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/inverse_gaussian.hpp (19760B)

//  Copyright John Maddock 2010.
//  Copyright Paul A. Bristow 2010.

//  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)

#ifndef BOOST_STATS_INVERSE_GAUSSIAN_HPP
#define BOOST_STATS_INVERSE_GAUSSIAN_HPP

#ifdef _MSC_VER
#pragma warning(disable: 4512) // assignment operator could not be generated
#endif

// http://en.wikipedia.org/wiki/Normal-inverse_Gaussian_distribution
// http://mathworld.wolfram.com/InverseGaussianDistribution.html

// The normal-inverse Gaussian distribution
// also called the Wald distribution (some sources limit this to when mean = 1).

// It is the continuous probability distribution
// that is defined as the normal variance-mean mixture where the mixing density is the 
// inverse Gaussian distribution. The tails of the distribution decrease more slowly
// than the normal distribution. It is therefore suitable to model phenomena
// where numerically large values are more probable than is the case for the normal distribution.

// The Inverse Gaussian distribution was first studied in relationship to Brownian motion.
// In 1956 M.C.K. Tweedie used the name 'Inverse Gaussian' because there is an inverse 
// relationship between the time to cover a unit distance and distance covered in unit time.

// Examples are returns from financial assets and turbulent wind speeds. 
// The normal-inverse Gaussian distributions form
// a subclass of the generalised hyperbolic distributions.

// See also

// http://en.wikipedia.org/wiki/Normal_distribution
// http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
// Also:
// Weisstein, Eric W. "Normal Distribution."
// From MathWorld--A Wolfram Web Resource.
// http://mathworld.wolfram.com/NormalDistribution.html

// http://www.jstatsoft.org/v26/i04/paper General class of inverse Gaussian distributions.
// ig package - withdrawn but at http://cran.r-project.org/src/contrib/Archive/ig/

// http://www.stat.ucl.ac.be/ISdidactique/Rhelp/library/SuppDists/html/inverse_gaussian.html
// R package for dinverse_gaussian, ...

// http://www.statsci.org/s/inverse_gaussian.s  and http://www.statsci.org/s/inverse_gaussian.html

//#include <boost/math/distributions/fwd.hpp>
#include <boost/math/special_functions/erf.hpp> // for erf/erfc.
#include <boost/math/distributions/complement.hpp>
#include <boost/math/distributions/detail/common_error_handling.hpp>
#include <boost/math/distributions/normal.hpp>
#include <boost/math/distributions/gamma.hpp> // for gamma function
// using boost::math::gamma_p;

#include <boost/math/tools/tuple.hpp>
//using std::tr1::tuple;
//using std::tr1::make_tuple;
#include <boost/math/tools/roots.hpp>
//using boost::math::tools::newton_raphson_iterate;

#include <utility>

namespace boost{ namespace math{

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

inverse_gaussian_distribution(RealType l_mean = 1, RealType l_scale = 1)
      : m_mean(l_mean), m_scale(l_scale)
   { // Default is a 1,1 inverse_gaussian distribution.
     static const char* function = "boost::math::inverse_gaussian_distribution<%1%>::inverse_gaussian_distribution";

RealType result;
     detail::check_scale(function, l_scale, &result, Policy());
     detail::check_location(function, l_mean, &result, Policy());
     detail::check_x_gt0(function, l_mean, &result, Policy());
   }

RealType mean()const
   { // alias for location.
      return m_mean; // aka mu
   }

// Synonyms, provided to allow generic use of find_location and find_scale.
   RealType location()const
   { // location, aka mu.
      return m_mean;
   }
   RealType scale()const
   { // scale, aka lambda.
      return m_scale;
   }

