528 lines
20 KiB
Plaintext
528 lines
20 KiB
Plaintext
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// boost\math\distributions\poisson.hpp
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// Copyright John Maddock 2006.
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// Copyright Paul A. Bristow 2007.
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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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// Poisson distribution is a discrete probability distribution.
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// It expresses the probability of a number (k) of
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// events, occurrences, failures or arrivals occurring in a fixed time,
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// assuming these events occur with a known average or mean rate (lambda)
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// and are independent of the time since the last event.
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// The distribution was discovered by Simeon-Denis Poisson (1781-1840).
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// Parameter lambda is the mean number of events in the given time interval.
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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 Poisson distribution
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// (like others including the binomial, negative binomial & 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/Poisson_distribution
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// http://documents.wolfram.com/v5/Add-onsLinks/StandardPackages/Statistics/DiscreteDistributions.html
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#ifndef BOOST_MATH_SPECIAL_POISSON_HPP
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#define BOOST_MATH_SPECIAL_POISSON_HPP
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#include <boost/math/distributions/fwd.hpp>
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#include <boost/math/special_functions/gamma.hpp> // for incomplete gamma. gamma_q
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#include <boost/math/special_functions/trunc.hpp> // for incomplete gamma. gamma_q
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#include <boost/math/distributions/complement.hpp> // complements
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#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks
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#include <boost/math/special_functions/fpclassify.hpp> // isnan.
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#include <boost/math/special_functions/factorials.hpp> // factorials.
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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 <utility>
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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 poisson_detail
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{
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// Common error checking routines for Poisson distribution functions.
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// These are convoluted, & apparently redundant, to try to ensure that
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// checks are always performed, even if exceptions are not enabled.
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template <class RealType, class Policy>
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inline bool check_mean(const char* function, const RealType& mean, RealType* result, const Policy& pol)
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{
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if(!(boost::math::isfinite)(mean) || (mean < 0))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"Mean argument is %1%, but must be >= 0 !", mean, pol);
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return false;
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}
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return true;
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} // bool check_mean
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template <class RealType, class Policy>
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inline bool check_mean_NZ(const char* function, const RealType& mean, RealType* result, const Policy& pol)
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{ // mean == 0 is considered an error.
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if( !(boost::math::isfinite)(mean) || (mean <= 0))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"Mean argument is %1%, but must be > 0 !", mean, pol);
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return false;
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}
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return true;
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} // bool check_mean_NZ
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template <class RealType, class Policy>
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inline bool check_dist(const char* function, const RealType& mean, RealType* result, const Policy& pol)
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{ // Only one check, so this is redundant really but should be optimized away.
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return check_mean_NZ(function, mean, result, pol);
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} // bool check_dist
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template <class RealType, class Policy>
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inline bool check_k(const char* function, const RealType& k, RealType* result, const Policy& pol)
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{
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if((k < 0) || !(boost::math::isfinite)(k))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"Number of events k 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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} // bool check_k
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template <class RealType, class Policy>
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inline bool check_dist_and_k(const char* function, RealType mean, RealType k, RealType* result, const Policy& pol)
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{
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if((check_dist(function, mean, result, pol) == false) ||
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(check_k(function, k, 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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} // bool check_dist_and_k
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template <class RealType, class Policy>
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inline bool check_prob(const char* function, const RealType& p, RealType* result, const Policy& pol)
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{ // Check 0 <= p <= 1
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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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"Probability 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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} // bool check_prob
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template <class RealType, class Policy>
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inline bool check_dist_and_prob(const char* function, RealType mean, RealType p, RealType* result, const Policy& pol)
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{
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if((check_dist(function, mean, result, pol) == false) ||
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(check_prob(function, p, 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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} // bool check_dist_and_prob
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} // namespace poisson_detail
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template <class RealType = double, class Policy = policies::policy<> >
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class poisson_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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poisson_distribution(RealType l_mean = 1) : m_l(l_mean) // mean (lambda).
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{ // Expected mean number of events that occur during the given interval.
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RealType r;
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poisson_detail::check_dist(
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"boost::math::poisson_distribution<%1%>::poisson_distribution",
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m_l,
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&r, Policy());
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} // poisson_distribution constructor.
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RealType mean() const
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{ // Private data getter function.
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return m_l;
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}
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private:
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// Data member, initialized by constructor.
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RealType m_l; // mean number of occurrences.
