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- [section:extreme_dist Extreme Value Distribution]
- ``#include <boost/math/distributions/extreme.hpp>``
- template <class RealType = double,
- class ``__Policy`` = ``__policy_class`` >
- class extreme_value_distribution;
- typedef extreme_value_distribution<> extreme_value;
- template <class RealType, class ``__Policy``>
- class extreme_value_distribution
- {
- public:
- typedef RealType value_type;
- extreme_value_distribution(RealType location = 0, RealType scale = 1);
- RealType scale()const;
- RealType location()const;
- };
- There are various
- [@http://mathworld.wolfram.com/ExtremeValueDistribution.html extreme value distributions]
- : this implementation represents the maximum case,
- and is variously known as a Fisher-Tippett distribution,
- a log-Weibull distribution or a Gumbel distribution.
- Extreme value theory is important for assessing risk for highly unusual events,
- such as 100-year floods.
- More information can be found on the
- [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm NIST],
- [@http://en.wikipedia.org/wiki/Extreme_value_distribution Wikipedia],
- [@http://mathworld.wolfram.com/ExtremeValueDistribution.html Mathworld],
- and [@http://en.wikipedia.org/wiki/Extreme_value_theory Extreme value theory]
- websites.
- The relationship of the types of extreme value distributions, of which this is but one, is
- discussed by
- [@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications
- Samuel Kotz & Saralees Nadarajah].
- The distribution has a PDF given by:
- [expression f(x) = (1/scale) e[super -(x-location)/scale] e[super -e[super -(x-location)/scale]]]
- which in the standard case (scale = 1, location = 0) reduces to:
- [expression f(x) = e[super -x]e[super -e[super -x]]]
- The following graph illustrates how the PDF varies with the location parameter:
- [graph extreme_value_pdf1]
- And this graph illustrates how the PDF varies with the shape parameter:
- [graph extreme_value_pdf2]
- [h4 Member Functions]
- extreme_value_distribution(RealType location = 0, RealType scale = 1);
-
- Constructs an Extreme Value distribution with the specified location and scale
- parameters.
- Requires `scale > 0`, otherwise calls __domain_error.
- RealType location()const;
-
- Returns the location parameter of the distribution.
-
- RealType scale()const;
-
- Returns the scale parameter of the 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 parameter is \[-[infin], +[infin]\].
- [h4 Accuracy]
- The extreme value distribution is implemented in terms of the
- standard library `exp` and `log` functions and as such should have very low
- error rates.
- [h4 Implementation]
- In the following table:
- /a/ is the location parameter, /b/ is the scale parameter,
- /x/ is the random variate, /p/ is the probability and /q = 1-p/.
- [table
- [[Function][Implementation Notes]]
- [[pdf][Using the relation: pdf = exp((a-x)/b) * exp(-exp((a-x)/b)) / b ]]
- [[cdf][Using the relation: p = exp(-exp((a-x)/b)) ]]
- [[cdf complement][Using the relation: q = -expm1(-exp((a-x)/b)) ]]
- [[quantile][Using the relation: a - log(-log(p)) * b]]
- [[quantile from the complement][Using the relation: a - log(-log1p(-q)) * b]]
- [[mean][a + [@http://en.wikipedia.org/wiki/Euler-Mascheroni_constant Euler-Mascheroni-constant] * b]]
- [[standard deviation][pi * b / sqrt(6)]]
- [[mode][The same as the location parameter /a/.]]
- [[skewness][12 * sqrt(6) * zeta(3) / pi[super 3] ]]
- [[kurtosis][27 / 5]]
- [[kurtosis excess][kurtosis - 3 or 12 / 5]]
- ]
- [endsect] [/section:extreme_dist Extreme Value]
- [/ extreme_value.qbk
- Copyright 2006 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).
- ]
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