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142 lines
4.1 KiB
Plaintext
142 lines
4.1 KiB
Plaintext
[section:laplace_dist Laplace Distribution]
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``#include <boost/math/distributions/laplace.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class laplace_distribution;
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typedef laplace_distribution<> laplace;
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template <class RealType, class ``__Policy``>
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class laplace_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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// Construct:
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laplace_distribution(RealType location = 0, RealType scale = 1);
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// Accessors:
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RealType location()const;
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RealType scale()const;
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};
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}} // namespaces
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Laplace distribution is the distribution of differences between two independent variates
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with identical exponential distributions (Abramowitz and Stegun 1972, p. 930).
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It is also called the double exponential distribution.
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[/ Wikipedia definition is The difference between two independent identically distributed
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exponential random variables is governed by a Laplace distribution.]
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For location parameter [mu][space] and scale parameter [sigma][space], it is defined by the
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probability density function:
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[equation laplace_pdf]
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The location and scale parameters are equivalent to the mean and
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standard deviation of the normal or Gaussian distribution.
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The following graph illustrates the effect of the
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parameters [mu][space] and [sigma][space] on the PDF.
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Note that the domain of the random variable remains
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\[-[infin],+[infin]\] irrespective of the value of the location parameter:
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[graph laplace_pdf]
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[h4 Member Functions]
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laplace_distribution(RealType location = 0, RealType scale = 1);
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Constructs a laplace distribution with location /location/ and
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scale /scale/.
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The location parameter is the same as the mean of the random variate.
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The scale parameter is proportional to the standard deviation of the random variate.
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Requires that the scale parameter is greater than zero, otherwise calls
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__domain_error.
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RealType location()const;
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Returns the /location/ parameter of this distribution.
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RealType scale()const;
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Returns the /scale/ parameter of this distribution.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
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distributions are supported: __usual_accessors.
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The domain of the random variable is \[-[infin],+[infin]\].
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[h4 Accuracy]
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The laplace distribution is implemented in terms of the
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standard library log and exp functions and as such should have very small errors.
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[h4 Implementation]
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In the following table [mu] is the location parameter of the distribution,
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[sigma] is its scale parameter, /x/ is the random variate, /p/ is the probability
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and its complement /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Using the relation: pdf = e[super -abs(x-[mu]) \/ [sigma]] \/ (2 * [sigma]) ]]
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[[cdf][Using the relations:
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x < [mu] : p = e[super (x-[mu])/[sigma] ] \/ [sigma]
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x >= [mu] : p = 1 - e[super ([mu]-x)/[sigma] ] \/ [sigma]
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]]
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[[cdf complement][Using the relation:
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-x < [mu] : q = e[super (-x-[mu])/[sigma] ] \/ [sigma]
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-x >= [mu] : q = 1 - e[super ([mu]+x)/[sigma] ] \/ [sigma]
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]]
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[[quantile][Using the relations:
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p < 0.5 : x = [mu] + [sigma] * log(2*p)
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p >= 0.5 : x = [mu] - [sigma] * log(2-2*p)
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]]
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[[quantile from the complement][Using the relation:
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q > 0.5: x = [mu] + [sigma]*log(2-2*q)
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q <=0.5: x = [mu] - [sigma]*log( 2*q )
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]]
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[[mean][[mu]]]
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[[variance][2 * [sigma][super 2] ]]
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[[mode][[mu]]]
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[[skewness][0]]
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[[kurtosis][6]]
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[[kurtosis excess][3]]
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]
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[h4 References]
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* [@http://mathworld.wolfram.com/LaplaceDistribution.html Weisstein, Eric W. "Laplace Distribution."] From MathWorld--A Wolfram Web Resource.
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* [@http://en.wikipedia.org/wiki/Laplace_distribution Laplace Distribution]
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* M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, p. 930.
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[endsect][/section:laplace_dist laplace]
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[/
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Copyright 2008, 2009 John Maddock, Paul A. Bristow and M.A. (Thijs) van den Berg.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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