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}; // template <class RealType, class Policy> class poisson_distribution
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typedef poisson_distribution<double> poisson; // Reserved name of type double.
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// Non-member functions to give properties of the distribution.
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> range(const poisson_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 poisson_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>());
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}
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template <class RealType, class Policy>
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inline RealType mean(const poisson_distribution<RealType, Policy>& dist)
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{ // Mean of poisson distribution = lambda.
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return dist.mean();
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} // mean
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template <class RealType, class Policy>
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inline RealType mode(const poisson_distribution<RealType, Policy>& dist)
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{ // mode.
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BOOST_MATH_STD_USING // ADL of std functions.
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return floor(dist.mean());
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}
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//template <class RealType, class Policy>
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//inline RealType median(const poisson_distribution<RealType, Policy>& dist)
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//{ // median = approximately lambda + 1/3 - 0.2/lambda
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// RealType l = dist.mean();
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// return dist.mean() + static_cast<RealType>(0.3333333333333333333333333333333333333333333333)
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// - static_cast<RealType>(0.2) / l;
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//} // BUT this formula appears to be out-by-one compared to quantile(half)
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// Query posted on Wikipedia.
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// Now implemented via quantile(half) in derived accessors.
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template <class RealType, class Policy>
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inline RealType variance(const poisson_distribution<RealType, Policy>& dist)
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{ // variance.
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return dist.mean();
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}
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// RealType standard_deviation(const poisson_distribution<RealType, Policy>& dist)
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// standard_deviation provided by derived accessors.
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template <class RealType, class Policy>
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inline RealType skewness(const poisson_distribution<RealType, Policy>& dist)
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{ // skewness = sqrt(l).
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BOOST_MATH_STD_USING // ADL of std functions.
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return 1 / sqrt(dist.mean());
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}
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template <class RealType, class Policy>
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inline RealType kurtosis_excess(const poisson_distribution<RealType, Policy>& dist)
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{ // skewness = sqrt(l).
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return 1 / dist.mean(); // kurtosis_excess 1/mean from Wiki & MathWorld eq 31.
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// http://mathworld.wolfram.com/Kurtosis.html explains that the kurtosis excess
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// is more convenient because the kurtosis excess of a normal distribution is zero
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// whereas the true kurtosis is 3.
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} // RealType kurtosis_excess
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template <class RealType, class Policy>
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inline RealType kurtosis(const poisson_distribution<RealType, Policy>& dist)
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{ // kurtosis is 4th moment about the mean = u4 / sd ^ 4
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// http://en.wikipedia.org/wiki/Curtosis
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// kurtosis can range from -2 (flat top) to +infinity (sharp peak & heavy tails).
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// http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
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return 3 + 1 / dist.mean(); // NIST.
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// http://mathworld.wolfram.com/Kurtosis.html explains that the kurtosis excess
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// is more convenient because the kurtosis excess of a normal distribution is zero
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// whereas the true kurtosis is 3.
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} // RealType kurtosis
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template <class RealType, class Policy>
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RealType pdf(const poisson_distribution<RealType, Policy>& dist, const RealType& k)
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{ // Probability Density/Mass Function.
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// Probability that there are EXACTLY k occurrences (or arrivals).
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BOOST_FPU_EXCEPTION_GUARD
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BOOST_MATH_STD_USING // for ADL of std functions.
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RealType mean = dist.mean();
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// Error check:
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RealType result = 0;
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if(false == poisson_detail::check_dist_and_k(
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"boost::math::pdf(const poisson_distribution<%1%>&, %1%)",
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mean,
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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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// Special case of mean zero, regardless of the number of events k.
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if (mean == 0)
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{ // Probability for any k is zero.
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return 0;
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}
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if (k == 0)
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{ // mean ^ k = 1, and k! = 1, so can simplify.
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return exp(-mean);
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}
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return boost::math::gamma_p_derivative(k+1, mean, Policy());
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} // pdf
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template <class RealType, class Policy>
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RealType cdf(const poisson_distribution<RealType, Policy>& dist, const RealType& k)
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{ // Cumulative Distribution Function Poisson.
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// The random variate k is the number of 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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// Returns the sum of the terms 0 through k of the Poisson Probability Density or Mass (pdf).
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// But note that the Poisson distribution
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// (like others including the binomial, negative binomial & Bernoulli)
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// is strictly defined as a discrete function: only integral values of k are envisaged.
