| // 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| // ig package - withdrawn but at http://cran.r-project.org/src/contrib/Archive/ig/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 mean = 1, RealType scale = 1)
|
| : m_mean(mean), m_scale(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, scale, &result, Policy());
|
| detail::check_location(function, 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();
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| RealType result;
|
| 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_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;
|
| 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_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>
|
| struct inverse_gaussian_quantile_functor
|
| {
|
|
|
| inverse_gaussian_quantile_functor(const boost::math::inverse_gaussian_distribution<RealType> 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> distribution;
|
| RealType prob;
|
| };
|
|
|
| template <class RealType>
|
| struct inverse_gaussian_quantile_complement_functor
|
| {
|
| inverse_gaussian_quantile_complement_functor(const boost::math::inverse_gaussian_distribution<RealType> 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> 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_distribution<RealType, no_overthrow_policy> 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;
|
| 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_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>(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;
|
| 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(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;
|
| if(false == detail::check_scale(function, scale, &result, Policy()))
|
| return result;
|
| if(false == detail::check_location(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>(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
|
|
|
|
|