[Boost-commit] svn:boost r78017 - in trunk/libs/math/doc/sf_and_dist: . distributions graphs

Subject: [Boost-commit] svn:boost r78017 - in trunk/libs/math/doc/sf_and_dist: . distributions graphs
From: pbristow_at_[hidden]
Date: 2012-04-16 13:20:27

Author: pbristow
Date: 2012-04-16 13:20:26 EDT (Mon, 16 Apr 2012)
New Revision: 78017
URL: http://svn.boost.org/trac/boost/changeset/78017

Log:
Updated for owens and skewnormal graphs
Added:
   trunk/libs/math/doc/sf_and_dist/distributions/skew_normal.qbk   (contents, props changed)
   trunk/libs/math/doc/sf_and_dist/graphs/owens_integration_area.png   (contents, props changed)
   trunk/libs/math/doc/sf_and_dist/graphs/owens_integration_area.svg   (contents, props changed)
   trunk/libs/math/doc/sf_and_dist/plot_owens_3d_xyp.png   (contents, props changed)
   trunk/libs/math/doc/sf_and_dist/plot_owens_t.png   (contents, props changed)
Text files modified: 
   trunk/libs/math/doc/sf_and_dist/distributions/normal.qbk |    14 ++++++++++++--                          
   1 files changed, 12 insertions(+), 2 deletions(-)

Modified: trunk/libs/math/doc/sf_and_dist/distributions/normal.qbk
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/distributions/normal.qbk	(original)
+++ trunk/libs/math/doc/sf_and_dist/distributions/normal.qbk	2012-04-16 13:20:26 EDT (Mon, 16 Apr 2012)
@@ -35,13 +35,22 @@
 
 Given mean [mu][space]and standard deviation [sigma][space]it has the PDF:
 
-[equation normal_ref1]
+[space] [equation normal_ref1]
 
 The variation the PDF with its parameters is illustrated
 in the following graph:
 
 [graph normal_pdf]
 
+The cumulative distribution function is given by
+
+[space] [equation normal_cdf]
+
+and illustrated by this graph
+
+[graph normal_cdf]
+
+
 [h4 Member Functions]
 
    normal_distribution(RealType mean = 0, RealType sd = 1);
@@ -94,6 +103,7 @@
 [[quantile from the complement][Using the relation: x = m + s * sqrt(2) * __erfc_inv(2*p)]]
 [[mean and standard deviation][The same as `dist.mean()` and `dist.standard_deviation()`]]
 [[mode][The same as the mean.]]
+[[median][The same as the mean.]]
 [[skewness][0]]
 [[kurtosis][3]]
 [[kurtosis excess][0]]
@@ -102,7 +112,7 @@
 [endsect][/section:normal_dist Normal]
 
 [/ normal.qbk
-  Copyright 2006, 2007 John Maddock and Paul A. Bristow.
+  Copyright 2006, 2007, 2012 John Maddock and Paul A. Bristow.
   Distributed under the Boost Software License, Version 1.0.
   (See accompanying file LICENSE_1_0.txt or copy at
   http://www.boost.org/LICENSE_1_0.txt).

