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Added non-central F docs, plus updated nc beta docs.
[SVN r43124]
This commit is contained in:
John Maddock
@@ -17,6 +17,7 @@
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[include distributions/negative_binomial.qbk]
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[include distributions/nc_beta.qbk]
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[include distributions/nc_chi_squared.qbk]
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[include distributions/nc_f.qbk]
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[include distributions/normal.qbk]
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[include distributions/pareto.qbk]
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[include distributions/poisson.qbk]
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@@ -80,10 +80,10 @@ Returns the parameter /lambda/ from which this object was constructed.
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Most of the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions]
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are supported: __cdf, __pdf, __quantile,
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__hazard, __chf, __range and __support.
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__median, __mode, __hazard, __chf, __range and __support.
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However, the following are not currently implemented:
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__mean, __median, __mode, __variance, __sd, __skewness,
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__mean, __variance, __sd, __skewness,
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__kurtosis and __kurtosis_excess.
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The domain of the random variable is \[0, 1\].
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188
doc/sf_and_dist/distributions/nc_f.qbk
Normal file
188
doc/sf_and_dist/distributions/nc_f.qbk
Normal file
@@ -0,0 +1,188 @@
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[section:nc_f_dist Non Central F Distribution]
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``#include <boost/math/distributions/non_central_f.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 non_central_f_distribution;
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typedef non_central_f_distribution<> non_central_f;
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template <class RealType, class ``__Policy``>
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class non_central_f_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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non_central_f_distribution(RealType v1, RealType v2, RealType lambda);
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// Accessor to parameters:
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RealType degrees_of_freedom1()const;
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RealType degrees_of_freedom2()const;
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// Accessor to non centrality parameter:
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RealType non_centrality()const;
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};
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}} // namespaces
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The non-central F distribution is a generalization of the
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[link math_toolkit.dist.dist_ref.dists.f_dist Fisher F distribution].
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It is defined as the ratio
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F = (X/v1) / (Y/v2)
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where X is a non-central [chi][super 2]
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random variable with /v1/ degrees of freedom and non centrality parameter [lambda],
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and Y is a central [chi][super 2] random variable with /v2/ degrees of freedom.
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This gives the following PDF:
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[equation nc_f_ref1]
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where L[sub a][super b](c) is a generalised Laguerre polynomial and B(a,b) is the
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beta function, or
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[equation nc_f_ref2]
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The following graph illustrates how the distribution changes
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for different values of [lambda]:
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[$../graphs/nc_f_pdf.png]
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[h4 Member Functions]
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non_central_f_distribution(RealType v1, RealType v2, RealType lambda);
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Constructs a non-central beta distribution with parameters /v1/ and /v2/
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and non-centrality parameter /lambda/.
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Requires v1 > 0, v2 > 0 and lambda >= 0, otherwise calls __domain_error.
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RealType degrees_of_freedom1()const;
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Returns the parameter /v1/ from which this object was constructed.
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RealType degrees_of_freedom2()const;
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Returns the parameter /v2/ from which this object was constructed.
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RealType non_centrality()const;
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Returns the parameter /lambda/ from which this object was constructed.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist.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 \[0, +[infin]\].
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[h4 Accuracy]
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This distribution is implemented in terms of the
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[link math_toolkit.dist.dist_ref.dists.nc_beta_dist non-central beta distribution]:
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refer to that distribution for accuracy data.
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[h4 Tests]
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Since this distribution is implemented by adapting another distribution,
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the tests consist of basic sanity checks computed by the R statistical
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package and the pbeta and dbeta functions.
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[h4 Implementation]
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In the following table /v1/ and /v2/ are the first and second
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degrees of freedom parameters of the distribution, [lambda]
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is the non-centrality parameter,
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/x/ is the random variate, /p/ is the probability, and /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Implemented in terms of the non-central beta PDF using the relation:
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f(x;v1,v2;[lambda]) = (v1\/v2) / ((1+y)*(1+y)) * g(y\/(1+y);v1\/2,v2\/2;[lambda])
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where g(x; a, b; [lambda]) is the non central beta PDF, and:
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y = x * v1 \/ v2
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]]
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[[cdf][Using the relation:
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p = B[sub y](v1\/2, v2\/2; [lambda])
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where B[sub x](a, b; [lambda]) is the non-central beta distribution CDF and
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y = x * v1 \/ v2
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]]
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[[cdf complement][Using the relation:
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q = 1 - B[sub y](v1\/2, v2\/2; [lambda])
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where 1 - B[sub x](a, b; [lambda]) is the complement of the
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non-central beta distribution CDF and
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y = x * v1 \/ v2
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]]
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[[quantile][Using the relation:
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x = (bx \/ (1-bx)) * (v1 \/ v2)
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where
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bx = Q[sub p][super -1](v1\/2, v2\/2; [lambda])
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and
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Q[sub p][super -1](v1\/2, v2\/2; [lambda])
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is the non-central beta quantile.
