boost/math/distributions/binomial.hpp
// boost\math\distributions\binomial.hpp // Copyright John Maddock 2006. // Copyright Paul A. Bristow 2007. // 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/binomial_distribution // Binomial distribution is the discrete probability distribution of // the number (k) of successes, in a sequence of // n independent (yes or no, success or failure) Bernoulli trials. // It expresses the probability of a number of events occurring in a fixed time // if these events occur with a known average rate (probability of success), // and are independent of the time since the last event. // The number of cars that pass through a certain point on a road during a given period of time. // The number of spelling mistakes a secretary makes while typing a single page. // The number of phone calls at a call center per minute. // The number of times a web server is accessed per minute. // The number of light bulbs that burn out in a certain amount of time. // The number of roadkill found per unit length of road // http://en.wikipedia.org/wiki/binomial_distribution // Given a sample of N measured values k[i], // we wish to estimate the value of the parameter x (mean) // of the binomial population from which the sample was drawn. // To calculate the maximum likelihood value = 1/N sum i = 1 to N of k[i] // Also may want a function for EXACTLY k. // And probability that there are EXACTLY k occurrences is // exp(-x) * pow(x, k) / factorial(k) // where x is expected occurrences (mean) during the given interval. // For example, if events occur, on average, every 4 min, // and we are interested in number of events occurring in 10 min, // then x = 10/4 = 2.5 // http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm // The binomial distribution is used when there are // exactly two mutually exclusive outcomes of a trial. // These outcomes are appropriately labeled "success" and "failure". // The binomial distribution is used to obtain // the probability of observing x successes in N trials, // with the probability of success on a single trial denoted by p. // The binomial distribution assumes that p is fixed for all trials. // P(x, p, n) = n!/(x! * (n-x)!) * p^x * (1-p)^(n-x) // http://mathworld.wolfram.com/BinomialCoefficient.html // The binomial coefficient (n; k) is the number of ways of picking // k unordered outcomes from n possibilities, // also known as a combination or combinatorial number. // The symbols _nC_k and (n; k) are used to denote a binomial coefficient, // and are sometimes read as "n choose k." // (n; k) therefore gives the number of k-subsets possible out of a set of n distinct items. // For example: // The 2-subsets of {1,2,3,4} are the six pairs {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, and {3,4}, so (4; 2)==6. // http://functions.wolfram.com/GammaBetaErf/Binomial/ for evaluation. // But note that the binomial distribution // (like others including the poisson, negative binomial & Bernoulli) // is strictly defined as a discrete function: only integral values of k are envisaged. // However because of the method of calculation using a continuous gamma function, // it is convenient to treat it as if a continous function, // and permit non-integral values of k. // To enforce the strict mathematical model, users should use floor or ceil functions // on k outside this function to ensure that k is integral. #ifndef BOOST_MATH_SPECIAL_BINOMIAL_HPP #define BOOST_MATH_SPECIAL_BINOMIAL_HPP #include <boost/math/distributions/fwd.hpp> #include <boost/math/special_functions/beta.hpp> // for incomplete beta. #include <boost/math/distributions/complement.hpp> // complements #include <boost/math/distributions/detail/common_error_handling.hpp> // error checks #include <boost/math/distributions/detail/inv_discrete_quantile.hpp> // error checks #include <boost/math/special_functions/fpclassify.hpp> // isnan. #include <boost/math/tools/roots.hpp> // for root finding. #include <utility> namespace boost { namespace math { template <class RealType, class Policy> class binomial_distribution; namespace binomial_detail{ // common error checking routines for binomial distribution functions: template <class RealType, class Policy> inline bool check_N(const char* function, const RealType& N, RealType* result, const Policy& pol) { if((N < 0) || !