribution
|
| // gives formulae in section on parameter estimation.
|
| // Also NIST EDA page 3 & 4 give the same.
|
| // http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm
|
| // http://www.epi.ucdavis.edu/diagnostictests/betabuster.html
|
|
|
| static RealType find_alpha(
|
| RealType mean, // Expected value of mean.
|
| RealType variance) // Expected value of variance.
|
| {
|
| static const char* function = "boost::math::beta_distribution<%1%>::find_alpha";
|
| RealType result; // of error checks.
|
| if(false ==
|
| beta_detail::check_mean(
|
| function, mean, &result, Policy())
|
| &&
|
| beta_detail::check_variance(
|
| function, variance, &result, Policy())
|
| )
|
| {
|
| return result;
|
| }
|
| return mean * (( (mean * (1 - mean)) / variance)- 1);
|
| } // RealType find_alpha
|
|
|
| static RealType find_beta(
|
| RealType mean, // Expected value of mean.
|
| RealType variance) // Expected value of variance.
|
| {
|
| static const char* function = "boost::math::beta_distribution<%1%>::find_beta";
|
| RealType result; // of error checks.
|
| if(false ==
|
| beta_detail::check_mean(
|
| function, mean, &result, Policy())
|
| &&
|
| beta_detail::check_variance(
|
| function, variance, &result, Policy())
|
| )
|
| {
|
| return result;
|
| }
|
| return (1 - mean) * (((mean * (1 - mean)) /variance)-1);
|
| } // RealType find_beta
|
|
|
| // Estimate alpha & beta from either alpha or beta, and x and probability.
|
| // Uses for these parameter estimators are unclear.
|
|
|
| static RealType find_alpha(
|
| RealType beta, // from beta.
|
| RealType x, // x.
|
| RealType probability) // cdf
|
| {
|
| static const char* function = "boost::math::beta_distribution<%1%>::find_alpha";
|
| RealType result; // of error checks.
|
| if(false ==
|
| beta_detail::check_prob(
|
| function, probability, &result, Policy())
|
| &&
|
| beta_detail::check_beta(
|
| function, beta, &result, Policy())
|
| &&
|
| beta_detail::check_x(
|
| function, x, &result, Policy())
|
| )
|
| {
|
| return result;
|
| }
|
| return ibeta_inva(beta, x, probability, Policy());
|
| } // RealType find_alpha(beta, a, probability)
|
|
|
| static RealType find_beta(
|
| // ibeta_invb(T b, T x, T p); (alpha, x, cdf,)
|
| RealType alpha, // alpha.
|
| RealType x, // probability x.
|
| RealType probability) // probability cdf.
|
| {
|
| static const char* function = "boost::math::beta_distribution<%1%>::find_beta";
|
| RealType result; // of error checks.
|
| if(false ==
|
| beta_detail::check_prob(
|
| function, probability, &result, Policy())
|
| &&
|
| beta_detail::check_alpha(
|
| function, alpha, &result, Policy())
|
| &&
|
| beta_detail::check_x(
|
| function, x, &result, Policy())
|
| )
|
| {
|
| return result;
|
| }
|
| return ibeta_invb(alpha, x, probability, Policy());
|
| } // RealType find_beta(alpha, x, probability)
|
|
|
| private:
|
| RealType m_alpha; // Two parameters of the beta distribution.
|
| RealType m_beta;
|
| }; // template <class RealType, class Policy> class beta_distribution
|
|
|
| template <class RealType, class Policy>
|
| inline const std::pair<RealType, RealType> range(const beta_distribution<RealType, Policy>& /* dist */)
|
| { // Range of permissible values for random variable x.
|
| using boost::math::tools::max_value;
|
| return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline const std::pair<RealType, RealType> support(const beta_distribution<RealType, Policy>& /* dist */)
|
| { // Range of supported values for random variable x.
|
| // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
|
| return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType mean(const beta_distribution<RealType, Policy>& dist)
|
| { // Mean of beta distribution = np.
|
| return dist.alpha() / (dist.alpha() + dist.beta());
|
| } // mean
|
|
|
| template <class RealType, class Policy>
|
| inline RealType variance(const beta_distribution<RealType, Policy>& dist)
|
| { // Variance of beta distribution = np(1-p).
