FM / distributions

/usr/include/boost/math/distributions

Name	Size	Mode	Actions
detail/	-	0755	rm
arcsine.hpp	18797	0644	edit dl rm
bernoulli.hpp	12143	0644	edit dl rm
beta.hpp	18807	0644	edit dl rm
binomial.hpp	28654	0644	edit dl rm
cauchy.hpp	12310	0644	edit dl rm
chi_squared.hpp	12242	0644	edit dl rm
complement.hpp	5978	0644	edit dl rm
empirical_cumulative_distribution_function.hpp	1886	0644	edit dl rm
exponential.hpp	8944	0644	edit dl rm
extreme_value.hpp	10491	0644	edit dl rm
find_location.hpp	6713	0644	edit dl rm
find_scale.hpp	9674	0644	edit dl rm
fisher_f.hpp	14391	0644	edit dl rm
fwd.hpp	5461	0644	edit dl rm
gamma.hpp	10660	0644	edit dl rm
geometric.hpp	21156	0644	edit dl rm
hyperexponential.hpp	21887	0644	edit dl rm
hypergeometric.hpp	12220	0644	edit dl rm
inverse_chi_squared.hpp	15202	0644	edit dl rm
inverse_gamma.hpp	15389	0644	edit dl rm
inverse_gaussian.hpp	19760	0644	edit dl rm
laplace.hpp	10580	0644	edit dl rm
logistic.hpp	11273	0644	edit dl rm
lognormal.hpp	11714	0644	edit dl rm
negative_binomial.hpp	26139	0644	edit dl rm
non_central_beta.hpp	35837	0644	edit dl rm
non_central_chi_squared.hpp	40704	0644	edit dl rm
non_central_f.hpp	15848	0644	edit dl rm
non_central_t.hpp	47973	0644	edit dl rm
normal.hpp	10768	0644	edit dl rm
pareto.hpp	15516	0644	edit dl rm
poisson.hpp	20267	0644	edit dl rm
rayleigh.hpp	10344	0644	edit dl rm
skew_normal.hpp	25646	0644	edit dl rm
students_t.hpp	18476	0644	edit dl rm
triangular.hpp	18407	0644	edit dl rm
uniform.hpp	12963	0644	edit dl rm
weibull.hpp	12081	0644	edit dl rm

Edit: /usr/include/boost/math/distributions/poisson.hpp (20267B)

// boost\math\distributions\poisson.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)

// Poisson distribution is a discrete probability distribution.
// It expresses the probability of a number (k) of
// events, occurrences, failures or arrivals occurring in a fixed time,
// assuming these events occur with a known average or mean rate (lambda)
// and are independent of the time since the last event.
// The distribution was discovered by Simeon-Denis Poisson (1781-1840).

// Parameter lambda is the mean number of events in the given time interval.
// The random variate k is the number of events, occurrences or arrivals.
// k argument may be integral, signed, or unsigned, or floating point.
// If necessary, it has already been promoted from an integral type.

// Note that the Poisson distribution
// (like others including the binomial, negative binomial & Bernoulli)
// is strictly defined as a discrete function:
// only integral values of k are envisaged.
// However because the method of calculation uses a continuous gamma function,
// it is convenient to treat it as if a continuous 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.

// See http://en.wikipedia.org/wiki/Poisson_distribution
// http://documents.wolfram.com/v5/Add-onsLinks/StandardPackages/Statistics/DiscreteDistributions.html

#ifndef BOOST_MATH_SPECIAL_POISSON_HPP
#define BOOST_MATH_SPECIAL_POISSON_HPP

#include <boost/math/distributions/fwd.hpp>
#include <boost/math/special_functions/gamma.hpp> // for incomplete gamma. gamma_q
#include <boost/math/special_functions/trunc.hpp> // for incomplete gamma. gamma_q
#include <boost/math/distributions/complement.hpp> // complements
#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks
#include <boost/math/special_functions/fpclassify.hpp> // isnan.
#include <boost/math/special_functions/factorials.hpp> // factorials.
#include <boost/math/tools/roots.hpp> // for root finding.
#include <boost/math/distributions/detail/inv_discrete_quantile.hpp>

#include <utility>

namespace boost
{
  namespace math
  {
    namespace poisson_detail
    {
      // Common error checking routines for Poisson distribution functions.
      // These are convoluted, & apparently redundant, to try to ensure that
      // checks are always performed, even if exceptions are not enabled.