(boost::math::isfinite)(N)) { *result = policies::raise_domain_error<RealType>( function, "Number of Trials argument is %1%, but must be >= 0 !", N, pol); return false; } return true; } template <class RealType, class Policy> inline bool check_success_fraction(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, "Success fraction argument is %1%, but must be >= 0 and <= 1 !", p, pol); return false; } return true; } template <class RealType, class Policy> inline bool check_dist(const char* function, const RealType& N, const RealType& p, RealType* result, const Policy& pol) { return check_success_fraction( function, p, result, pol) && check_N( function, N, result, pol); } template <class RealType, class Policy> inline bool check_dist_and_k(const char* function, const RealType& N, const RealType& p, RealType k, RealType* result, const Policy& pol) { if(check_dist(function, N, p, result, pol) == false) return false; if((k < 0) || !(boost::math::isfinite)(k)) { *result = policies::raise_domain_error<RealType>( function, "Number of Successes argument is %1%, but must be >= 0 !", k, pol); return false; } if(k > N) { *result = policies::raise_domain_error<RealType>( function, "Number of Successes argument is %1%, but must be <= Number of Trials !", k, pol); return false; } return true; } template <class RealType, class Policy> inline bool check_dist_and_prob(const char* function, const RealType& N, RealType p, RealType prob, RealType* result, const Policy& pol) { if((check_dist(function, N, p, result, pol) && detail::check_probability(function, prob, result, pol)) == false) return false; return true; } template <class T, class Policy> T inverse_binomial_cornish_fisher(T n, T sf, T p, T q, const Policy& pol) { BOOST_MATH_STD_USING // mean: T m = n * sf; // standard deviation: T sigma = sqrt(n * sf * (1 - sf)); // skewness T sk = (1 - 2 * sf) / sigma; // kurtosis: // T k = (1 - 6 * sf * (1 - sf) ) / (n * sf * (1 - sf)); // Get the inverse of a std normal distribution: T x = boost::math::erfc_inv(p > q ? 2 * q : 2 * p, pol) * constants::root_two<T>(); // Set the sign: if(p < 0.5) x = -x; T x2 = x * x; // w is correction term due to skewness T w = x + sk * (x2 - 1) / 6; /* // Add on correction due to kurtosis. // Disabled for now, seems to make things worse? // if(n >= 10) w += k * x * (x2 - 3) / 24 + sk * sk * x * (2 * x2 - 5) / -36; */ w = m + sigma * w; if(w < tools::min_value<T>()) return sqrt(tools::min_value<T>()); if(w > n) return n; return w; } template <class RealType, class Policy> RealType quantile_imp(const binomial_distribution<RealType, Policy>& dist, const RealType& p, const RealType& q, bool comp) { // Quantile or Percent Point Binomial function. // Return the number of expected successes k, // for a given probability p. // // Error checks: BOOST_MATH_STD_USING // ADL of std names RealType result = 0; RealType trials = dist.trials(); RealType success_fraction = dist.success_fraction(); if(false == binomial_detail::check_dist_and_prob( "boost::math::quantile(binomial_distribution<%1%> const&, %1%)", trials, success_fraction, p, &result, Policy())) { return result; } // Special cases: // if(p == 0) { // There may actually be no answer to this question, // since the probability of zero successes may be non-zero, // but zero is the best we can do: return 0; } if(p == 1) { // Probability of n or fewer successes is always one, // so n is the most sensible answer here: return trials; } if (p <= pow(1 - success_fraction, trials)) { // p <= pdf(dist, 0) == cdf(dist, 0) return 0; // So the only reasonable result is zero. } // And root finder would fail otherwise. if(success_fraction == 1) { // our formulae break down in this case: return p > 0.5f ? trials : 0; } // Solve for quantile numerically: // RealType guess = binomial_detail::inverse_binomial_cornish_fisher(trials, success_fraction, p, q, Policy()); RealType factor = 8; if(trials > 100) factor = 1.01f; // guess is pretty accurate else if((trials > 10) && (trials - 1 > guess) && (guess > 3)) factor = 1.15f; // less accurate but OK. else if(trials < 10) { // pretty inaccurate guess in this area: if(guess > trials / 64) { guess = trials / 4; factor = 2; } else guess = trials / 1024; } else factor = 2; // trials largish, but in far tails. typedef typename Policy::discrete_quantile_type discrete_quantile_type; boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); return detail::inverse_discrete_quantile( dist, comp ? q : p, comp, guess, factor, RealType(1), discrete_quantile_type(), max_iter); } // quantile } template <class RealType = double, class Policy = policies::policy<> > class binomial_distribution { public: typedef RealType value_type; typedef Policy policy_type; binomial_distribution(RealType n = 1, RealType p = 0.5) : m_n(n), m_p(p) { // Default