RealType shape()const
   { // shape, aka phi = lambda/mu.
      return m_scale / m_mean;
   }

private:
   //
   // Data members:
   //
   RealType m_mean;  // distribution mean or location, aka mu.
   RealType m_scale;    // distribution standard deviation or scale, aka lambda.
}; // class normal_distribution

typedef inverse_gaussian_distribution<double> inverse_gaussian;

template <class RealType, class Policy>
inline const std::pair<RealType, RealType> range(const inverse_gaussian_distribution<RealType, Policy>& /*dist*/)
{ // Range of permissible values for random variable x, zero to max.
   using boost::math::tools::max_value;
   return std::pair<RealType, RealType>(static_cast<RealType>(0.), max_value<RealType>()); // - to + max value.
}

template <class RealType, class Policy>
inline const std::pair<RealType, RealType> support(const inverse_gaussian_distribution<RealType, Policy>& /*dist*/)
{ // Range of supported values for random variable x, zero to max.
  // 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>()); // - to + max value.
}

template <class RealType, class Policy>
inline RealType pdf(const inverse_gaussian_distribution<RealType, Policy>& dist, const RealType& x)
{ // Probability Density Function
   BOOST_MATH_STD_USING  // for ADL of std functions

RealType scale = dist.scale();
   RealType mean = dist.mean();
   RealType result = 0;
   static const char* function = "boost::math::pdf(const inverse_gaussian_distribution<%1%>&, %1%)";
   if(false == detail::check_scale(function, scale, &result, Policy()))
   {
      return result;
   }
   if(false == detail::check_location(function, mean, &result, Policy()))
   {
      return result;
   }
   if(false == detail::check_x_gt0(function, mean, &result, Policy()))
   {
      return result;
   }
   if(false == detail::check_positive_x(function, x, &result, Policy()))
   {
      return result;
   }

if (x == 0)
   {
     return 0; // Convenient, even if not defined mathematically.
   }

result =
     sqrt(scale / (constants::two_pi<RealType>() * x * x * x))
    * exp(-scale * (x - mean) * (x - mean) / (2 * x * mean * mean));
   return result;
} // pdf

template <class RealType, class Policy>
inline RealType cdf(const inverse_gaussian_distribution<RealType, Policy>& dist, const RealType& x)
{ // Cumulative Density Function.
   BOOST_MATH_STD_USING  // for ADL of std functions.

RealType scale = dist.scale();
   RealType mean = dist.mean();
   static const char* function = "boost::math::cdf(const inverse_gaussian_distribution<%1%>&, %1%)";
   RealType result = 0;
   if(false == detail::check_scale(function, scale, &result, Policy()))
   {
      return result;
   }
   if(false == detail::check_location(function, mean, &result, Policy()))
   {
      return result;
   }
   if (false == detail::check_x_gt0(function, mean, &result, Policy()))
   {
      return result;
   }
   if(false == detail::check_positive_x(function, x, &result, Policy()))
   {
     return result;
   }
   if (x == 0)
   {
     return 0; // Convenient, even if not defined mathematically.
   }
   // Problem with this formula for large scale > 1000 or small x, 
   //result = 0.5 * (erf(sqrt(scale / x) * ((x / mean) - 1) / constants::root_two<RealType>(), Policy()) + 1)
   //  + exp(2 * scale / mean) / 2 
   //  * (1 - erf(sqrt(scale / x) * (x / mean + 1) / constants::root_two<RealType>(), Policy()));
   // so use normal distribution version:
   // Wikipedia CDF equation http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution.

normal_distribution<RealType> n01;

RealType n0 = sqrt(scale / x);
   n0 *= ((x / mean) -1);
   RealType n1 = cdf(n01, n0);
   RealType expfactor = exp(2 * scale / mean);
   RealType n3 = - sqrt(scale / x);
   n3 *= (x / mean) + 1;
   RealType n4 = cdf(n01, n3);
   result = n1 + expfactor * n4;
   return result;
} // cdf

template <class RealType, class Policy>
struct inverse_gaussian_quantile_functor
{