Added: trunk/libs/math/doc/sf_and_dist/distributions/skew_normal.qbk
==============================================================================
--- (empty file)
+++ trunk/libs/math/doc/sf_and_dist/distributions/skew_normal.qbk	2012-04-16 13:20:26 EDT (Mon, 16 Apr 2012)
@@ -0,0 +1,191 @@
+[section:skew_normal_dist Skew Normal Distribution]
+
+``#include <boost/math/distributions/skew_normal.hpp>``
+
+   namespace boost{ namespace math{ 
+      
+   template <class RealType = double, 
+             class ``__Policy``   = ``__policy_class`` >
+   class skew_normal_distribution;
+   
+   typedef skew_normal_distribution<> normal;
+   
+   template <class RealType, class ``__Policy``>
+   class skew_normal_distribution
+   {
+   public:
+      typedef RealType value_type;
+      typedef Policy   policy_type;
+      // Constructor:
+      skew_normal_distribution(RealType location = 0, RealType scale = 1, RealType shape = 0);
+      // Accessors:
+      RealType location()const; // mean if normal.
+      RealType scale()const; // width, standard deviation if normal.
+      RealType shape()const; // The distribution is right skewed if shape > 0 and is left skewed if shape < 0.
+                             // The distribution is normal if shape is zero.
+   };
+   
+   }} // namespaces
+   
+The skew normal distribution is a variant of the most well known
+Gaussian statistical distribution.  
+
+The skew normal distribution with shape zero resembles the
+[@http://en.wikipedia.org/wiki/Normal_distribution Normal Distribution],
+hence the latter can be regarded as a special case of the more generic skew normal distribution.
+
+If the standard (mean = 0, scale = 1) normal distribution probability density function is 
+
+[space][space][equation normal01_pdf]
+
+and the cumulative distribution function
+
+[space][space][equation normal01_cdf]
+
+then the [@http://en.wikipedia.org/wiki/Probability_density_function PDF]
+of the [@http://en.wikipedia.org/wiki/Skew_normal_distribution skew normal distribution]
+with shape parameter [alpha], defined by O'Hagan and Leonhard (1976) is
+
+[space][space][equation skew_normal_pdf0]
+
+Given [@http://en.wikipedia.org/wiki/Location_parameter location] [xi],
+[@http://en.wikipedia.org/wiki/Scale_parameter scale] [omega],
+and [@http://en.wikipedia.org/wiki/Shape_parameter shape] [alpha],
+it can be 
+[@http://en.wikipedia.org/wiki/Skew_normal_distribution transformed],
+to the form:
+
+[space][space][equation skew_normal_pdf]
+
+and [@http://en.wikipedia.org/wiki/Cumulative_distribution_function CDF]:
+
+[space][space][equation skew_normal_cdf]
+
+where ['T(h,a)] is Owen's T function, and ['[Phi](x)] is the normal distribution.
+
+The variation the PDF and CDF with its parameters is illustrated
+in the following graphs:
+
+[graph skew_normal_pdf]
+[graph skew_normal_cdf]
+
+[h4 Member Functions]
+
+   skew_normal_distribution(RealType location = 0, RealType scale = 1, RealType shape = 0);
+   
+Constructs a skew_normal distribution with location [xi],
+scale [omega] and shape [alpha].
+
+Requires scale > 0, otherwise __domain_error is called.
+
+   RealType location()const;   
+   
+returns the location [xi] of this distribution,
+   
+   RealType scale()const;
+
+returns the scale [omega] of this distribution,
+   
+   RealType shape()const;
+
+returns the shape [alpha] of this distribution.
+     
+(Location and scale function match other similar distributions,
+allowing the functions `find_location` and `find_scale` to be used generically).
+
+[note While the shape parameter may be chosen arbitrarily (finite),
+the resulting [*skewness] of the distribution is in fact limited to about (-1, 1);
+strictly, the interval is (-0.9952717, 0.9952717).
+
+A parameter [delta] is related to the shape [alpha] by
+[delta] = [alpha] / (1 + [alpha][pow2]),
+and used in the expression for skewness
+[equation skew_normal_skewness]
+] [/note]
+
+[h4 References]
+
+* [@http://azzalini.stat.unipd.it/SN/ Skew-Normal Probability Distribution] for many links and bibliography.
+* [@http://azzalini.stat.unipd.it/SN/Intro/intro.html A very brief introduction to the skew-normal distribution]
+by Adelchi Azzalini (2005-11-2).
+* See a [@http://www.tri.org.au/azzalini.html skew-normal function animation].