n = 1 is the Bernoulli distribution // with equal probability of 'heads' or 'tails. RealType r; binomial_detail::check_dist( "boost::math::binomial_distribution<%1%>::binomial_distribution", m_n, m_p, &r, Policy()); } // binomial_distribution constructor. RealType success_fraction() const { // Probability. return m_p; } RealType trials() const { // Total number of trials. return m_n; } enum interval_type{ clopper_pearson_exact_interval, jeffreys_prior_interval }; // // Estimation of the success fraction parameter. // The best estimate is actually simply successes/trials, // these functions are used // to obtain confidence intervals for the success fraction. // static RealType find_lower_bound_on_p( RealType trials, RealType successes, RealType probability, interval_type t = clopper_pearson_exact_interval) { static const char* function = "boost::math::binomial_distribution<%1%>::find_lower_bound_on_p"; // Error checks: RealType result = 0; if(false == binomial_detail::check_dist_and_k( function, trials, RealType(0), successes, &result, Policy()) && binomial_detail::check_dist_and_prob( function, trials, RealType(0), probability, &result, Policy())) { return result; } if(successes == 0) return 0; // NOTE!!! The Clopper Pearson formula uses "successes" not // "successes+1" as usual to get the lower bound, // see http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm return (t == clopper_pearson_exact_interval) ? ibeta_inv(successes, trials - successes + 1, probability, static_cast<RealType*>(0), Policy()) : ibeta_inv(successes + 0.5f, trials - successes + 0.5f, probability, static_cast<RealType*>(0), Policy()); } static RealType find_upper_bound_on_p( RealType trials, RealType successes, RealType probability, interval_type t = clopper_pearson_exact_interval) { static const char* function = "boost::math::binomial_distribution<%1%>::find_upper_bound_on_p"; // Error checks: RealType result = 0; if(false == binomial_detail::check_dist_and_k( function, trials, RealType(0), successes, &result, Policy()) && binomial_detail::check_dist_and_prob( function, trials, RealType(0), probability, &result, Policy())) { return result; } if(trials == successes) return 1; return (t == clopper_pearson_exact_interval) ? ibetac_inv(successes + 1, trials - successes, probability, static_cast<RealType*>(0), Policy()) : ibetac_inv(successes + 0.5f, trials - successes + 0.5f, probability, static_cast<RealType*>(0), Policy()); } // Estimate number of trials parameter: // // "How many trials do I need to be P% sure of seeing k events?" // or // "How many trials can I have to be P% sure of seeing fewer than k events?" // static RealType find_minimum_number_of_trials( RealType k, // number of events RealType p, // success fraction RealType alpha) // risk level { static const char* function = "boost::math::binomial_distribution<%1%>::find_minimum_number_of_trials"; // Error checks: RealType result = 0; if(false == binomial_detail::check_dist_and_k( function, k, p, k, &result, Policy()) && binomial_detail::check_dist_and_prob( function, k, p, alpha, &result, Policy())) { return result; } result = ibetac_invb(k + 1, p, alpha, Policy()); // returns n - k return result + k; } static RealType find_maximum_number_of_trials( RealType k, // number of events RealType p, // success fraction RealType alpha) // risk level { static const char* function = "boost::math::binomial_distribution<%1%>::find_maximum_number_of_trials"; // Error checks: RealType result = 0; if(false == binomial_detail::check_dist_and_k( function, k, p, k, &result, Policy()) && binomial_detail::check_dist_and_prob( function, k, p, alpha, &result, Policy())) { return result; } result = ibeta_invb(k + 1, p, alpha, Policy()); // returns n - k return result + k; } private: RealType m_n; // Not sure if this shouldn't be an int? RealType m_p; // success_fraction }; // template <class RealType, class Policy> class binomial_distribution typedef binomial_distribution<> binomial; // typedef binomial_distribution<double> binomial; // IS now included since no longer a name clash with function binomial. //typedef binomial_distribution<double> binomial; // Reserved name of type double. template <class RealType, class Policy> const std::pair<RealType, RealType> range(const binomial_distribution<RealType, Policy>& dist) { // Range of permissible values for random variable k. using boost::math::tools::max_value; return std::pair<RealType, RealType>(static_cast<RealType>(0), dist.trials()); } template <class RealType, class Policy> const std::pair<RealType, RealType> support(const