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194 lines
7.0 KiB
Plaintext
194 lines
7.0 KiB
Plaintext
[section:skew_normal_dist Skew Normal Distribution]
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``#include <boost/math/distributions/skew_normal.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 skew_normal_distribution;
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typedef skew_normal_distribution<> normal;
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template <class RealType, class ``__Policy``>
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class skew_normal_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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// Constructor:
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skew_normal_distribution(RealType location = 0, RealType scale = 1, RealType shape = 0);
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// Accessors:
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RealType location()const; // mean if normal.
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RealType scale()const; // width, standard deviation if normal.
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RealType shape()const; // The distribution is right skewed if shape > 0 and is left skewed if shape < 0.
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// The distribution is normal if shape is zero.
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};
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}} // namespaces
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The skew normal distribution is a variant of the most well known
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Gaussian statistical distribution.
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The skew normal distribution with shape zero resembles the
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[@http://en.wikipedia.org/wiki/Normal_distribution Normal Distribution],
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hence the latter can be regarded as a special case of the more generic skew normal distribution.
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If the standard (mean = 0, scale = 1) normal distribution probability density function is
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[space][space][equation normal01_pdf]
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and the cumulative distribution function
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[space][space][equation normal01_cdf]
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then the [@http://en.wikipedia.org/wiki/Probability_density_function PDF]
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of the [@http://en.wikipedia.org/wiki/Skew_normal_distribution skew normal distribution]
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with shape parameter [alpha], defined by O'Hagan and Leonhard (1976) is
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[space][space][equation skew_normal_pdf0]
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Given [@http://en.wikipedia.org/wiki/Location_parameter location] [xi],
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[@http://en.wikipedia.org/wiki/Scale_parameter scale] [omega],
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and [@http://en.wikipedia.org/wiki/Shape_parameter shape] [alpha],
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it can be
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[@http://en.wikipedia.org/wiki/Skew_normal_distribution transformed],
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to the form:
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[space][space][equation skew_normal_pdf]
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and [@http://en.wikipedia.org/wiki/Cumulative_distribution_function CDF]:
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[space][space][equation skew_normal_cdf]
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where ['T(h,a)] is Owen's T function, and ['[Phi](x)] is the normal distribution.
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The variation the PDF and CDF with its parameters is illustrated
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in the following graphs:
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[graph skew_normal_pdf]
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[graph skew_normal_cdf]
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[h4 Member Functions]
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skew_normal_distribution(RealType location = 0, RealType scale = 1, RealType shape = 0);
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Constructs a skew_normal distribution with location [xi],
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scale [omega] and shape [alpha].
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Requires scale > 0, otherwise __domain_error is called.
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RealType location()const;
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returns the location [xi] of this distribution,
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RealType scale()const;
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returns the scale [omega] of this distribution,
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RealType shape()const;
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returns the shape [alpha] of this distribution.
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(Location and scale function match other similar distributions,
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allowing the functions `find_location` and `find_scale` to be used generically).
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[note While the shape parameter may be chosen arbitrarily (finite),
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the resulting [*skewness] of the distribution is in fact limited to about (-1, 1);
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strictly, the interval is (-0.9952717, 0.9952717).
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A parameter [delta] is related to the shape [alpha] by
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[delta] = [alpha] / (1 + [alpha][pow2]),
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and used in the expression for skewness
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[equation skew_normal_skewness]
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] [/note]
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[h4 References]
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* [@http://azzalini.stat.unipd.it/SN/ Skew-Normal Probability Distribution] for many links and bibliography.
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* [@http://azzalini.stat.unipd.it/SN/Intro/intro.html A very brief introduction to the skew-normal distribution]
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by Adelchi Azzalini (2005-11-2).
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* See a [@http://www.tri.org.au/azzalini.html skew-normal function animation].
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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]
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that are generic to all distributions are supported: __usual_accessors.
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The domain of the random variable is ['-[max_value], +[min_value]].
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Infinite values are not supported.
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There are no [@http://en.wikipedia.org/wiki/Closed-form_expression closed-form expression]
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known for the mode and median, but these are computed for the
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* mode - by finding the maximum of the PDF.
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* median - by computing `quantile(1/2)`.
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The maximum of the PDF is sought through searching the root of f'(x)=0.
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Both involve iterative methods that will have lower accuracy than other estimates.
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[h4 Testing]
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__R using library(sn) described at
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[@http://azzalini.stat.unipd.it/SN/ Skew-Normal Probability Distribution],
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and at [@http://cran.r-project.org/web/packages/sn/sn.pd R skew-normal(sn) package].
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Package sn provides functions related to the skew-normal (SN)
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and the skew-t (ST) probability distributions,
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both for the univariate and for the the multivariate case,
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including regression models.
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__Mathematica was also used to generate some more accurate spot test data.
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[h4 Accuracy]
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The skew_normal distribution with shape = zero is implemented as a special case,
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equivalent to the normal distribution in terms of the
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[link math_toolkit.sf_erf.error_function error function],
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and therefore should have excellent accuracy.
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The PDF and mean, variance, skewness and kurtosis are also accurately evaluated using
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[@http://en.wikipedia.org/wiki/Analytical_expression analytical expressions].
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The CDF requires [@http://en.wikipedia.org/wiki/Owen%27s_T_function Owen's T function]
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that is evaluated using a Boost C++ __owens_t implementation of the algorithms of
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M. Patefield and D. Tandy, Journal of Statistical Software, 5(5), 1-25 (2000);
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the complicated accuracy of this function is discussed in detail at __owens_t.
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The median and mode are calculated by iterative root finding, and both will be less accurate.
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[h4 Implementation]
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In the following table, [xi] is the location of the distribution,
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and [omega] is its scale, and [alpha] is its shape.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Using:[equation skew_normal_pdf] ]]
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[[cdf][Using: [equation skew_normal_cdf][br]
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where ['T(h,a)] is Owen's T function, and ['[Phi](x)] is the normal distribution. ]]
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[[cdf complement][Using: complement of normal distribution + 2 * Owens_t]]
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[[quantile][Maximum of the pdf is sought through searching the root of f'(x)=0]]
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[[quantile from the complement][-quantile(SN(-location [xi], scale [omega], -shape[alpha]), p)]]
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[[location][location [xi]]]
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[[scale][scale [omega]]]
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[[shape][shape [alpha]]]
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[[median][quantile(1/2)]]
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[[mean][[equation skew_normal_mean]]]
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[[mode][Maximum of the pdf is sought through searching the root of f'(x)=0]]
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[[variance][[equation skew_normal_variance] ]]
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[[skewness][[equation skew_normal_skewness] ]]
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[[kurtosis][kurtosis excess-3]]
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[[kurtosis excess] [ [equation skew_normal_kurt_ex] ]]
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] [/table]
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[endsect] [/section:skew_normal_dist skew_Normal]
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[/ skew_normal.qbk
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Copyright 2012 Bejamin Sobotta, John Maddock and Paul A. Bristow.
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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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