inverse_gaussian_quantile_functor(const boost::math::inverse_gaussian_distribution<RealType, Policy> dist, RealType const& p)
    : distribution(dist), prob(p)
  {
  }
  boost::math::tuple<RealType, RealType> operator()(RealType const& x)
  {
    RealType c = cdf(distribution, x);
    RealType fx = c - prob;  // Difference cdf - value - to minimize.
    RealType dx = pdf(distribution, x); // pdf is 1st derivative.
    // return both function evaluation difference f(x) and 1st derivative f'(x).
    return boost::math::make_tuple(fx, dx);
  }
  private:
  const boost::math::inverse_gaussian_distribution<RealType, Policy> distribution;
  RealType prob; 
};

template <class RealType, class Policy>
struct inverse_gaussian_quantile_complement_functor
{ 
    inverse_gaussian_quantile_complement_functor(const boost::math::inverse_gaussian_distribution<RealType, Policy> dist, RealType const& p)
    : distribution(dist), prob(p)
  {
  }
  boost::math::tuple<RealType, RealType> operator()(RealType const& x)
  {
    RealType c = cdf(complement(distribution, x));
    RealType fx = c - prob;  // Difference cdf - value - to minimize.
    RealType dx = -pdf(distribution, x); // pdf is 1st derivative.
    // return both function evaluation difference f(x) and 1st derivative f'(x).
    //return std::tr1::make_tuple(fx, dx); if available.
    return boost::math::make_tuple(fx, dx);
  }
  private:
  const boost::math::inverse_gaussian_distribution<RealType, Policy> distribution;
  RealType prob; 
};

namespace detail
{
  template <class RealType>
  inline RealType guess_ig(RealType p, RealType mu = 1, RealType lambda = 1)
  { // guess at random variate value x for inverse gaussian quantile.
      BOOST_MATH_STD_USING
      using boost::math::policies::policy;
      // Error type.
      using boost::math::policies::overflow_error;
      // Action.
      using boost::math::policies::ignore_error;

typedef policy<
        overflow_error<ignore_error> // Ignore overflow (return infinity)
      > no_overthrow_policy;

RealType x; // result is guess at random variate value x.
    RealType phi = lambda / mu;
    if (phi > 2.)
    { // Big phi, so starting to look like normal Gaussian distribution.
      //    x=(qnorm(p,0,1,true,false) - 0.5 * sqrt(mu/lambda)) / sqrt(lambda/mu);
      // Whitmore, G.A. and Yalovsky, M.
      // A normalising logarithmic transformation for inverse Gaussian random variables,
      // Technometrics 20-2, 207-208 (1978), but using expression from
      // V Seshadri, Inverse Gaussian distribution (1998) ISBN 0387 98618 9, page 6.
 
      normal_distribution<RealType, no_overthrow_policy> n01;
      x = mu * exp(quantile(n01, p) / sqrt(phi) - 1/(2 * phi));
     }
    else
    { // phi < 2 so much less symmetrical with long tail,
      // so use gamma distribution as an approximation.
      using boost::math::gamma_distribution;

// Define the distribution, using gamma_nooverflow:
      typedef gamma_distribution<RealType, no_overthrow_policy> gamma_nooverflow;

gamma_nooverflow g(static_cast<RealType>(0.5), static_cast<RealType>(1.));

// gamma_nooverflow g(static_cast<RealType>(0.5), static_cast<RealType>(1.));
      // R qgamma(0.2, 0.5, 1)  0.0320923
      RealType qg = quantile(complement(g, p));
      //RealType qg1 = qgamma(1.- p, 0.5, 1.0, true, false);
      x = lambda / (qg * 2);
      // 
      if (x > mu/2) // x > mu /2?
      { // x too large for the gamma approximation to work well.
        //x = qgamma(p, 0.5, 1.0); // qgamma(0.270614, 0.5, 1) = 0.05983807
        RealType q = quantile(g, p);
       // x = mu * exp(q * static_cast<RealType>(0.1));  // Said to improve at high p
       // x = mu * x;  // Improves at high p?
        x = mu * exp(q / sqrt(phi) - 1/(2 * phi));
      }
    }
    return x;
  }  // guess_ig
} // namespace detail

template <class RealType, class Policy>
inline RealType quantile(const inverse_gaussian_distribution<RealType, Policy>& dist, const RealType& p)
{
   BOOST_MATH_STD_USING  // for ADL of std functions.
   // No closed form exists so guess and use Newton Raphson iteration.

