1[section:extreme_dist Extreme Value Distribution] 2 3``#include <boost/math/distributions/extreme.hpp>`` 4 5 template <class RealType = double, 6 class ``__Policy`` = ``__policy_class`` > 7 class extreme_value_distribution; 8 9 typedef extreme_value_distribution<> extreme_value; 10 11 template <class RealType, class ``__Policy``> 12 class extreme_value_distribution 13 { 14 public: 15 typedef RealType value_type; 16 17 extreme_value_distribution(RealType location = 0, RealType scale = 1); 18 19 RealType scale()const; 20 RealType location()const; 21 }; 22 23There are various 24[@http://mathworld.wolfram.com/ExtremeValueDistribution.html extreme value distributions] 25: this implementation represents the maximum case, 26and is variously known as a Fisher-Tippett distribution, 27a log-Weibull distribution or a Gumbel distribution. 28 29Extreme value theory is important for assessing risk for highly unusual events, 30such as 100-year floods. 31 32More information can be found on the 33[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm NIST], 34[@http://en.wikipedia.org/wiki/Extreme_value_distribution Wikipedia], 35[@http://mathworld.wolfram.com/ExtremeValueDistribution.html Mathworld], 36and [@http://en.wikipedia.org/wiki/Extreme_value_theory Extreme value theory] 37websites. 38 39The relationship of the types of extreme value distributions, of which this is but one, is 40discussed by 41[@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications 42Samuel Kotz & Saralees Nadarajah]. 43 44The distribution has a PDF given by: 45 46[expression f(x) = (1/scale) e[super -(x-location)/scale] e[super -e[super -(x-location)/scale]]] 47 48which in the standard case (scale = 1, location = 0) reduces to: 49 50[expression f(x) = e[super -x]e[super -e[super -x]]] 51 52The following graph illustrates how the PDF varies with the location parameter: 53 54[graph extreme_value_pdf1] 55 56And this graph illustrates how the PDF varies with the shape parameter: 57 58[graph extreme_value_pdf2] 59 60[h4 Member Functions] 61 62 extreme_value_distribution(RealType location = 0, RealType scale = 1); 63 64Constructs an Extreme Value distribution with the specified location and scale 65parameters. 66 67Requires `scale > 0`, otherwise calls __domain_error. 68 69 RealType location()const; 70 71Returns the location parameter of the distribution. 72 73 RealType scale()const; 74 75Returns the scale parameter of the distribution. 76 77[h4 Non-member Accessors] 78 79All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] 80that are generic to all distributions are supported: __usual_accessors. 81 82The domain of the random parameter is \[-[infin], +[infin]\]. 83 84[h4 Accuracy] 85 86The extreme value distribution is implemented in terms of the 87standard library `exp` and `log` functions and as such should have very low 88error rates. 89 90[h4 Implementation] 91 92In the following table: 93/a/ is the location parameter, /b/ is the scale parameter, 94/x/ is the random variate, /p/ is the probability and /q = 1-p/. 95 96[table 97[[Function][Implementation Notes]] 98[[pdf][Using the relation: pdf = exp((a-x)/b) * exp(-exp((a-x)/b)) / b ]] 99[[cdf][Using the relation: p = exp(-exp((a-x)/b)) ]] 100[[cdf complement][Using the relation: q = -expm1(-exp((a-x)/b)) ]] 101[[quantile][Using the relation: a - log(-log(p)) * b]] 102[[quantile from the complement][Using the relation: a - log(-log1p(-q)) * b]] 103[[mean][a + [@http://en.wikipedia.org/wiki/Euler-Mascheroni_constant Euler-Mascheroni-constant] * b]] 104[[standard deviation][pi * b / sqrt(6)]] 105[[mode][The same as the location parameter /a/.]] 106[[skewness][12 * sqrt(6) * zeta(3) / pi[super 3] ]] 107[[kurtosis][27 / 5]] 108[[kurtosis excess][kurtosis - 3 or 12 / 5]] 109] 110 111[endsect] [/section:extreme_dist Extreme Value] 112 113[/ extreme_value.qbk 114 Copyright 2006 John Maddock and Paul A. Bristow. 115 Distributed under the Boost Software License, Version 1.0. 116 (See accompanying file LICENSE_1_0.txt or copy at 117 http://www.boost.org/LICENSE_1_0.txt). 118] 119 120