ceph/src/boost/libs/math/doc/distributions/extreme_value.qbk

   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
  23 There are various
  24 [@http://mathworld.wolfram.com/ExtremeValueDistribution.html extreme value distributions]
  25 : this implementation represents the maximum case,
  26 and is variously known as a Fisher-Tippett distribution,
  27 a log-Weibull distribution or a Gumbel distribution.
  28
  29 Extreme value theory is important for assessing risk for highly unusual events,
  30 such as 100-year floods.
  31
  32 More 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],
  36 and [@http://en.wikipedia.org/wiki/Extreme_value_theory Extreme value theory]
  37 websites.
  38
  39 The relationship of the types of extreme value distributions, of which this is but one, is
  40 discussed by
  41 [@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications
  42 Samuel Kotz & Saralees Nadarajah].
  43
  44 The distribution has a PDF given by:
  45
  46 f(x) = (1/scale) e[super -(x-location)/scale] e[super -e[super -(x-location)/scale]]
  47
  48 Which in the standard case (scale = 1, location = 0) reduces to:
  49
  50 f(x) = e[super -x]e[super -e[super -x]]
  51
  52 The following graph illustrates how the PDF varies with the location parameter:
  53
  54 [graph extreme_value_pdf1]
  55
  56 And 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
  64 Constructs an Extreme Value distribution with the specified location and scale
  65 parameters.
  66
  67 Requires `scale > 0`, otherwise calls __domain_error.
  68
  69    RealType location()const;
  70
  71 Returns the location parameter of the distribution.
  72
  73    RealType scale()const;
  74
  75 Returns the scale parameter of the distribution.
  76
  77 [h4 Non-member Accessors]
  78
  79 All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
  80 that are generic to all distributions are supported: __usual_accessors.
  81
  82 The domain of the random parameter is \[-[infin], +[infin]\].
  83
  84 [h4 Accuracy]
  85
  86 The extreme value distribution is implemented in terms of the
  87 standard library `exp` and `log` functions and as such should have very low
  88 error rates.
  89
  90 [h4 Implementation]
  91
  92 In 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