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| 7c673cae FG |
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 |