binomial_distribution<RealType, Policy>& dist) { // Range of supported values for random variable k. // 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), dist.trials()); } template <class RealType, class Policy> inline RealType mean(const binomial_distribution<RealType, Policy>& dist) { // Mean of Binomial distribution = np. return dist.trials() * dist.success_fraction(); } // mean template <class RealType, class Policy> inline RealType variance(const binomial_distribution<RealType, Policy>& dist) { // Variance of Binomial distribution = np(1-p). return dist.trials() * dist.success_fraction() * (1 - dist.success_fraction()); } // variance template <class RealType, class Policy> RealType pdf(const binomial_distribution<RealType, Policy>& dist, const RealType& k) { // Probability Density/Mass Function. BOOST_FPU_EXCEPTION_GUARD BOOST_MATH_STD_USING // for ADL of std functions RealType n = dist.trials(); // Error check: RealType result = 0; // initialization silences some compiler warnings if(false == binomial_detail::check_dist_and_k( "boost::math::pdf(binomial_distribution<%1%> const&, %1%)", n, dist.success_fraction(), k, &result, Policy())) { return result; } // Special cases of success_fraction, regardless of k successes and regardless of n trials. if (dist.success_fraction() == 0) { // probability of zero successes is 1: return static_cast<RealType>(k == 0 ? 1 : 0); } if (dist.success_fraction() == 1) { // probability of n successes is 1: return static_cast<RealType>(k == n ? 1 : 0); } // k argument may be integral, signed, or unsigned, or floating point. // If necessary, it has already been promoted from an integral type. if (n == 0) { return 1; // Probability = 1 = certainty. } if (k == 0) { // binomial coeffic (n 0) = 1, // n ^ 0 = 1 return pow(1 - dist.success_fraction(), n); } if (k == n) { // binomial coeffic (n n) = 1, // n ^ 0 = 1 return pow(dist.success_fraction(), k); // * pow((1 - dist.success_fraction()), (n - k)) = 1 } // Probability of getting exactly k successes // if C(n, k) is the binomial coefficient then: // // f(k; n,p) = C(n, k) * p^k * (1-p)^(n-k) // = (n!/(k!(n-k)!)) * p^k * (1-p)^(n-k) // = (tgamma(n+1) / (tgamma(k+1)*tgamma(n-k+1))) * p^k * (1-p)^(n-k) // = p^k (1-p)^(n-k) / (beta(k+1, n-k+1) * (n+1)) // = ibeta_derivative(k+1, n-k+1, p) / (n+1) // using boost::math::ibeta_derivative; // a, b, x return ibeta_derivative(k+1, n-k+1, dist.success_fraction(), Policy()) / (n+1); } // pdf template <class RealType, class Policy> inline RealType cdf(const binomial_distribution<RealType, Policy>& dist, const RealType& k) { // Cumulative Distribution Function Binomial. // The random variate k is the number of successes in n trials. // k argument may be integral, signed, or unsigned, or floating point. // If necessary, it has already been promoted from an integral type. // Returns the sum of the terms 0 through k of the Binomial Probability Density/Mass: // // i=k // -- ( n ) i n-i // > | | p (1-p) // -- ( i ) // i=0 // The terms are not summed directly instead // the incomplete beta integral is employed, // according to the formula: // P = I[1-p]( n-k, k+1). // = 1 - I[p](k + 1, n - k) BOOST_MATH_STD_USING // for ADL of std functions RealType n = dist.trials(); RealType p = dist.success_fraction(); // Error check: RealType result = 0; if(false == binomial_detail::check_dist_and_k( "boost::math::cdf(binomial_distribution<%1%> const&, %1%)", n, p, k, &result, Policy())) { return result; } if (k == n) { return 1; } // Special cases, regardless of k. if (p == 0) { // This need explanation: // the pdf is zero for all cases except when k == 0. // For zero p the probability of zero successes is one. // Therefore the cdf is always 1: // the probability of k or *fewer* successes is always 1 // if there are never any successes! return 1; } if (p == 1) { // This is correct but needs explanation: // when k = 1 // all the cdf and pdf values are zero *except* when k == n, // and that case has been handled above already. return 0; } // // P = I[1-p](n - k, k + 1) // = 1 - I[p](k + 1, n - k) // Use of ibetac here prevents cancellation errors in calculating // 1-p if p is very small, perhaps smaller than machine epsilon. // // Note that we do not use a finite sum here, since the incomplete // beta uses a finite sum internally for integer arguments, so // we'll just let it take care of the necessary logic. // return ibetac(k + 1, n - k, p, Policy()); } // binomial cdf template <class RealType, class Policy> inline RealType cdf(const