RealType mean = dist.mean();
   RealType scale = dist.scale();
   static const char* function = "boost::math::quantile(const inverse_gaussian_distribution<%1%>&, %1%)";

RealType result = 0;
   if(false == detail::check_scale(function, scale, &result, Policy()))
      return result;
   if(false == detail::check_location(function, mean, &result, Policy()))
      return result;
   if (false == detail::check_x_gt0(function, mean, &result, Policy()))
      return result;
   if(false == detail::check_probability(function, p, &result, Policy()))
      return result;
   if (p == 0)
   {
     return 0; // Convenient, even if not defined mathematically?
   }
   if (p == 1)
   { // overflow 
      result = policies::raise_overflow_error<RealType>(function,
        "probability parameter is 1, but must be < 1!", Policy());
      return result; // std::numeric_limits<RealType>::infinity();
   }

RealType guess = detail::guess_ig(p, dist.mean(), dist.scale());
  using boost::math::tools::max_value;

RealType min = 0.; // Minimum possible value is bottom of range of distribution.
  RealType max = max_value<RealType>();// Maximum possible value is top of range. 
  // int digits = std::numeric_limits<RealType>::digits; // Maximum possible binary digits accuracy for type T.
  // digits used to control how accurate to try to make the result.
  // To allow user to control accuracy versus speed,
  int get_digits = policies::digits<RealType, Policy>();// get digits from policy, 
  boost::uintmax_t m = policies::get_max_root_iterations<Policy>(); // and max iterations.
  using boost::math::tools::newton_raphson_iterate;
  result =
    newton_raphson_iterate(inverse_gaussian_quantile_functor<RealType, Policy>(dist, p), guess, min, max, get_digits, m);
   return result;
} // quantile

template <class RealType, class Policy>
inline RealType cdf(const complemented2_type<inverse_gaussian_distribution<RealType, Policy>, RealType>& c)
{
   BOOST_MATH_STD_USING  // for ADL of std functions.

RealType scale = c.dist.scale();
   RealType mean = c.dist.mean();
   RealType x = c.param;
   static const char* function = "boost::math::cdf(const complement(inverse_gaussian_distribution<%1%>&), %1%)";
   // infinite arguments not supported.
   //if((boost::math::isinf)(x))
   //{
   //  if(x < 0) return 1; // cdf complement -infinity is unity.
   //  return 0; // cdf complement +infinity is zero
   //}
   // These produce MSVC 4127 warnings, so the above used instead.
   //if(std::numeric_limits<RealType>::has_infinity && x == std::numeric_limits<RealType>::infinity())
   //{ // cdf complement +infinity is zero.
   //  return 0;
   //}
   //if(std::numeric_limits<RealType>::has_infinity && x == -std::numeric_limits<RealType>::infinity())
   //{ // cdf complement -infinity is unity.
   //  return 1;
   //}
   RealType result = 0;
   if(false == detail::check_scale(function, scale, &result, Policy()))
      return result;
   if(false == detail::check_location(function, mean, &result, Policy()))
      return result;
   if (false == detail::check_x_gt0(function, mean, &result, Policy()))
      return result;
   if(false == detail::check_positive_x(function, x, &result, Policy()))
      return result;

normal_distribution<RealType> n01;
   RealType n0 = sqrt(scale / x);
   n0 *= ((x / mean) -1);
   RealType cdf_1 = cdf(complement(n01, n0));

RealType expfactor = exp(2 * scale / mean);
   RealType n3 = - sqrt(scale / x);
   n3 *= (x / mean) + 1;

