517 lines
21 KiB
Plaintext
517 lines
21 KiB
Plaintext
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// boost\math\distributions\geometric.hpp
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// Copyright John Maddock 2010.
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// Copyright Paul A. Bristow 2010.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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// geometric distribution is a discrete probability distribution.
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// It expresses the probability distribution of the number (k) of
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// events, occurrences, failures or arrivals before the first success.
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// supported on the set {0, 1, 2, 3...}
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// Note that the set includes zero (unlike some definitions that start at one).
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// The random variate k is the number of events, occurrences or arrivals.
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// k argument may be integral, signed, or unsigned, or floating point.
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// If necessary, it has already been promoted from an integral type.
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// Note that the geometric distribution
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// (like others including the binomial, geometric & Bernoulli)
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// is strictly defined as a discrete function:
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// only integral values of k are envisaged.
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// However because the method of calculation uses a continuous gamma function,
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// it is convenient to treat it as if a continous function,
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// and permit non-integral values of k.
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// To enforce the strict mathematical model, users should use floor or ceil functions
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// on k outside this function to ensure that k is integral.
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// See http://en.wikipedia.org/wiki/geometric_distribution
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// http://documents.wolfram.com/v5/Add-onsLinks/StandardPackages/Statistics/DiscreteDistributions.html
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// http://mathworld.wolfram.com/GeometricDistribution.html
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#ifndef BOOST_MATH_SPECIAL_GEOMETRIC_HPP
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#define BOOST_MATH_SPECIAL_GEOMETRIC_HPP
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#include <boost/math/distributions/fwd.hpp>
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#include <boost/math/special_functions/beta.hpp> // for ibeta(a, b, x) == Ix(a, b).
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#include <boost/math/distributions/complement.hpp> // complement.
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#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks domain_error & logic_error.
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#include <boost/math/special_functions/fpclassify.hpp> // isnan.
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#include <boost/math/tools/roots.hpp> // for root finding.
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#include <boost/math/distributions/detail/inv_discrete_quantile.hpp>
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#include <boost/type_traits/is_floating_point.hpp>
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#include <boost/type_traits/is_integral.hpp>
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#include <boost/type_traits/is_same.hpp>
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#include <boost/mpl/if.hpp>
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#include <limits> // using std::numeric_limits;
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#include <utility>
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#if defined (BOOST_MSVC)
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# pragma warning(push)
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// This believed not now necessary, so commented out.
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//# pragma warning(disable: 4702) // unreachable code.
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// in domain_error_imp in error_handling.
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#endif
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namespace boost
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{
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namespace math
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{
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namespace geometric_detail
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{
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// Common error checking routines for geometric distribution function:
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template <class RealType, class Policy>
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inline bool check_success_fraction(const char* function, const RealType& p, RealType* result, const Policy& pol)
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{
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if( !(boost::math::isfinite)(p) || (p < 0) || (p > 1) )
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"Success fraction argument is %1%, but must be >= 0 and <= 1 !", p, pol);
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return false;
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}
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return true;
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}
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template <class RealType, class Policy>
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inline bool check_dist(const char* function, const RealType& p, RealType* result, const Policy& pol)
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{
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return check_success_fraction(function, p, result, pol);
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}
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template <class RealType, class Policy>
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inline bool check_dist_and_k(const char* function, const RealType& p, RealType k, RealType* result, const Policy& pol)
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{
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if(check_dist(function, p, result, pol) == false)
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{
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return false;
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}
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if( !(boost::math::isfinite)(k) || (k < 0) )
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{ // Check k failures.
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*result = policies::raise_domain_error<RealType>(
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function,
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"Number of failures argument is %1%, but must be >= 0 !", k, pol);
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return false;
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}
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return true;
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} // Check_dist_and_k
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template <class RealType, class Policy>
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inline bool check_dist_and_prob(const char* function, RealType p, RealType prob, RealType* result, const Policy& pol)
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{
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if((check_dist(function, p, result, pol) && detail::check_probability(function, prob, result, pol)) == false)
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{
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return false;
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}
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return true;
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} // check_dist_and_prob
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} // namespace geometric_detail
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template <class RealType = double, class Policy = policies::policy<> >
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class geometric_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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geometric_distribution(RealType p) : m_p(p)
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{ // Constructor stores success_fraction p.