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// However because of the method of calculation using a continuous gamma function,
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// it is convenient to treat it as if it is 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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// outside this function to ensure that k is integral.
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// The terms are not summed directly (at least for larger k)
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// instead the incomplete gamma integral is employed,
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BOOST_MATH_STD_USING // for ADL of std function exp.
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RealType mean = dist.mean();
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// Error checks:
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RealType result = 0;
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if(false == poisson_detail::check_dist_and_k(
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"boost::math::cdf(const poisson_distribution<%1%>&, %1%)",
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mean,
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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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// Special cases:
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if (mean == 0)
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{ // Probability for any k is zero.
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return 0;
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}
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if (k == 0)
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{ // return pdf(dist, static_cast<RealType>(0));
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// but mean (and k) have already been checked,
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// so this avoids unnecessary repeated checks.
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return exp(-mean);
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}
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// For small integral k could use a finite sum -
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// it's cheaper than the gamma function.
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// BUT this is now done efficiently by gamma_q function.
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// Calculate poisson cdf using the gamma_q function.
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return gamma_q(k+1, mean, Policy());
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} // binomial cdf
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template <class RealType, class Policy>
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RealType cdf(const complemented2_type<poisson_distribution<RealType, Policy>, RealType>& c)
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{ // Complemented Cumulative Distribution Function Poisson
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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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// But note that the Poisson distribution
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// (like others including the binomial, negative binomial & Bernoulli)
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// is strictly defined as a discrete function: only integral values of k are envisaged.
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// However because of the method of calculation using a continuous gamma function,
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// it is convenient to treat it as is it is 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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// outside this function to ensure that k is integral.
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// Returns the sum of the terms k+1 through inf of the Poisson Probability Density/Mass (pdf).
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// The terms are not summed directly (at least for larger k)
|
||
|
|
// instead the incomplete gamma integral is employed,
|
||
|
|
|
||
|
|
RealType const& k = c.param;
|
||
|
|
poisson_distribution<RealType, Policy> const& dist = c.dist;
|
||
|
|
|
||
|
|
RealType mean = dist.mean();
|
||
|
|
|
||
|
|
// Error checks:
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == poisson_detail::check_dist_and_k(
|
||
|
|
"boost::math::cdf(const poisson_distribution<%1%>&, %1%)",
|
||
|
|
mean,
|
||
|
|
k,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
// Special case of mean, regardless of the number of events k.
|
||
|
|
if (mean == 0)
|
||
|
|
{ // Probability for any k is unity, complement of zero.
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
if (k == 0)
|
||
|
|
{ // Avoid repeated checks on k and mean in gamma_p.
|
||
|
|
return -boost::math::expm1(-mean, Policy());
|
||
|
|
}
|
||
|
|
// Unlike un-complemented cdf (sum from 0 to k),
|
||
|
|
// can't use finite sum from k+1 to infinity for small integral k,
|
||
|
|
// anyway it is now done efficiently by gamma_p.
|
||
|
|
return gamma_p(k + 1, mean, Policy()); // Calculate Poisson cdf using the gamma_p function.
|
||
|
|
// CCDF = gamma_p(k+1, lambda)
|
||
|
|
} // poisson ccdf
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType quantile(const poisson_distribution<RealType, Policy>& dist, const RealType& p)
|
||
|
|
{ // Quantile (or Percent Point) Poisson function.
|
||
|
|
// Return the number of expected events k for a given probability p.
|
||
|
|
static const char* function = "boost::math::quantile(const poisson_distribution<%1%>&, %1%)";
|
||
|
|
RealType result = 0; // of Argument checks:
|
||
|
|
if(false == poisson_detail::check_prob(
|
||
|
|
function,
|
||
|
|
p,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
// Special case:
|
||
|
|
if (dist.mean() == 0)
|
||
|
|
{ // if mean = 0 then p = 0, so k can be anything?
