
[library Boost.Accumulators
    [quickbook 1.3]
    [authors [Niebler, Eric]]
    [copyright 2005 2006 Eric Niebler]
    [category math]
    [id accumulators]
    [dirname accumulators]
    [purpose 
        Incremental accumulation framework and statistical accumulator library.
    ]
    [license
        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])
    ]
]

[/  Images   ]

[def _note_                  [$images/note.png]]
[def _alert_                 [$images/caution.png]]
[def _detail_                [$images/note.png]]
[def _tip_                   [$images/tip.png]]

[/  Links   ]

[def _sample_type_           '''<replaceable>sample-type</replaceable>''']
[def _weight_type_           '''<replaceable>weight-type</replaceable>''']
[def _variate_type_          '''<replaceable>variate-type</replaceable>''']
[def _variate_tag_           '''<replaceable>variate-tag</replaceable>''']
[def _left_or_right_         '''<replaceable>left-or-right</replaceable>''']
[def _implementation_defined_ '''<replaceable>implementation-defined</replaceable>''']
[def _boost_                 [@http://www.boost.org Boost]]
[def _mpl_                   [@../../libs/mpl/index.html MPL]]
[def _mpl_lambda_expression_ [@../../libs/mpl/doc/refmanual/lambda-expression.html MPL Lambda Expression]]
[def _parameter_             [@../../libs/parameter/index.html Boost.Parameter]]
[def _accumulator_set_       [classref boost::accumulators::accumulator_set `accumulator_set<>`]]
[def _accumulator_base_      [classref boost::accumulators::accumulator_base `accumulator_base`]]
[def _depends_on_            [classref boost::accumulators::depends_on `depends_on<>`]]
[def _feature_of_            [classref boost::accumulators::feature_of `feature_of<>`]]
[def _as_feature_            [classref boost::accumulators::as_feature `as_feature<>`]]
[def _features_              [classref boost::accumulators::features `features<>`]]
[def _external_              [classref boost::accumulators::external `external<>`]]
[def _droppable_             [classref boost::accumulators::droppable `droppable<>`]]
[def _droppable_accumulator_ [classref boost::accumulators::droppable_accumulator `droppable_accumulator<>`]]
[def _extractor_             [classref boost::accumulators::tag::extractor `extractor<>`]]
[def _tail_                  [classref boost::accumulators::tag::tail `tail`]]
[def _tail_variate_          [classref boost::accumulators::tag::tail_variate `tail_variate<>`]]
[def _extract_result_        [funcref boost::accumulators::extract_result `extract_result()`]]
[def _ZKB_                   [@http://www.zkb.com Z'''&uuml;'''rcher Kantonalbank]]

[section Preface]

[:["It is better to be approximately right than exactly wrong.]\n['-- Old adage]]

[h2 Description]

Boost.Accumulators is both a library for incremental statistical computation as
well as an extensible framework for incremental calculation in general. The library
deals primarily with the concept of an ['accumulator], which is a primitive
computational entity that accepts data one sample at a time and maintains some 
internal state. These accumulators may offload some of their computations on other
accumulators, on which they depend. Accumulators are grouped within an ['accumulator
set]. Boost.Accumulators resolves the inter-dependencies between accumulators in a
set and ensures that accumulators are processed in the proper order.

[endsect]

[section User's Guide]

This section describes how to use the Boost.Accumulators framework to create new
accumulators and how to use the existing statistical accumulators to perform incremental
statistical computation. For detailed information regarding specific components in
Boost.Accumulators, check the [link accumulators_framework_reference Reference] section.

[h2 Hello, World!]

Below is a complete example of how to use the Accumulators Framework and the 
Statistical Accumulators to perform an incremental statistical calculation. It
calculates the mean and 2nd moment of a sequence of doubles.

    #include <iostream>
    #include <boost/accumulators/accumulators.hpp>
    #include <boost/accumulators/statistics/stats.hpp>
    #include <boost/accumulators/statistics/mean.hpp>
    #include <boost/accumulators/statistics/moment.hpp>
    using namespace boost::accumulators;

    int main()
    {
        // Define an accumulator set for calculating the mean and the
        // 2nd moment ...
        accumulator_set<double, stats<tag::mean, tag::moment<2> > > acc;

        // push in some data ...
        acc(1.2);
        acc(2.3);
        acc(3.4);
        acc(4.5);

        // Display the results ...
        std::cout << "Mean:   " << mean(acc) << std::endl;
        std::cout << "Moment: " << moment<2>(acc) << std::endl;

        return 0;
    }

This program displays the following:

[pre
Mean:   2.85
Moment: 9.635
]

[section The Accumulators Framework]

The Accumulators Framework is framework for performing incremental calculations. Usage
of the framework follows the following pattern:

* Users build a computational object, called an ['_accumulator_set_], by selecting
  the computations in which they are interested, or authoring their own computational
  primitives which fit within the framework.
* Users push data into the _accumulator_set_ object one sample at a time.
* The _accumulator_set_ computes the requested quantities in the most efficient method
  possible, resolving dependencies between requested calculations, possibly cacheing
  intermediate results.