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RealType result;
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geometric_detail::check_dist(
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"geometric_distribution<%1%>::geometric_distribution",
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m_p, // Check success_fraction 0 <= p <= 1.
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&result, Policy());
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} // geometric_distribution constructor.
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// Private data getter class member functions.
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RealType success_fraction() const
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{ // Probability of success as fraction in range 0 to 1.
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return m_p;
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}
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RealType successes() const
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{ // Total number of successes r = 1 (for compatibility with negative binomial?).
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return 1;
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}
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// Parameter estimation.
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// (These are copies of negative_binomial distribution with successes = 1).
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static RealType find_lower_bound_on_p(
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RealType trials,
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RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.
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{
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static const char* function = "boost::math::geometric<%1%>::find_lower_bound_on_p";
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RealType result = 0; // of error checks.
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RealType successes = 1;
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RealType failures = trials - successes;
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if(false == detail::check_probability(function, alpha, &result, Policy())
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&& geometric_detail::check_dist_and_k(
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function, RealType(0), failures, &result, Policy()))
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{
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return result;
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}
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// Use complement ibeta_inv function for lower bound.
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// This is adapted from the corresponding binomial formula
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// here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
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// This is a Clopper-Pearson interval, and may be overly conservative,
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// see also "A Simple Improved Inferential Method for Some
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// Discrete Distributions" Yong CAI and K. KRISHNAMOORTHY
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// http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
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//
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return ibeta_inv(successes, failures + 1, alpha, static_cast<RealType*>(0), Policy());
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} // find_lower_bound_on_p
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static RealType find_upper_bound_on_p(
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RealType trials,
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RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.
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{
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static const char* function = "boost::math::geometric<%1%>::find_upper_bound_on_p";
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RealType result = 0; // of error checks.
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RealType successes = 1;
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RealType failures = trials - successes;
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if(false == geometric_detail::check_dist_and_k(
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function, RealType(0), failures, &result, Policy())
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&& detail::check_probability(function, alpha, &result, Policy()))
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{
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return result;
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}
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if(failures == 0)
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{
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return 1;
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}// Use complement ibetac_inv function for upper bound.
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// Note adjusted failures value: *not* failures+1 as usual.
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// This is adapted from the corresponding binomial formula
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// here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
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// This is a Clopper-Pearson interval, and may be overly conservative,
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// see also "A Simple Improved Inferential Method for Some
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// Discrete Distributions" Yong CAI and K. Krishnamoorthy
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// http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
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//
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return ibetac_inv(successes, failures, alpha, static_cast<RealType*>(0), Policy());
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} // find_upper_bound_on_p
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// Estimate number of trials :
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// "How many trials do I need to be P% sure of seeing k or fewer failures?"
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static RealType find_minimum_number_of_trials(
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RealType k, // number of failures (k >= 0).
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RealType p, // success fraction 0 <= p <= 1.
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RealType alpha) // risk level threshold 0 <= alpha <= 1.
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{
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static const char* function = "boost::math::geometric<%1%>::find_minimum_number_of_trials";
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// Error checks:
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RealType result = 0;
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if(false == geometric_detail::check_dist_and_k(
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function, p, k, &result, Policy())
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&& detail::check_probability(function, alpha, &result, Policy()))
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{
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return result;
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}
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result = ibeta_inva(k + 1, p, alpha, Policy()); // returns n - k
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return result + k;
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} // RealType find_number_of_failures
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static RealType find_maximum_number_of_trials(
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RealType k, // number of failures (k >= 0).
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RealType p, // success fraction 0 <= p <= 1.
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RealType alpha) // risk level threshold 0 <= alpha <= 1.
