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boost/accumulators/statistics/variance.hpp

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

#ifndef BOOST_ACCUMULATORS_STATISTICS_VARIANCE_HPP_EAN_28_10_2005
#define BOOST_ACCUMULATORS_STATISTICS_VARIANCE_HPP_EAN_28_10_2005

#include <boost/mpl/placeholders.hpp>
#include <boost/accumulators/framework/accumulator_base.hpp>
#include <boost/accumulators/framework/extractor.hpp>
#include <boost/accumulators/numeric/functional.hpp>
#include <boost/accumulators/framework/parameters/sample.hpp>
#include <boost/accumulators/framework/depends_on.hpp>
#include <boost/accumulators/statistics_fwd.hpp>
#include <boost/accumulators/statistics/count.hpp>
#include <boost/accumulators/statistics/sum.hpp>
#include <boost/accumulators/statistics/mean.hpp>
#include <boost/accumulators/statistics/moment.hpp>

namespace boost { namespace accumulators
{

namespace impl
{
//! Lazy calculation of variance.
/*!
Default sample variance implementation based on the second moment \f$M_n^{(2)} \f$ moment<2>, mean and count.
\f[
\sigma_n^2 = M_n^{(2)} - \mu_n^2.
\f]
where
\f[
\mu_n = \frac{1}{n} \sum_{i = 1}^n x_i.
\f]
is the estimate of the sample mean and \f$n\f$ is the number of samples.
*/
template<typename Sample, typename MeanFeature>
struct lazy_variance_impl
: accumulator_base
{
// for boost::result_of
typedef typename numeric::functional::fdiv<Sample, std::size_t>::result_type result_type;

lazy_variance_impl(dont_care) {}

template<typename Args>
result_type result(Args const &args) const
{
extractor<MeanFeature> mean;
result_type tmp = mean(args);
return accumulators::moment<2>(args) - tmp * tmp;
}
};

//! Iterative calculation of variance.
/*!
Iterative calculation of sample variance \f$\sigma_n^2\f$ according to the formula
\f[
\sigma_n^2 = \frac{1}{n} \sum_{i = 1}^n (x_i - \mu_n)^2 = \frac{n-1}{n} \sigma_{n-1}^2 + \frac{1}{n-1}(x_n - \mu_n)^2.
\f]
where
\f[
\mu_n = \frac{1}{n} \sum_{i = 1}^n x_i.
\f]
is the estimate of the sample mean and \f$n\f$ is the number of samples.

Note that the sample variance is not defined for \f$n <= 1\f$.

A simplification can be obtained by the approximate recursion
\f[
\sigma_n^2 \approx \frac{n-1}{n} \sigma_{n-1}^2 + \frac{1}{n}(x_n - \mu_n)^2.
\f]
because the difference
\f[
\left(\frac{1}{n-1} - \frac{1}{n}\right)(x_n - \mu_n)^2 = \frac{1}{n(n-1)}(x_n - \mu_n)^2.
\f]
converges to zero as \f$n \rightarrow \infty\f$. However, for small \f$n \f$ the difference
can be non-negligible.
*/
template<typename Sample, typename MeanFeature, typename Tag>
struct variance_impl
: accumulator_base
{
// for boost::result_of
typedef typename numeric::functional::fdiv<Sample, std::size_t>::result_type result_type;

template<typename Args>
variance_impl(Args const &args)
: variance(numeric::fdiv(args[sample | Sample()], numeric::one<std::size_t>::value))
{
}

template<typename Args>
void operator ()(Args const &args)
{
std::size_t cnt = count(args);

if(cnt > 1)
{
extractor<MeanFeature> mean;
result_type tmp = args[parameter::keyword<Tag>::get()] - mean(args);
this->variance =
numeric::fdiv(this->variance * (cnt - 1), cnt)
+ numeric::fdiv(tmp * tmp, cnt - 1);
}
}

result_type result(dont_care) const
{
return this->variance;
}

private:
result_type variance;
};

} // namespace impl

///////////////////////////////////////////////////////////////////////////////
// tag::variance
// tag::immediate_variance
//
namespace tag
{
struct lazy_variance
: depends_on<moment<2>, mean>
{
/// INTERNAL ONLY
///
typedef accumulators::impl::lazy_variance_impl<mpl::_1, mean> impl;
};

struct variance
: depends_on<count, immediate_mean>
{
/// INTERNAL ONLY
///
typedef accumulators::impl::variance_impl<mpl::_1, mean, sample> impl;
};
}

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

BOOST_ACCUMULATORS_IGNORE_GLOBAL(lazy_variance)
BOOST_ACCUMULATORS_IGNORE_GLOBAL(variance)
}

using extract::lazy_variance;
using extract::variance;

// variance(lazy) -> lazy_variance
template<>
struct as_feature<tag::variance(lazy)>
{
typedef tag::lazy_variance type;
};

// variance(immediate) -> variance
template<>
struct as_feature<tag::variance(immediate)>
{
typedef tag::variance type;
};

// for the purposes of feature-based dependency resolution,
// immediate_variance provides the same feature as variance
template<>
struct feature_of<tag::lazy_variance>
: feature_of<tag::variance>
{
};

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

// for the purposes of feature-based dependency resolution,
// weighted_variance provides the same feature as variance
template<>
struct feature_of<tag::weighted_variance>
: feature_of<tag::variance>
{
};

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

// for the purposes of feature-based dependency resolution,
// immediate_weighted_variance provides the same feature as immediate_variance
template<>
struct feature_of<tag::lazy_weighted_variance>
: feature_of<tag::lazy_variance>
{
};

////////////////////////////////////////////////////////////////////////////
//// droppable_accumulator<variance_impl>
////  need to specialize droppable lazy variance to cache the result at the
////  point the accumulator is dropped.
///// INTERNAL ONLY
/////
//template<typename Sample, typename MeanFeature>
//struct droppable_accumulator<impl::variance_impl<Sample, MeanFeature> >
//  : droppable_accumulator_base<
//        with_cached_result<impl::variance_impl<Sample, MeanFeature> >
//    >
//{
//    template<typename Args>
//    droppable_accumulator(Args const &args)
//      : droppable_accumulator::base(args)
//    {
//    }
//};

}} // namespace boost::accumulators

#endif