|
||
|
|
if (false == poisson_detail::check_mean_NZ(
|
||
|
|
function,
|
||
|
|
dist.mean(),
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
if(p == 0)
|
||
|
|
{
|
||
|
|
return 0; // Exact result regardless of discrete-quantile Policy
|
||
|
|
}
|
||
|
|
if(p == 1)
|
||
|
|
{
|
||
|
|
return policies::raise_overflow_error<RealType>(function, 0, Policy());
|
||
|
|
}
|
||
|
|
typedef typename Policy::discrete_quantile_type discrete_type;
|
||
|
|
boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
|
||
|
|
RealType guess, factor = 8;
|
||
|
|
RealType z = dist.mean();
|
||
|
|
if(z < 1)
|
||
|
|
guess = z;
|
||
|
|
else
|
||
|
|
guess = boost::math::detail::inverse_poisson_cornish_fisher(z, p, RealType(1-p), Policy());
|
||
|
|
if(z > 5)
|
||
|
|
{
|
||
|
|
if(z > 1000)
|
||
|
|
factor = 1.01f;
|
||
|
|
else if(z > 50)
|
||
|
|
factor = 1.1f;
|
||
|
|
else if(guess > 10)
|
||
|
|
factor = 1.25f;
|
||
|
|
else
|
||
|
|
factor = 2;
|
||
|
|
if(guess < 1.1)
|
||
|
|
factor = 8;
|
||
|
|
}
|
||
|
|
|
||
|
|
return detail::inverse_discrete_quantile(
|
||
|
|
dist,
|
||
|
|
p,
|
||
|
|
false,
|
||
|
|
guess,
|
||
|
|
factor,
|
||
|
|
RealType(1),
|
||
|
|
discrete_type(),
|
||
|
|
max_iter);
|
||
|
|
} // quantile
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType quantile(const complemented2_type<poisson_distribution<RealType, Policy>, RealType>& c)
|
||
|
|
{ // Quantile (or Percent Point) of Poisson function.
|
||
|
|
// Return the number of expected events k for a given
|
||
|
|
// complement of the probability q.
|
||
|
|
//
|
||
|
|
// Error checks:
|
||
|
|
static const char* function = "boost::math::quantile(complement(const poisson_distribution<%1%>&, %1%))";
|
||
|
|
RealType q = c.param;
|
||
|
|
const poisson_distribution<RealType, Policy>& dist = c.dist;
|
||
|
|
RealType result = 0; // of argument checks.
|
||
|
|
if(false == poisson_detail::check_prob(
|
||
|
|
function,
|
||
|
|
q,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
// Special case:
|
||
|
|
if (dist.mean() == 0)
|
||
|
|
{ // if mean = 0 then p = 0, so k can be anything?
|
||
|
|
if (false == poisson_detail::check_mean_NZ(
|
||
|
|
function,
|
||
|
|
dist.mean(),
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
if(q == 0)
|
||
|
|
{
|
||
|
|
return policies::raise_overflow_error<RealType>(function, 0, Policy());
|
||
|
|
}
|
||
|
|
if(q == 1)
|
||
|
|
{
|
||
|
|
return 0; // Exact result regardless of discrete-quantile Policy
|
||
|
|
}
|
||
|
|
typedef typename Policy::discrete_quantile_type discrete_type;
|
||
|
|
boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
|
||
|
|
RealType guess, factor = 8;
|
||
|
|
RealType z = dist.mean();
|
||
|
|
if(z < 1)
|
||
|
|
guess = z;
|
||
|
|
else
|
||
|
|
guess = boost::math::detail::inverse_poisson_cornish_fisher(z, RealType(1-q), q, Policy());
|
||
|
|
if(z > 5)
|
||
|
|
{
|
||
|
|
if(z > 1000)
|
||
|
|
factor = 1.01f;
|
||
|
|
else if(z > 50)
|
||
|
|
factor = 1.1f;
|
||
|
|
else if(guess > 10)
|
||
|
|
factor = 1.25f;
|
||
|
|
else
|
||
|
|
factor = 2;
|
||
|
|
if(guess < 1.1)
|
||
|
|
factor = 8;
|
||
|
|
}
|
||
|
|
|
||
|
|
return detail::inverse_discrete_quantile(
|
||
|
|
dist,
|
||
|
|
q,
|
||
|
|
true,
|
||
|
|
guess,
|
||
|
|
factor,
|
||
|
|
RealType(1),
|
||
|
|
discrete_type(),
|
||
|
|
max_iter);
|
||
|
|
} // 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>
|
||
|
|
#include <boost/math/distributions/detail/inv_discrete_quantile.hpp>
|
||
|
|
|
||
|
|
#endif // BOOST_MATH_SPECIAL_POISSON_HPP
|
||
|
|
|
||
|
|
|
||
|
|
|