template <class RealType, class Policy>
      inline bool check_mean(const char* function, const RealType& mean, RealType* result, const Policy& pol)
      {
        if(!(boost::math::isfinite)(mean) || (mean < 0))
        {
          *result = policies::raise_domain_error<RealType>(
            function,
            "Mean argument is %1%, but must be >= 0 !", mean, pol);
          return false;
        }
        return true;
      } // bool check_mean

template <class RealType, class Policy>
      inline bool check_mean_NZ(const char* function, const RealType& mean, RealType* result, const Policy& pol)
      { // mean == 0 is considered an error.
        if( !(boost::math::isfinite)(mean) || (mean <= 0))
        {
          *result = policies::raise_domain_error<RealType>(
            function,
            "Mean argument is %1%, but must be > 0 !", mean, pol);
          return false;
        }
        return true;
      } // bool check_mean_NZ

template <class RealType, class Policy>
      inline bool check_dist(const char* function, const RealType& mean, RealType* result, const Policy& pol)
      { // Only one check, so this is redundant really but should be optimized away.
        return check_mean_NZ(function, mean, result, pol);
      } // bool check_dist

template <class RealType, class Policy>
      inline bool check_k(const char* function, const RealType& k, RealType* result, const Policy& pol)
      {
        if((k < 0) || !(boost::math::isfinite)(k))
        {
          *result = policies::raise_domain_error<RealType>(
            function,
            "Number of events k argument is %1%, but must be >= 0 !", k, pol);
          return false;
        }
        return true;
      } // bool check_k

template <class RealType, class Policy>
      inline bool check_dist_and_k(const char* function, RealType mean, RealType k, RealType* result, const Policy& pol)
      {
        if((check_dist(function, mean, result, pol) == false) ||
          (check_k(function, k, result, pol) == false))
        {
          return false;
        }
        return true;
      } // bool check_dist_and_k

template <class RealType, class Policy>
      inline bool check_prob(const char* function, const RealType& p, RealType* result, const Policy& pol)
      { // Check 0 <= p <= 1
        if(!(boost::math::isfinite)(p) || (p < 0) || (p > 1))
        {
          *result = policies::raise_domain_error<RealType>(
            function,
            "Probability argument is %1%, but must be >= 0 and <= 1 !", p, pol);
          return false;
        }
        return true;
      } // bool check_prob

template <class RealType, class Policy>
      inline bool check_dist_and_prob(const char* function, RealType mean,  RealType p, RealType* result, const Policy& pol)
      {
        if((check_dist(function, mean, result, pol) == false) ||
          (check_prob(function, p, result, pol) == false))
        {
          return false;
        }
        return true;
      } // bool check_dist_and_prob

} // namespace poisson_detail

template <class RealType = double, class Policy = policies::policy<> >
    class poisson_distribution
    {
    public:
      typedef RealType value_type;
      typedef Policy policy_type;

poisson_distribution(RealType l_mean = 1) : m_l(l_mean) // mean (lambda).
      { // Expected mean number of events that occur during the given interval.
        RealType r;
        poisson_detail::check_dist(
           "boost::math::poisson_distribution<%1%>::poisson_distribution",
          m_l,
          &r, Policy());
      } // poisson_distribution constructor.

RealType mean() const
      { // Private data getter function.
        return m_l;
      }
    private:
      // Data member, initialized by constructor.
      RealType m_l; // mean number of occurrences.
    }; // template <class RealType, class Policy> class poisson_distribution

typedef poisson_distribution<double> poisson; // Reserved name of type double.

// Non-member functions to give properties of the distribution.

template <class RealType, class Policy>
    inline const std::pair<RealType, RealType> range(const poisson_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), max_value<RealType>()); // Max integer?
    }

template <class RealType, class Policy>
    inline const std::pair<RealType, RealType> support(const poisson_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.
       using boost::math::tools::max_value;
       return std::pair<RealType, RealType>(static_cast<RealType>(0),  max_value<RealType>());
    }

template <class RealType, class Policy>
    inline RealType mean(const poisson_distribution<RealType, Policy>& dist)
    { // Mean of poisson distribution = lambda.
      return dist.mean();
    } // mean

template <class RealType, class Policy>
    inline RealType mode(const poisson_distribution<RealType, Policy>& dist)
    { // mode.
      BOOST_MATH_STD_USING // ADL of std functions.
      return floor(dist.mean());
    }

//template <class RealType, class Policy>
    //inline RealType median(const poisson_distribution<RealType, Policy>& dist)
    //{ // median = approximately lambda + 1/3 - 0.2/lambda
    //  RealType l = dist.mean();
    //  return dist.mean() + static_cast<RealType>(0.3333333333333333333333333333333333333333333333)
    //   - static_cast<RealType>(0.2) / l;
    //} // BUT this formula appears to be out-by-one compared to quantile(half)
    // Query posted on Wikipedia.
    // Now implemented via quantile(half) in derived accessors.