complemented2_type<binomial_distribution<RealType, Policy>, RealType>& c) { // Complemented Cumulative Distribution Function Binomial. // The random variate k is the number of successes in n trials. // k argument may be integral, signed, or unsigned, or floating point. // If necessary, it has already been promoted from an integral type. // Returns the sum of the terms k+1 through n of the Binomial Probability Density/Mass: // // i=n // -- ( n ) i n-i // > | | p (1-p) // -- ( i ) // i=k+1 // The terms are not summed directly instead // the incomplete beta integral is employed, // according to the formula: // Q = 1 -I[1-p]( n-k, k+1). // = I[p](k + 1, n - k) BOOST_MATH_STD_USING // for ADL of std functions RealType const& k = c.param; binomial_distribution<RealType, Policy> const& dist = c.dist; RealType n = dist.trials(); RealType p = dist.success_fraction(); // Error checks: RealType result = 0; if(false == binomial_detail::check_dist_and_k( "boost::math::cdf(binomial_distribution<%1%> const&, %1%)", n, p, k, &result, Policy())) { return result; } if (k == n) { // Probability of greater than n successes is necessarily zero: return 0; } // Special cases, regardless of k. if (p == 0) { // This need explanation: the pdf is zero for all // cases except when k == 0. For zero p the probability // of zero successes is one. Therefore the cdf is always // 1: the probability of *more than* k successes is always 0 // if there are never any successes! return 0; } if (p == 1) { // This needs explanation, when p = 1 // we always have n successes, so the probability // of more than k successes is 1 as long as k < n. // The k == n case has already been handled above. return 1; } // // Calculate cdf binomial using the incomplete beta function. // Q = 1 -I[1-p](n - k, k + 1) // = I[p](k + 1, n - k) // Use of ibeta here prevents cancellation errors in calculating // 1-p if p is very small, perhaps smaller than machine epsilon. // // Note that we do not use a finite sum here, since the incomplete // beta uses a finite sum internally for integer arguments, so // we'll just let it take care of the necessary logic. // return ibeta(k + 1, n - k, p, Policy()); } // binomial cdf template <class RealType, class Policy> inline RealType quantile(const binomial_distribution<RealType, Policy>& dist, const RealType& p) { return binomial_detail::quantile_imp(dist, p, RealType(1-p), false); } // quantile template <class RealType, class Policy> RealType quantile(const complemented2_type<binomial_distribution<RealType, Policy>, RealType>& c) { return binomial_detail::quantile_imp(c.dist, RealType(1-c.param), c.param, true); } // quantile template <class RealType, class Policy> inline RealType mode(const binomial_distribution<RealType, Policy>& dist) { BOOST_MATH_STD_USING // ADL of std functions. RealType p = dist.success_fraction(); RealType n = dist.trials(); return floor(p * (n + 1)); } template <class RealType, class Policy> inline RealType median(const binomial_distribution<RealType, Policy>& dist) { // Bounds for the median of the negative binomial distribution // VAN DE VEN R. ; WEBER N. C. ; // Univ. Sydney, school mathematics statistics, Sydney N.S.W. 2006, AUSTRALIE // Metrika (Metrika) ISSN 0026-1335 CODEN MTRKA8 // 1993, vol. 40, no3-4, pp. 185-189 (4 ref.) // Bounds for median and 50 percetage point of binomial and negative binomial distribution // Metrika, ISSN 0026-1335 (Print) 1435-926X (Online) // Volume 41, Number 1 / December, 1994, DOI 10.1007/BF01895303 BOOST_MATH_STD_USING // ADL of std functions. RealType p = dist.success_fraction(); RealType n = dist.trials(); // Wikipedia says one of floor(np) -1, floor (np), floor(np) +1 return floor(p * n); // Chose the middle value. } template <class RealType, class Policy> inline RealType skewness(const binomial_distribution<RealType, Policy>& dist) { BOOST_MATH_STD_USING // ADL of std functions. RealType p = dist.success_fraction(); RealType n = dist.trials(); return (1 - 2 * p) / sqrt(n * p * (1 - p)); } template <class RealType, class Policy> inline RealType kurtosis(const binomial_distribution<RealType, Policy>& dist) { RealType p = dist.success_fraction(); RealType n = dist.trials(); return 3 - 6 / n + 1 / (n * p * (1 - p)); } template <class RealType, class Policy> inline RealType kurtosis_excess(const binomial_distribution<RealType, Policy>& dist) { RealType p = dist.success_fraction(); RealType q = 1 - p; RealType n = dist.trials(); return (1 - 6 * p * q) / (n * p * q); } } // 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> #endif // BOOST_MATH_SPECIAL_BINOMIAL_HPP