weighted_kurtosis.hpp

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///////////////////////////////////////////////////////////////////////////////
// weighted_kurtosis.hpp
//
//  Copyright 2006 Olivier Gygi, Daniel Egloff. 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)

#ifndef BOOST_ACCUMULATORS_STATISTICS_WEIGHTED_KURTOSIS_HPP_EAN_28_10_2005
#define BOOST_ACCUMULATORS_STATISTICS_WEIGHTED_KURTOSIS_HPP_EAN_28_10_2005

#include <limits>
#include <boost/mpl/placeholders.hpp>
#include <boost/accumulators/framework/accumulator_base.hpp>
#include <boost/accumulators/framework/extractor.hpp>
#include <boost/accumulators/framework/parameters/sample.hpp>
#include <boost/accumulators/numeric/functional.hpp>
#include <boost/accumulators/framework/depends_on.hpp>
#include <boost/accumulators/statistics_fwd.hpp>
#include <boost/accumulators/statistics/weighted_moment.hpp>
#include <boost/accumulators/statistics/weighted_mean.hpp>

namespace boost { namespace accumulators
{

namespace impl
{
    ///////////////////////////////////////////////////////////////////////////////
    // weighted_kurtosis_impl
    /**
        @brief Kurtosis estimation for weighted samples

        The kurtosis of a sample distribution is defined as the ratio of the 4th central moment and the square of the 2nd central
        moment (the variance) of the samples, minus 3. The term \f$ -3 \f$ is added in order to ensure that the normal distribution
        has zero kurtosis. The kurtosis can also be expressed by the simple moments:

        \f[
            \hat{g}_2 =
                \frac
                {\widehat{m}_n^{(4)}-4\widehat{m}_n^{(3)}\hat{\mu}_n+6\widehat{m}_n^{(2)}\hat{\mu}_n^2-3\hat{\mu}_n^4}
                {\left(\widehat{m}_n^{(2)} - \hat{\mu}_n^{2}\right)^2} - 3,
        \f]

        where \f$ \widehat{m}_n^{(i)} \f$ are the \f$ i \f$-th moment and \f$ \hat{\mu}_n \f$ the mean (first moment) of the
        \f$ n \f$ samples.

        The kurtosis estimator for weighted samples is formally identical to the estimator for unweighted samples, except that
        the weighted counterparts of all measures it depends on are to be taken.
    */
    template<typename Sample, typename Weight>
    struct weighted_kurtosis_impl
      : accumulator_base
    {
        typedef typename numeric::functional::multiplies<Sample, Weight>::result_type weighted_sample;
        // for boost::result_of
        typedef typename numeric::functional::average<weighted_sample, weighted_sample>::result_type result_type;

        weighted_kurtosis_impl(dont_care)
        {
        }

        template<typename Args>
        result_type result(Args const &args) const
        {
            return numeric::average(
                        weighted_moment<4>(args)
                        - 4. * weighted_moment<3>(args) * weighted_mean(args)
                        + 6. * weighted_moment<2>(args) * weighted_mean(args) * weighted_mean(args)
                        - 3. * weighted_mean(args) * weighted_mean(args) * weighted_mean(args) * weighted_mean(args)
                      , ( weighted_moment<2>(args) - weighted_mean(args) * weighted_mean(args) )
                        * ( weighted_moment<2>(args) - weighted_mean(args) * weighted_mean(args) )
                   ) - 3.;
        }
    };

} // namespace impl

///////////////////////////////////////////////////////////////////////////////
// tag::weighted_kurtosis
//
namespace tag
{
    struct weighted_kurtosis
      : depends_on<weighted_mean, weighted_moment<2>, weighted_moment<3>, weighted_moment<4> >
    {
        /// INTERNAL ONLY
        ///
        typedef accumulators::impl::weighted_kurtosis_impl<mpl::_1, mpl::_2> impl;
    };
}

///////////////////////////////////////////////////////////////////////////////
// extract::weighted_kurtosis
//
namespace extract
{
    extractor<tag::weighted_kurtosis> const weighted_kurtosis = {};
}

using extract::weighted_kurtosis;

}} // namespace boost::accumulators

#endif