| [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: |
| |
| 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: |
| |
| 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.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). |
| ] |
| |