|
| RealType a = dist.alpha();
|
| RealType b = dist.beta();
|
| return (a * b) / ((a + b ) * (a + b) * (a + b + 1));
|
| } // variance
|
|
|
| template <class RealType, class Policy>
|
| inline RealType mode(const beta_distribution<RealType, Policy>& dist)
|
| {
|
| static const char* function = "boost::math::mode(beta_distribution<%1%> const&)";
|
|
|
| RealType result;
|
| if ((dist.alpha() <= 1))
|
| {
|
| result = policies::raise_domain_error<RealType>(
|
| function,
|
| "mode undefined for alpha = %1%, must be > 1!", dist.alpha(), Policy());
|
| return result;
|
| }
|
|
|
| if ((dist.beta() <= 1))
|
| {
|
| result = policies::raise_domain_error<RealType>(
|
| function,
|
| "mode undefined for beta = %1%, must be > 1!", dist.beta(), Policy());
|
| return result;
|
| }
|
| RealType a = dist.alpha();
|
| RealType b = dist.beta();
|
| return (a-1) / (a + b - 2);
|
| } // mode
|
|
|
| //template <class RealType, class Policy>
|
| //inline RealType median(const beta_distribution<RealType, Policy>& dist)
|
| //{ // Median of beta distribution is not defined.
|
| // return tools::domain_error<RealType>(function, "Median is not implemented, result is %1%!", std::numeric_limits<RealType>::quiet_NaN());
|
| //} // median
|
|
|
| //But WILL be provided by the derived accessor as quantile(0.5).
|
|
|
| template <class RealType, class Policy>
|
| inline RealType skewness(const beta_distribution<RealType, Policy>& dist)
|
| {
|
| BOOST_MATH_STD_USING // ADL of std functions.
|
| RealType a = dist.alpha();
|
| RealType b = dist.beta();
|
| return (2 * (b-a) * sqrt(a + b + 1)) / ((a + b + 2) * sqrt(a * b));
|
| } // skewness
|
|
|
| template <class RealType, class Policy>
|
| inline RealType kurtosis_excess(const beta_distribution<RealType, Policy>& dist)
|
| {
|
| RealType a = dist.alpha();
|
| RealType b = dist.beta();
|
| RealType a_2 = a * a;
|
| RealType n = 6 * (a_2 * a - a_2 * (2 * b - 1) + b * b * (b + 1) - 2 * a * b * (b + 2));
|
| RealType d = a * b * (a + b + 2) * (a + b + 3);
|
| return n / d;
|
| } // kurtosis_excess
|
|
|
| template <class RealType, class Policy>
|
| inline RealType kurtosis(const beta_distribution<RealType, Policy>& dist)
|
| {
|
| return 3 + kurtosis_excess(dist);
|
| } // kurtosis
|
|
|
| template <class RealType, class Policy>
|
| inline RealType pdf(const beta_distribution<RealType, Policy>& dist, const RealType& x)
|
| { // Probability Density/Mass Function.
|
| BOOST_FPU_EXCEPTION_GUARD
|
|
|
| static const char* function = "boost::math::pdf(beta_distribution<%1%> const&, %1%)";
|
|
|
| BOOST_MATH_STD_USING // for ADL of std functions
|
|
|
| RealType a = dist.alpha();
|
| RealType b = dist.beta();
|
|
|
| // Argument checks:
|
| RealType result;
|
| if(false == beta_detail::check_dist_and_x(
|
| function,
|
| a, b, x,
|
| &result, Policy()))
|
| {
|
| return result;
|
| }
|
| using boost::math::beta;
|
| return ibeta_derivative(a, b, x, Policy());
|
| } // pdf
|
|
|
| template <class RealType, class Policy>
|
| inline RealType cdf(const beta_distribution<RealType, Policy>& dist, const RealType& x)
|
| { // Cumulative Distribution Function beta.
|
| BOOST_MATH_STD_USING // for ADL of std functions
|
|
|
| static const char* function = "boost::math::cdf(beta_distribution<%1%> const&, %1%)";
|
|
|
| RealType a = dist.alpha();
|
| RealType b = dist.beta();
|
|
|
| // Argument checks:
|
| RealType result;
|
| if(false == beta_detail::check_dist_and_x(
|
| function,
|
| a, b, x,
|
| &result, Policy()))
|
| {
|
| return result;
|
| }
|
| // Special cases:
|
| if (x == 0)
|
| {
|
| return 0;
|
| }
|
| else if (x == 1)
|
| {
|
| return 1;
|
| }
|
| return ibeta(a, b, x, Policy());
|
| } // beta cdf
|
|
|
| template <class RealType, class Policy>
|
| inline RealType cdf(const complemented2_type<beta_distribution<RealType, Policy>, RealType>& c)
|
| { // Complemented Cumulative Distribution Function beta.
