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// 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)
&& beta_detail::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 l_alpha = 1, RealType l_beta = 1) : m_alpha(l_alpha), m_beta(l_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 = 0; // 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 = 0; // 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 = 0; // 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 = 0; // 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 = 0;
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 = 0;
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 = 0;
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 = 0; // 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 = 0;
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