//RealType n5 = +sqrt(scale/x) * ((x /mean) + 1); // note now positive sign.
   RealType n6 = cdf(complement(n01, +sqrt(scale/x) * ((x /mean) + 1)));
   // RealType n4 = cdf(n01, n3); // = 
   result = cdf_1 - expfactor * n6; 
   return result;
} // cdf complement

template <class RealType, class Policy>
inline RealType quantile(const complemented2_type<inverse_gaussian_distribution<RealType, Policy>, RealType>& c)
{
   BOOST_MATH_STD_USING  // for ADL of std functions

RealType scale = c.dist.scale();
   RealType mean = c.dist.mean();
   static const char* function = "boost::math::quantile(const complement(inverse_gaussian_distribution<%1%>&), %1%)";
   RealType result = 0;
   if(false == detail::check_scale(function, scale, &result, Policy()))
      return result;
   if(false == detail::check_location(function, mean, &result, Policy()))
      return result;
   if (false == detail::check_x_gt0(function, mean, &result, Policy()))
      return result;
   RealType q = c.param;
   if(false == detail::check_probability(function, q, &result, Policy()))
      return result;

RealType guess = detail::guess_ig(q, mean, scale);
   // Complement.
   using boost::math::tools::max_value;

RealType min = 0.; // Minimum possible value is bottom of range of distribution.
  RealType max = max_value<RealType>();// Maximum possible value is top of range. 
  // int digits = std::numeric_limits<RealType>::digits; // Maximum possible binary digits accuracy for type T.
  // digits used to control how accurate to try to make the result.
  int get_digits = policies::digits<RealType, Policy>();
  boost::uintmax_t m = policies::get_max_root_iterations<Policy>();
  using boost::math::tools::newton_raphson_iterate;
  result =
    newton_raphson_iterate(inverse_gaussian_quantile_complement_functor<RealType, Policy>(c.dist, q), guess, min, max, get_digits, m);
   return result;
} // quantile

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

template <class RealType, class Policy>
inline RealType scale(const inverse_gaussian_distribution<RealType, Policy>& dist)
{ // aka lambda
   return dist.scale();
}

template <class RealType, class Policy>
inline RealType shape(const inverse_gaussian_distribution<RealType, Policy>& dist)
{ // aka phi
   return dist.shape();
}

template <class RealType, class Policy>
inline RealType standard_deviation(const inverse_gaussian_distribution<RealType, Policy>& dist)
{
  BOOST_MATH_STD_USING
  RealType scale = dist.scale();
  RealType mean = dist.mean();
  RealType result = sqrt(mean * mean * mean / scale);
  return result;
}

template <class RealType, class Policy>
inline RealType mode(const inverse_gaussian_distribution<RealType, Policy>& dist)
{
  BOOST_MATH_STD_USING
  RealType scale = dist.scale();
  RealType  mean = dist.mean();
  RealType result = mean * (sqrt(1 + (9 * mean * mean)/(4 * scale * scale)) 
      - 3 * mean / (2 * scale));
  return result;
}

template <class RealType, class Policy>
inline RealType skewness(const inverse_gaussian_distribution<RealType, Policy>& dist)
{
  BOOST_MATH_STD_USING
  RealType scale = dist.scale();
  RealType  mean = dist.mean();
  RealType result = 3 * sqrt(mean/scale);
  return result;
}

template <class RealType, class Policy>
inline RealType kurtosis(const inverse_gaussian_distribution<RealType, Policy>& dist)
{
  RealType scale = dist.scale();
  RealType  mean = dist.mean();
  RealType result = 15 * mean / scale -3;
  return result;
}

template <class RealType, class Policy>
inline RealType kurtosis_excess(const inverse_gaussian_distribution<RealType, Policy>& dist)
{
  RealType scale = dist.scale();
  RealType  mean = dist.mean();
  RealType result = 15 * mean / scale;
  return result;
}

} // 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_STATS_INVERSE_GAUSSIAN_HPP

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