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