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

   1 [section:laplace_dist Laplace Distribution]
   2
   3 ``#include <boost/math/distributions/laplace.hpp>``
   4
   5    namespace boost{ namespace math{
   6
   7    template <class RealType = double,
   8              class ``__Policy``   = ``__policy_class`` >
   9    class laplace_distribution;
  10
  11    typedef laplace_distribution<> laplace;
  12
  13    template <class RealType, class ``__Policy``>
  14    class laplace_distribution
  15    {
  16    public:
  17       typedef RealType value_type;
  18       typedef Policy   policy_type;
  19       // Construct:
  20       laplace_distribution(RealType location = 0, RealType scale = 1);
  21       // Accessors:
  22       RealType location()const;
  23       RealType scale()const;
  24    };
  25
  26    }} // namespaces
  27
  28 Laplace distribution is the distribution of differences between two independent variates
  29 with identical exponential distributions (Abramowitz and Stegun 1972, p. 930).
  30 It is also called the double exponential distribution.
  31
  32 [/ Wikipedia definition is The difference between two independent identically distributed
  33 exponential random variables is governed by a Laplace distribution.]
  34
  35 For location parameter [mu][space] and scale parameter [sigma][space], it is defined by the
  36 probability density function:
  37
  38 [equation laplace_pdf]
  39
  40 The location and scale parameters are equivalent to the mean and
  41 standard deviation of the normal or Gaussian distribution.
  42
  43 The following graph illustrates the effect of the
  44 parameters [mu][space] and [sigma][space] on the PDF.
  45 Note that the domain of the random variable remains
  46 \[-[infin],+[infin]\] irrespective of the value of the location parameter:
  47
  48 [graph laplace_pdf]
  49
  50 [h4 Member Functions]
  51
  52    laplace_distribution(RealType location = 0, RealType scale = 1);
  53
  54 Constructs a laplace distribution with location /location/ and
  55 scale /scale/.
  56
  57 The location parameter is the same as the mean of the random variate.
  58
  59 The scale parameter is proportional to the standard deviation of the random variate.
  60
  61 Requires that the scale parameter is greater than zero, otherwise calls
  62 __domain_error.
  63
  64    RealType location()const;
  65
  66 Returns the /location/ parameter of this distribution.
  67
  68    RealType scale()const;
  69
  70 Returns the /scale/ parameter of this distribution.
  71
  72 [h4 Non-member Accessors]
  73
  74 All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
  75 distributions are supported: __usual_accessors.
  76
  77 The domain of the random variable is \[-[infin],+[infin]\].
  78
  79 [h4 Accuracy]
  80
  81 The laplace distribution is implemented in terms of the
  82 standard library log and exp functions and as such should have very small errors.
  83
  84 [h4 Implementation]
  85
  86 In the following table [mu] is the location parameter of the distribution,
  87 [sigma] is its scale parameter, /x/ is the random variate, /p/ is the probability
  88 and its complement /q = 1-p/.
  89
  90 [table
  91 [[Function][Implementation Notes]]
  92 [[pdf][Using the relation: pdf = e[super -abs(x-[mu]) \/ [sigma]] \/ (2 * [sigma]) ]]
  93 [[cdf][Using the relations:
  94
  95 x <  [mu] : p =  e[super (x-[mu])/[sigma] ] \/ [sigma]
  96
  97 x >= [mu] : p =  1 - e[super ([mu]-x)/[sigma] ] \/ [sigma]
  98 ]]
  99 [[cdf complement][Using the relation:
 100
 101 -x <  [mu] : q =  e[super (-x-[mu])/[sigma] ] \/ [sigma]
 102
 103 -x >= [mu] : q =  1 - e[super ([mu]+x)/[sigma] ] \/ [sigma]
 104 ]]
 105 [[quantile][Using the relations:
 106
 107 p <  0.5 : x = [mu] + [sigma] * log(2*p)
 108
 109 p >= 0.5 : x = [mu] - [sigma] * log(2-2*p)
 110 ]]
 111 [[quantile from the complement][Using the relation:
 112
 113 q > 0.5: x = [mu] + [sigma]*log(2-2*q)
 114
 115 q <=0.5: x = [mu] - [sigma]*log( 2*q )
 116 ]]
 117 [[mean][[mu]]]
 118 [[variance][2 * [sigma][super 2] ]]
 119 [[mode][[mu]]]
 120 [[skewness][0]]
 121 [[kurtosis][6]]
 122 [[kurtosis excess][3]]
 123 ]
 124
 125 [h4 References]
 126
 127 * [@http://mathworld.wolfram.com/LaplaceDistribution.html Weisstein, Eric W. "Laplace Distribution."] From MathWorld--A Wolfram Web Resource.
 128
 129 * [@http://en.wikipedia.org/wiki/Laplace_distribution Laplace Distribution]
 130
 131 * M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, p. 930.
 132
 133 [endsect][/section:laplace_dist laplace]
 134
 135 [/
 136   Copyright 2008, 2009 John Maddock, Paul A. Bristow and M.A. (Thijs) van den Berg.
 137   Distributed under the Boost Software License, Version 1.0.
 138   (See accompanying file LICENSE_1_0.txt or copy at
 139   http://www.boost.org/LICENSE_1_0.txt).
 140 ]
 141