# Boost C++ Libraries

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### boost/accumulators/statistics/kurtosis.hpp

///////////////////////////////////////////////////////////////////////////////
// kurtosis.hpp
//
//  Software License, Version 1.0. (See accompanying file

#ifndef BOOST_ACCUMULATORS_STATISTICS_KURTOSIS_HPP_EAN_28_10_2005
#define BOOST_ACCUMULATORS_STATISTICS_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/mean.hpp>
#include <boost/accumulators/statistics/moment.hpp>

namespace boost { namespace accumulators
{

namespace impl
{
///////////////////////////////////////////////////////////////////////////////
// kurtosis_impl
/**
@brief Kurtosis estimation

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.
*/
template<typename Sample>
struct kurtosis_impl
: accumulator_base
{
// for boost::result_of
typedef typename numeric::functional::average<Sample, Sample>::result_type result_type;

kurtosis_impl(dont_care) {}

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

} // namespace impl

///////////////////////////////////////////////////////////////////////////////
// tag::kurtosis
//
namespace tag
{
struct kurtosis
: depends_on<mean, moment<2>, moment<3>, moment<4> >
{
/// INTERNAL ONLY
///
typedef accumulators::impl::kurtosis_impl<mpl::_1> impl;
};
}

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

BOOST_ACCUMULATORS_IGNORE_GLOBAL(kurtosis)
}

using extract::kurtosis;

// So that kurtosis can be automatically substituted with
// weighted_kurtosis when the weight parameter is non-void
template<>
struct as_weighted_feature<tag::kurtosis>
{
typedef tag::weighted_kurtosis type;
};

template<>
struct feature_of<tag::weighted_kurtosis>
: feature_of<tag::kurtosis>
{
};

}} // namespace boost::accumulators

#endif