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{
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static const char* function = "boost::math::geometric<%1%>::find_maximum_number_of_trials";
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// Error checks:
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RealType result = 0;
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if(false == geometric_detail::check_dist_and_k(
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function, p, k, &result, Policy())
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&& detail::check_probability(function, alpha, &result, Policy()))
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{
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return result;
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}
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result = ibetac_inva(k + 1, p, alpha, Policy()); // returns n - k
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return result + k;
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} // RealType find_number_of_trials complemented
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private:
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//RealType m_r; // successes fixed at unity.
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RealType m_p; // success_fraction
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}; // template <class RealType, class Policy> class geometric_distribution
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typedef geometric_distribution<double> geometric; // Reserved name of type double.
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> range(const geometric_distribution<RealType, Policy>& /* dist */)
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{ // Range of permissible values for random variable k.
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using boost::math::tools::max_value;
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return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // max_integer?
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}
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> support(const geometric_distribution<RealType, Policy>& /* dist */)
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{ // Range of supported values for random variable k.
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// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
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using boost::math::tools::max_value;
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return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // max_integer?
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}
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template <class RealType, class Policy>
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inline RealType mean(const geometric_distribution<RealType, Policy>& dist)
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{ // Mean of geometric distribution = (1-p)/p.
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return (1 - dist.success_fraction() ) / dist.success_fraction();
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} // mean
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// median implemented via quantile(half) in derived accessors.
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template <class RealType, class Policy>
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inline RealType mode(const geometric_distribution<RealType, Policy>&)
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{ // Mode of geometric distribution = zero.
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BOOST_MATH_STD_USING // ADL of std functions.
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return 0;
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} // mode
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template <class RealType, class Policy>
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inline RealType variance(const geometric_distribution<RealType, Policy>& dist)
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{ // Variance of Binomial distribution = (1-p) / p^2.
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return (1 - dist.success_fraction())
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/ (dist.success_fraction() * dist.success_fraction());
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} // variance
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template <class RealType, class Policy>
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inline RealType skewness(const geometric_distribution<RealType, Policy>& dist)
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{ // skewness of geometric distribution = 2-p / (sqrt(r(1-p))
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BOOST_MATH_STD_USING // ADL of std functions.
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RealType p = dist.success_fraction();
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return (2 - p) / sqrt(1 - p);
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} // skewness
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template <class RealType, class Policy>
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inline RealType kurtosis(const geometric_distribution<RealType, Policy>& dist)
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{ // kurtosis of geometric distribution
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// http://en.wikipedia.org/wiki/geometric is kurtosis_excess so add 3
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RealType p = dist.success_fraction();
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return 3 + (p*p - 6*p + 6) / (1 - p);
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} // kurtosis
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template <class RealType, class Policy>
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inline RealType kurtosis_excess(const geometric_distribution<RealType, Policy>& dist)
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{ // kurtosis excess of geometric distribution
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// http://mathworld.wolfram.com/Kurtosis.html table of kurtosis_excess
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RealType p = dist.success_fraction();
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return (p*p - 6*p + 6) / (1 - p);
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} // kurtosis_excess
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// RealType standard_deviation(const geometric_distribution<RealType, Policy>& dist)
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// standard_deviation provided by derived accessors.
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// RealType hazard(const geometric_distribution<RealType, Policy>& dist)
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// hazard of geometric distribution provided by derived accessors.
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// RealType chf(const geometric_distribution<RealType, Policy>& dist)
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// chf of geometric distribution provided by derived accessors.
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template <class RealType, class Policy>
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inline RealType pdf(const geometric_distribution<RealType, Policy>& dist, const RealType& k)
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{ // Probability Density/Mass Function.
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BOOST_FPU_EXCEPTION_GUARD
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BOOST_MATH_STD_USING // For ADL of math functions.