template <class RealType, class Policy>
    inline RealType variance(const poisson_distribution<RealType, Policy>& dist)
    { // variance.
      return dist.mean();
    }

// RealType standard_deviation(const poisson_distribution<RealType, Policy>& dist)
    // standard_deviation provided by derived accessors.

template <class RealType, class Policy>
    inline RealType skewness(const poisson_distribution<RealType, Policy>& dist)
    { // skewness = sqrt(l).
      BOOST_MATH_STD_USING // ADL of std functions.
      return 1 / sqrt(dist.mean());
    }

template <class RealType, class Policy>
    inline RealType kurtosis_excess(const poisson_distribution<RealType, Policy>& dist)
    { // skewness = sqrt(l).
      return 1 / dist.mean(); // kurtosis_excess 1/mean from Wiki & MathWorld eq 31.
      // http://mathworld.wolfram.com/Kurtosis.html explains that the kurtosis excess
      // is more convenient because the kurtosis excess of a normal distribution is zero
      // whereas the true kurtosis is 3.
    } // RealType kurtosis_excess

template <class RealType, class Policy>
    inline RealType kurtosis(const poisson_distribution<RealType, Policy>& dist)
    { // kurtosis is 4th moment about the mean = u4 / sd ^ 4
      // http://en.wikipedia.org/wiki/Kurtosis
      // kurtosis can range from -2 (flat top) to +infinity (sharp peak & heavy tails).
      // http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
      return 3 + 1 / dist.mean(); // NIST.
      // http://mathworld.wolfram.com/Kurtosis.html explains that the kurtosis excess
      // is more convenient because the kurtosis excess of a normal distribution is zero
      // whereas the true kurtosis is 3.
    } // RealType kurtosis

template <class RealType, class Policy>
    RealType pdf(const poisson_distribution<RealType, Policy>& dist, const RealType& k)
    { // Probability Density/Mass Function.
      // Probability that there are EXACTLY k occurrences (or arrivals).
      BOOST_FPU_EXCEPTION_GUARD

BOOST_MATH_STD_USING // for ADL of std functions.

RealType mean = dist.mean();
      // Error check:
      RealType result = 0;
      if(false == poisson_detail::check_dist_and_k(
        "boost::math::pdf(const poisson_distribution<%1%>&, %1%)",
        mean,
        k,
        &result, Policy()))
      {
        return result;
      }

// Special case of mean zero, regardless of the number of events k.
      if (mean == 0)
      { // Probability for any k is zero.
        return 0;
      }
      if (k == 0)
      { // mean ^ k = 1, and k! = 1, so can simplify.
        return exp(-mean);
      }
      return boost::math::gamma_p_derivative(k+1, mean, Policy());
    } // pdf

template <class RealType, class Policy>
    RealType cdf(const poisson_distribution<RealType, Policy>& dist, const RealType& k)
    { // Cumulative Distribution Function Poisson.
      // The random variate k is the number of occurrences(or arrivals)
      // 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 Poisson Probability Density or Mass (pdf).

// But note that the Poisson distribution
      // (like others including the binomial, 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 it is a continuous function
      // and permit non-integral values of k.
      // To enforce the strict mathematical model, users should use floor or ceil functions
      // outside this function to ensure that k is integral.

// The terms are not summed directly (at least for larger k)
      // instead the incomplete gamma integral is employed,

BOOST_MATH_STD_USING // for ADL of std function exp.

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 cases:
      if (mean == 0)
      { // Probability for any k is zero.
        return 0;
      }
      if (k == 0)
      { // return pdf(dist, static_cast<RealType>(0));
        // but mean (and k) have already been checked,
        // so this avoids unnecessary repeated checks.
       return exp(-mean);
      }
      // For small integral k could use a finite sum -
      // it's cheaper than the gamma function.
      // BUT this is now done efficiently by gamma_q function.
      // Calculate poisson cdf using the gamma_q function.
      return gamma_q(k+1, mean, Policy());
    } // binomial cdf

template <class RealType, class Policy>
    RealType cdf(const complemented2_type<poisson_distribution<RealType, Policy>, RealType>& c)
    { // Complemented Cumulative Distribution Function Poisson
      // The random variate k is the number of events, occurrences or arrivals.
      // k argument may be integral, signed, or unsigned, or floating point.
      // If necessary, it has already been promoted from an integral type.
      // But note that the Poisson distribution
      // (like others including the binomial, 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 is it is a continuous function
      // and permit non-integral values of k.
      // To enforce the strict mathematical model, users should use floor or ceil functions
      // outside this function to ensure that k is integral.

// Returns the sum of the terms k+1 through inf of the Poisson Probability Density/Mass (pdf).
      // 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

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