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