|
|
|
| BOOST_MATH_STD_USING // for ADL of std functions
|
|
|
| static const char* function = "boost::math::cdf(beta_distribution<%1%> const&, %1%)";
|
|
|
| RealType const& x = c.param;
|
| beta_distribution<RealType, Policy> const& dist = c.dist;
|
| RealType a = dist.alpha();
|
| RealType b = dist.beta();
|
|
|
| // Argument checks:
|
| RealType result;
|
| if(false == beta_detail::check_dist_and_x(
|
| function,
|
| a, b, x,
|
| &result, Policy()))
|
| {
|
| return result;
|
| }
|
| if (x == 0)
|
| {
|
| return 1;
|
| }
|
| else if (x == 1)
|
| {
|
| return 0;
|
| }
|
| // Calculate cdf beta using the incomplete beta function.
|
| // Use of ibeta here prevents cancellation errors in calculating
|
| // 1 - x if x is very small, perhaps smaller than machine epsilon.
|
| return ibetac(a, b, x, Policy());
|
| } // beta cdf
|
|
|
| template <class RealType, class Policy>
|
| inline RealType quantile(const beta_distribution<RealType, Policy>& dist, const RealType& p)
|
| { // Quantile or Percent Point beta function or
|
| // Inverse Cumulative probability distribution function CDF.
|
| // Return x (0 <= x <= 1),
|
| // for a given probability p (0 <= p <= 1).
|
| // These functions take a probability as an argument
|
| // and return a value such that the probability that a random variable x
|
| // will be less than or equal to that value
|
| // is whatever probability you supplied as an argument.
|
|
|
| static const char* function = "boost::math::quantile(beta_distribution<%1%> const&, %1%)";
|
|
|
| RealType result; // of argument checks:
|
| RealType a = dist.alpha();
|
| RealType b = dist.beta();
|
| if(false == beta_detail::check_dist_and_prob(
|
| function,
|
| a, b, p,
|
| &result, Policy()))
|
| {
|
| return result;
|
| }
|
| // Special cases:
|
| if (p == 0)
|
| {
|
| return 0;
|
| }
|
| if (p == 1)
|
| {
|
| return 1;
|
| }
|
| return ibeta_inv(a, b, p, static_cast<RealType*>(0), Policy());
|
| } // quantile
|
|
|
| template <class RealType, class Policy>
|
| inline RealType quantile(const complemented2_type<beta_distribution<RealType, Policy>, RealType>& c)
|
| { // Complement Quantile or Percent Point beta function .
|
| // Return the number of expected x for a given
|
| // complement of the probability q.
|
|
|
| static const char* function = "boost::math::quantile(beta_distribution<%1%> const&, %1%)";
|
|
|
| //
|
| // Error checks:
|
| RealType q = c.param;
|
| const beta_distribution<RealType, Policy>& dist = c.dist;
|
| RealType result;
|
| RealType a = dist.alpha();
|
| RealType b = dist.beta();
|
| if(false == beta_detail::check_dist_and_prob(
|
| function,
|
| a,
|
| b,
|
| q,
|
| &result, Policy()))
|
| {
|
| return result;
|
| }
|
| // Special cases:
|
| if(q == 1)
|
| {
|
| return 0;
|
| }
|
| if(q == 0)
|
| {
|
| return 1;
|
| }
|
|
|
| return ibetac_inv(a, b, q, static_cast<RealType*>(0), Policy());
|
| } // Quantile Complement
|
|
|
| } // namespace math
|
| } // namespace boost
|
|
|
| // This include must be at the end, *after* the accessors
|
| // for this distribution have been defined, in order to
|
| // keep compilers that support two-phase lookup happy.
|
| #include <boost/math/distributions/detail/derived_accessors.hpp>
|
|
|
| #if defined (BOOST_MSVC)
|
| # pragma warning(pop)
|
| #endif
|
|
|
| #endif // BOOST_MATH_DIST_BETA_HPP
|
|
|
|
|