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static const char* function = "boost::math::pdf(const geometric_distribution<%1%>&, %1%)";
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RealType p = dist.success_fraction();
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RealType result = 0;
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if(false == geometric_detail::check_dist_and_k(
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function,
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p,
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k,
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&result, Policy()))
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{
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return result;
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}
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if (k == 0)
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{
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return p; // success_fraction
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}
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RealType q = 1 - p; // Inaccurate for small p?
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// So try to avoid inaccuracy for large or small p.
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// but has little effect > last significant bit.
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//cout << "p * pow(q, k) " << result << endl; // seems best whatever p
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//cout << "exp(p * k * log1p(-p)) " << p * exp(k * log1p(-p)) << endl;
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//if (p < 0.5)
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//{
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// result = p * pow(q, k);
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//}
|
||
|
|
//else
|
||
|
|
//{
|
||
|
|
// result = p * exp(k * log1p(-p));
|
||
|
|
//}
|
||
|
|
result = p * pow(q, k);
|
||
|
|
return result;
|
||
|
|
} // geometric_pdf
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType cdf(const geometric_distribution<RealType, Policy>& dist, const RealType& k)
|
||
|
|
{ // Cumulative Distribution Function of geometric.
|
||
|
|
static const char* function = "boost::math::cdf(const geometric_distribution<%1%>&, %1%)";
|
||
|
|
|
||
|
|
// k argument may be integral, signed, or unsigned, or floating point.
|
||
|
|
// If necessary, it has already been promoted from an integral type.
|
||
|
|
RealType p = dist.success_fraction();
|
||
|
|
// Error check:
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == geometric_detail::check_dist_and_k(
|
||
|
|
function,
|
||
|
|
p,
|
||
|
|
k,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
if(k == 0)
|
||
|
|
{
|
||
|
|
return p; // success_fraction
|
||
|
|
}
|
||
|
|
//RealType q = 1 - p; // Bad for small p
|
||
|
|
//RealType probability = 1 - std::pow(q, k+1);
|
||
|
|
|
||
|
|
RealType z = boost::math::log1p(-p, Policy()) * (k + 1);
|
||
|
|
RealType probability = -boost::math::expm1(z, Policy());
|
||
|
|
|
||
|
|
return probability;
|
||
|
|
} // cdf Cumulative Distribution Function geometric.
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType cdf(const complemented2_type<geometric_distribution<RealType, Policy>, RealType>& c)
|
||
|
|
{ // Complemented Cumulative Distribution Function geometric.
|
||
|
|
BOOST_MATH_STD_USING
|
||
|
|
static const char* function = "boost::math::cdf(const geometric_distribution<%1%>&, %1%)";
|
||
|
|
// k argument may be integral, signed, or unsigned, or floating point.
|
||
|
|
// If necessary, it has already been promoted from an integral type.
|
||
|
|
RealType const& k = c.param;
|
||
|
|
geometric_distribution<RealType, Policy> const& dist = c.dist;
|
||
|
|
RealType p = dist.success_fraction();
|
||
|
|
// Error check:
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == geometric_detail::check_dist_and_k(
|
||
|
|
function,
|
||
|
|
p,
|
||
|
|
k,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
RealType z = boost::math::log1p(-p, Policy()) * (k+1);
|
||
|
|
RealType probability = exp(z);
|
||
|
|
return probability;
|
||
|
|
} // cdf Complemented Cumulative Distribution Function geometric.
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType quantile(const geometric_distribution<RealType, Policy>& dist, const RealType& x)
|
||
|
|
{ // Quantile, percentile/100 or Percent Point geometric function.
|
||
|
|
// Return the number of expected failures k for a given probability p.
|
||
|
|
|
||
|
|
// Inverse cumulative Distribution Function or Quantile (percentile / 100) of geometric Probability.
|
||
|
|
// k argument may be integral, signed, or unsigned, or floating point.
|
||
|
|
|
||
|
|
static const char* function = "boost::math::quantile(const geometric_distribution<%1%>&, %1%)";
|
||
|
|
BOOST_MATH_STD_USING // ADL of std functions.
|
||
|
|
|
||
|
|
RealType success_fraction = dist.success_fraction();
|
||
|
|
// Check dist and x.
