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171 lines
7.4 KiB
Plaintext
171 lines
7.4 KiB
Plaintext
[section:inverse_gaussian_dist Inverse Gaussian (or Inverse Normal) Distribution]
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``#include <boost/math/distributions/inverse_gaussian.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class inverse_gaussian_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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inverse_gaussian_distribution(RealType mean = 1, RealType scale = 1);
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RealType mean()const; // mean default 1.
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RealType scale()const; // Optional scale, default 1 (unscaled).
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RealType shape()const; // Shape = scale/mean.
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};
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typedef inverse_gaussian_distribution<double> inverse_gaussian;
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}} // namespace boost // namespace math
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The Inverse Gaussian distribution distribution is a continuous probability distribution.
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The distribution is also called 'normal-inverse Gaussian distribution',
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and 'normal Inverse' distribution.
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It is also convenient to provide unity as default for both mean and scale.
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This is the Standard form for all distributions.
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The Inverse Gaussian distribution was first studied in relation to Brownian motion.
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In 1956 M.C.K. Tweedie used the name Inverse Gaussian because there is an inverse relationship
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between the time to cover a unit distance and distance covered in unit time.
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The inverse Gaussian is one of family of distributions that have been called the
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[@http://en.wikipedia.org/wiki/Tweedie_distributions Tweedie distributions].
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(So ['inverse] in the name may mislead: it does [*not] relate to the inverse of a distribution).
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The tails of the distribution decrease more slowly than the normal distribution.
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It is therefore suitable to model phenomena
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where numerically large values are more probable than is the case for the normal distribution.
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For stock market returns and prices, a key characteristic is that it models
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that extremely large variations from typical (crashes) can occur
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even when almost all (normal) variations are small.
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Examples are returns from financial assets and turbulent wind speeds.
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The normal-inverse Gaussian distributions form
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a subclass of the generalised hyperbolic distributions.
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See
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[@http://en.wikipedia.org/wiki/Normal-inverse_Gaussian_distribution distribution].
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[@http://mathworld.wolfram.com/InverseGaussianDistribution.html
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Weisstein, Eric W. "Inverse Gaussian Distribution." From MathWorld--A Wolfram Web Resource.]
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If you want a `double` precision inverse_gaussian distribution you can use
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``boost::math::inverse_gaussian_distribution<>``
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or, more conveniently, you can write
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using boost::math::inverse_gaussian;
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inverse_gaussian my_ig(2, 3);
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For mean parameters [mu] and scale (also called precision) parameter [lambda],
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and random variate x,
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the inverse_gaussian distribution is defined by the probability density function (PDF):
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__spaces f(x;[mu], [lambda]) = [sqrt]([lambda]/2[pi]x[super 3]) e[super -[lambda](x-[mu])[sup2]/2[mu][sup2]x]
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and Cumulative Density Function (CDF):
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__spaces F(x;[mu], [lambda]) = [Phi]{[sqrt]([lambda]/x) (x/[mu]-1)} + e[super 2[mu]/[lambda]] [Phi]{-[sqrt]([lambda]/[mu]) (1+x/[mu])}
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where [Phi] is the standard normal distribution CDF.
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The following graphs illustrate how the PDF and CDF of the inverse_gaussian distribution
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varies for a few values of parameters [mu] and [lambda]:
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[graph inverse_gaussian_pdf] [/.png or .svg]
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[graph inverse_gaussian_cdf]
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Tweedie also provided 3 other parameterisations where ([mu] and [lambda])
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are replaced by their ratio [phi] = [lambda]/[mu] and by 1/[mu]:
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these forms may be more suitable for Bayesian applications.
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These can be found on Seshadri, page 2 and are also discussed by Chhikara and Folks on page 105.
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Another related parameterisation, the __wald_distrib (where mean [mu] is unity) is also provided.
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[h4 Member Functions]
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inverse_gaussian_distribution(RealType df = 1, RealType scale = 1); // optionally scaled.
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Constructs an inverse_gaussian distribution with [mu] mean,
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and scale [lambda], with both default values 1.
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Requires that both the mean [mu] parameter and scale [lambda] are greater than zero,
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otherwise calls __domain_error.
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RealType mean()const;
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Returns the mean [mu] parameter of this distribution.
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RealType scale()const;
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Returns the scale [lambda] parameter of this distribution.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
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distributions are supported: __usual_accessors.
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The domain of the random variate is \[0,+[infin]).
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[note Unlike some definitions, this implementation supports a random variate
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equal to zero as a special case, returning zero for both pdf and cdf.]
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[h4 Accuracy]
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The inverse_gaussian distribution is implemented in terms of the
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exponential function and standard normal distribution ['N]0,1 [Phi] :
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refer to the accuracy data for those functions for more information.
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But in general, gamma (and thus inverse gamma) results are often accurate to a few epsilon,
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>14 decimal digits accuracy for 64-bit double.
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[h4 Implementation]
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In the following table [mu] is the mean parameter and
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[lambda] is the scale parameter of the inverse_gaussian distribution,
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/x/ is the random variate, /p/ is the probability and /q = 1-p/ its complement.
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Parameters [mu] for shape and [lambda] for scale
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are used for the inverse gaussian function.
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[table
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[[Function] [Implementation Notes] ]
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[[pdf] [ [sqrt]([lambda]/ 2[pi]x[super 3]) e[super -[lambda](x - [mu])[sup2]/ 2[mu][sup2]x]]]
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[[cdf][ [Phi]{[sqrt]([lambda]/x) (x/[mu]-1)} + e[super 2[mu]/[lambda]] [Phi]{-[sqrt]([lambda]/[mu]) (1+x/[mu])} ]]
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[[cdf complement] [using complement of [Phi] above.] ]
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[[quantile][No closed form known. Estimated using a guess refined by Newton-Raphson iteration.]]
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[[quantile from the complement][No closed form known. Estimated using a guess refined by Newton-Raphson iteration.]]
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[[mode][[mu] {[sqrt](1+9[mu][sup2]/4[lambda][sup2])[sup2] - 3[mu]/2[lambda]} ]]
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[[median][No closed form analytic equation is known, but is evaluated as quantile(0.5)]]
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[[mean][[mu]] ]
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[[variance][[mu][cubed]/[lambda]] ]
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[[skewness][3 [sqrt] ([mu]/[lambda])] ]
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[[kurtosis_excess][15[mu]/[lambda]] ]
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[[kurtosis][12[mu]/[lambda]] ]
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] [/table]
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[h4 References]
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#Wald, A. (1947). Sequential analysis. Wiley, NY.
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#The Inverse Gaussian distribution : theory, methodology, and applications, Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989).
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#The Inverse Gaussian distribution : statistical theory and applications, Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998).
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#[@http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.wald.html Numpy and Scipy Documentation].
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#[@http://bm2.genes.nig.ac.jp/RGM2/R_current/library/statmod/man/invgauss.html R statmod invgauss functions].
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#[@http://cran.r-project.org/web/packages/SuppDists/index.html R SuppDists invGauss functions].
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(Note that these R implementations names differ in case).
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#[@http://www.statsci.org/s/invgauss.html StatSci.org invgauss help].
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#[@http://www.statsci.org/s/invgauss.statSci.org invgauss R source].
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#[@http://www.biostat.wustl.edu/archives/html/s-news/2001-12/msg00144.html pwald, qwald].
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#[@http://www.brighton-webs.co.uk/distributions/wald.asp Brighton Webs wald].
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[endsect] [/section:inverse_gaussian_dist Inverse Gaussiann Distribution]
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[/
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Copyright 2010 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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] |