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]]
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[[quantile
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from the complement][
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Using the relation:
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x = (bx \/ (1-bx)) * (v1 \/ v2)
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where
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bx = QC[sub q][super -1](v1\/2, v2\/2; [lambda])
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and
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QC[sub q][super -1](v1\/2, v2\/2; [lambda])
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is the non-central beta quantile from the complement.]]
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[[mean][v2 * (v1 + l) \/ (v1 * (v2 - 2))]]
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[[mode][By numeric maximalisation of the PDF.]]
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[[variance][Refer to, [@http://mathworld.wolfram.com/NoncentralF-Distribution.html
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Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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[[skewness][Refer to, [@http://mathworld.wolfram.com/NoncentralF-Distribution.html
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Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.],
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and to the [@http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html
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Mathematica documentation] ]]
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[[kurtosis and kurtosis excess]
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[Refer to, [@http://mathworld.wolfram.com/NoncentralF-Distribution.html
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Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.],
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and to the [@http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html
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Mathematica documentation] ]]
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]
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[endsect][/section:nc_beta_dist]
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[/ nc_beta.qbk
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Copyright 2006 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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@@ -14,7 +14,7 @@ for mmlfile in $*; do
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pngfile=$(basename $svgfile .svg).png
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tempfile=temp.mml
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# strip html wrappers put in by MathCast:
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cat $svgfile | tr -d "\r\n" | sed -e 's/.*\(<math[^>]*>.*<\/math>\).*/\1/' > $tempfile
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cat $mmlfile | tr -d "\r\n" | sed -e 's/.*\(<math[^>]*>.*<\/math>\).*/\1/' > $tempfile
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echo Generating $svgfile
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$python $math2svg $tempfile > $svgfile
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@@ -24,3 +24,4 @@ for mmlfile in $*; do
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done
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314
doc/sf_and_dist/equations/nc_f_ref1.mml
Normal file
314
doc/sf_and_dist/equations/nc_f_ref1.mml
Normal file
@@ -0,0 +1,314 @@
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<?xml version='1.0'?>
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<!DOCTYPE html PUBLIC '-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN'
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'http://www.w3.org/TR/MathML2/dtd/xhtml-math11-f.dtd'
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[<!ENTITY mathml 'http://www.w3.org/1998/Math/MathML'>]>
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<html xmlns='http://www.w3.org/1999/xhtml'>
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<head><title>nc_f_ref1</title>
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<!-- MathML created with MathCast Equation Editor version 0.88 -->
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</head>
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<body>
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<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
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<mrow>
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<mi>f</mi>
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<mfenced>
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<mrow>
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<mi>x</mi>
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<mo>;</mo>
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<msub>
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<mi>ν</mi>
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<mn>1</mn>
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</msub>
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<mo>,</mo>
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<msub>
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<mi>ν</mi>
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<mn>2</mn>
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</msub>
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<mo>;</mo>
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<mi>λ</mi>
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</mrow>
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</mfenced>
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<mspace width="1em"/>
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<mo>=</mo>
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<mspace width="1em"/>
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<msup>
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<mi>e</mi>
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<mrow>
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<mrow>
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<mo>−</mo>
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<mfrac>
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<mi>λ</mi>
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<mn>2</mn>
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</mfrac>
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</mrow>
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<mo>+</mo>
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<mfrac>
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<mfenced>
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<mrow>
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<mi>λ</mi>
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<msub>
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</msub>
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</mrow>
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</mfenced>
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<mfenced>
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<mrow>
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<mrow>
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<msub>
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</mrow>
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</mfenced>
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</mrow>
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</mfenced>
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</mfrac>
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</mrow>
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</msup>
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<msup>
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<mrow>
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<msub>
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<mi>ν</mi>
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</msub>
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</mrow>
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<mrow>
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<mfrac>
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</mfrac>
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</msub>
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<mo>+</mo>
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<msub>
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<mi>ν</mi>
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<mn>1</mn>
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</msub>
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</mrow>
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</mfenced>
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<mrow>
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<mo>−</mo>
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<mfenced>
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<mrow>
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<mi>ν</mi>
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</msub>
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<mo>+</mo>
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</mrow>
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<mrow>
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</mfrac>
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<mi>ν</mi>
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<mn>1</mn>
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</msub>
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</mrow>
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<mrow>
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<mo>+</mo>
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<mfrac>
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<mn>2</mn>
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</mfrac>
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</mfenced>
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<msubsup>
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<mi>L</mi>
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<mrow>
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<mfrac>
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<msub>
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</mrow>
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</mfrac>
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</msub>
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<mn>2</mn>
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</msub>
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<mo>+</mo>
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<msub>
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<mn>1</mn>
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</msub>
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</mrow>
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</mfrac>
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</mrow>
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</mfenced>
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</mrow>
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<mrow>
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<mi>B</mi>
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<mfenced>
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<mrow>
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</mfrac>
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<mi>ν</mi>
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<mn>1</mn>
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</msub>
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</mrow>
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<mo>,</mo>
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<mfrac>
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<mn>2</mn>
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</msub>
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</mrow>
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</mrow>
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</mfenced>
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<mi>Γ</mi>
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<mfenced>
|
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Reference in New Issue
Block a user