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == geometric_detail::check_dist_and_prob
|
||
|
|
(function, success_fraction, x, &result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
// Special cases.
|
||
|
|
if (x == 1)
|
||
|
|
{ // Would need +infinity failures for total confidence.
|
||
|
|
result = policies::raise_overflow_error<RealType>(
|
||
|
|
function,
|
||
|
|
"Probability argument is 1, which implies infinite failures !", Policy());
|
||
|
|
return result;
|
||
|
|
// usually means return +std::numeric_limits<RealType>::infinity();
|
||
|
|
// unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR
|
||
|
|
}
|
||
|
|
if (x == 0)
|
||
|
|
{ // No failures are expected if P = 0.
|
||
|
|
return 0; // Total trials will be just dist.successes.
|
||
|
|
}
|
||
|
|
// if (P <= pow(dist.success_fraction(), 1))
|
||
|
|
if (x <= success_fraction)
|
||
|
|
{ // p <= pdf(dist, 0) == cdf(dist, 0)
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
if (x == 1)
|
||
|
|
{
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
// log(1-x) /log(1-success_fraction) -1; but use log1p in case success_fraction is small
|
||
|
|
result = boost::math::log1p(-x, Policy()) / boost::math::log1p(-success_fraction, Policy()) - 1;
|
||
|
|
// Subtract a few epsilons here too?
|
||
|
|
// to make sure it doesn't slip over, so ceil would be one too many.
|
||
|
|
return result;
|
||
|
|
} // RealType quantile(const geometric_distribution dist, p)
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType quantile(const complemented2_type<geometric_distribution<RealType, Policy>, RealType>& c)
|
||
|
|
{ // Quantile or Percent Point Binomial function.
|
||
|
|
// Return the number of expected failures k for a given
|
||
|
|
// complement of the probability Q = 1 - P.
|
||
|
|
static const char* function = "boost::math::quantile(const geometric_distribution<%1%>&, %1%)";
|
||
|
|
BOOST_MATH_STD_USING
|
||
|
|
// Error checks:
|
||
|
|
RealType x = c.param;
|
||
|
|
const geometric_distribution<RealType, Policy>& dist = c.dist;
|
||
|
|
RealType success_fraction = dist.success_fraction();
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == geometric_detail::check_dist_and_prob(
|
||
|
|
function,
|
||
|
|
success_fraction,
|
||
|
|
x,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
// Special cases:
|
||
|
|
if(x == 1)
|
||
|
|
{ // There may actually be no answer to this question,
|
||
|
|
// since the probability of zero failures may be non-zero,
|
||
|
|
return 0; // but zero is the best we can do:
|
||
|
|
}
|
||
|
|
if (-x <= boost::math::powm1(dist.success_fraction(), dist.successes(), Policy()))
|
||
|
|
{ // q <= cdf(complement(dist, 0)) == pdf(dist, 0)
|
||
|
|
return 0; //
|
||
|
|
}
|
||
|
|
if(x == 0)
|
||
|
|
{ // Probability 1 - Q == 1 so infinite failures to achieve certainty.
|
||
|
|
// Would need +infinity failures for total confidence.
|
||
|
|
result = policies::raise_overflow_error<RealType>(
|
||
|
|
function,
|
||
|
|
"Probability argument complement is 0, which implies infinite failures !", Policy());
|
||
|
|
return result;
|
||
|
|
// usually means return +std::numeric_limits<RealType>::infinity();
|
||
|
|
// unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR
|
||
|
|
}
|
||
|
|
// log(x) /log(1-success_fraction) -1; but use log1p in case success_fraction is small
|
||
|
|
result = log(x) / boost::math::log1p(-success_fraction, Policy()) - 1;
|
||
|
|
return result;
|
||
|
|
|
||
|
|
} // 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_SPECIAL_GEOMETRIC_HPP
|