+
+[h4 Non-member Accessors]
+
+All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions]
+that are generic to all distributions are supported: __usual_accessors.
+
+The domain of the random variable is ['-[max_value], +[min_value]].
+Infinite values are not supported.
+
+There are no [@http://en.wikipedia.org/wiki/Closed-form_expression closed-form expression]
+known for the mode and median, but these are computed for the
+
+* mode - by finding the maximum of the PDF.
+* median - by computing `quantile(1/2)`.
+
+The maximum of the PDF is sought through searching the root of f'(x)=0.
+
+Both involve iterative methods that will have lower accuracy than other estimates.
+
+[h4 testing]
+
+__R using library(sn) described at
+[@http://azzalini.stat.unipd.it/SN/  Skew-Normal Probability Distribution],
+and at [@http://cran.r-project.org/web/packages/sn/sn.pd R skew-normal(sn) package].
+
+Package sn provides functions related to the skew-normal (SN)
+and the skew-t (ST) probability distributions,
+both for the univariate and for the the multivariate case,
+including regression models. 
+
+[h4 Accuracy]
+
+The skew_normal distribution with shape = zero is implemented as a special case,
+equivalent to the normal distribution in terms of the
+[link math_toolkit.special.sf_erf.error_function error function], 
+and therefore should have excellent accuracy.
+
+The PDF and mean, variance, skewness and kurtosis are also accurately evaluated using
+[@http://en.wikipedia.org/wiki/Analytical_expression analytical expressions].
+The CDF requires [@http://en.wikipedia.org/wiki/Owen%27s_T_function Owen's T function]
+that is evaluated using a Boost C++ __owens_t implementation of the algorithms of
+M. Patefield and D. Tandy, Journal of Statistical Software, 5(5), 1-25 (2000);
+the complicated accuracy of this function is discussed in detail at __owens_t.
+
+The median and mode are calculated by iterative root finding, and both will be less accurate.
+
+[h4 Implementation]
+
+In the following table, [xi] is the location of the distribution,
+and [omega] is its scale, and [alpha] is its shape.
+
+[table
+[[Function][Implementation Notes]]
+[[pdf][Using:[equation skew_normal_pdf] ]]
+[[cdf][Using: [equation skew_normal_cdf]\n
+where ['T(h,a)] is Owen's T function, and ['[Phi](x)] is the normal distribution. ]]
+[[cdf complement][Using: complement of normal distribution + 2 * Owens_t]]
+[[quantile][Maximum of the pdf is sought through searching the root of f'(x)=0]]
+[[quantile from the complement][-quantile(SN(-location [xi], scale [omega], -shape[alpha]), p)]]
+[[location][location [xi]]]
+[[scale][scale [omega]]]
+[[shape][shape [alpha]]]
+[[median][quantile(1/2)]]
+[[mean][[equation skew_normal_mean]]]
+[[mode][Maximum of the pdf is sought through searching the root of f'(x)=0]]
+[[variance][[equation skew_normal_variance] ]]
+[[skewness][[equation skew_normal_skewness] ]]
+[[kurtosis][kurtosis excess-3]]
+[[kurtosis excess] [ [equation skew_normal_kurt_ex] ]]
+] [/table]
+
+[endsect] [/section:skew_normal_dist skew_Normal]
+
+[/ skew_normal.qbk
+  Copyright 2012 Bejamin Sobotta, John Maddock and Paul A. Bristow.
+  Distributed under the Boost Software License, Version 1.0.
+  (See accompanying file LICENSE_1_0.txt or copy at
+  http://www.boost.org/LICENSE_1_0.txt).
+]
+

Added: trunk/libs/math/doc/sf_and_dist/graphs/owens_integration_area.png
==============================================================================
Binary file. No diff available.

Added: trunk/libs/math/doc/sf_and_dist/graphs/owens_integration_area.svg
==============================================================================
--- (empty file)
+++ trunk/libs/math/doc/sf_and_dist/graphs/owens_integration_area.svg	2012-04-16 13:20:26 EDT (Mon, 16 Apr 2012)
@@ -0,0 +1,121 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+   xmlns:dc="http://purl.org/dc/elements/1.1/"
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+   xmlns:xlink="http://www.w3.org/1999/xlink"
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Added: trunk/libs/math/doc/sf_and_dist/plot_owens_3d_xyp.png
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Added: trunk/libs/math/doc/sf_and_dist/plot_owens_t.png
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