The Accumulators Framework defines the utilities needed for defining primitive
computational elements, called ['accumulators]. It also provides the _accumulator_set_
type, described above.

[h2 Terminology]

The following terms are used in the rest of the documentation.

[variablelist
    [[Sample]          [[#sample_type] A datum that is pushed into an _accumulator_set_.
                        The type of the sample is the ['sample type].]]
    [[Weight]          [[#weight_type] An optional scalar value passed along with the
                        sample specifying the weight of the sample. Conceptually, each
                        sample is multiplied with its weight. The type of the weight is
                        the ['weight type].]]
    [[Feature]         [An abstract primitive computational entity. When defining an
                        _accumulator_set_, users specify the features in which they are
                        interested, and the _accumulator_set_ figures out which
                        ['accumulators] would best provide those features. Features may
                        depend on other features. If they do, the accumulator set figures
                        out which accumulators to add to satisfy the dependencies.]]
    [[Accumulator]     [A concrete primitive computational entity. An accumulator is a
                        concrete implementation of a feature. It satisfies exactly one
                        abstract feature. Several different accumulators may provide the
                        same feature, but may represent different implementation strategies.]]
    [[Accumulator Set] [A collection of accumulators. An accumulator set is specified with
                        a sample type and a list of features. The accumulator set uses this
                        information to generate an ordered set of accumulators depending on
                        the feature dependency graph. An accumulator set accepts samples one
                        datum at a time, propogating them to each accumulator in order. At any
                        point, results can be extracted from the accumulator set.]]
    [[Extractor]       [A function or function object that can be used to extract a result
                        from an _accumulator_set_.]]
]

[h2 Overview]

Here is a list of the important types and functions in the Accumulator Framework and
a brief description of each.

[table Accumulators Toolbox
    [[Tool]              [Description]]
    [[_accumulator_set_] [This is the most important type in the Accumulators Framework.
                          It is a collection of accumulators. A datum pushed into an 
                          _accumulator_set_ is forwarded to each accumulator, in an order
                          determined by the dependency relationships between the
                          accumulators. Computational results can be extracted from an
                          accumulator at any time.]]
    [[_depends_on_     ] [Used to specify which other features a feature depends on.]]
    [[_feature_of_     ] [Trait used to tell the Accumulators Framework that, for the purpose
                          of feature-based dependency resolution, one feature should be
                          treated the same as another.]]
    [[_as_feature_     ] [Used to create an alias for a feature. For example, if there are
                          two features, fast_X and accurate_X, they can be mapped to
                          X(fast) and X(accurate) with _as_feature_. This is just syntactic
                          sugar.]]
    [[_features_       ] [An _mpl_ sequence. We can use _features_ as the second template
                          parameter when declaring an _accumulator_set_.]]
    [[_external_       ] [Used when declaring an _accumulator_set_. If the weight type is
                          specified with _external_, then the weight accumulators are
                          assumed to reside in a separate accumulator set which will be passed
                          in with a named parameter.]]
    [[_extractor_      ] [A class template useful for creating an extractor function object.
                          It is parameterized on a feature, and it has member functions for
                          extracting from an _accumulator_set_ the result corresponding to
                          that feature.]]
]

[section Using [^accumulator_set<>]]

Our tour of the _accumulator_set_ class template begins with the forward declaration:

    template< typename Sample, typename Features, typename Weight = void >
    struct accumulator_set;

The template parameters have the following meaning:

[variablelist
    [[`Sample`]     [The type of the data that will be accumulated.]]
    [[`Features`]   [An _mpl_ sequence of features to be calculated.]]
    [[`Weight`]     [The type of the (optional) weight paramter.]]
]

For example, the following line declares an _accumulator_set_ that will accept
a sequence of doubles one at a time and calculate the min and mean:

    accumulator_set< double, features< tag::min, tag::mean > > acc;

Notice that we use the _features_ template to specify a list of features to be calculated.
_features_ is an MPL sequence of features.

[note _features_ is a synonym of `mpl::vector<>`. In fact, we could use `mpl::vector<>`
 or any MPL sequence if we prefer, and the meaning would be the same.]

Once we have defined an _accumulator_set_, we can then push data into it,
and it will calculate the quantities you requested, as shown below.

    // push some data into the accumulator_set ...
    acc(1.2);
    acc(2.3);
    acc(3.4);

Since _accumulator_set_ defines its accumulate function to be the function call operator,
we might be tempted to use an _accumulator_set_ as a UnaryFunction to a standard
algorithm such as `std::for_each`. That's fine as long as we keep in mind that the standard
algorithms take UnaryFunction objects by value, which involves making a copy of the
_accumulator_set_ object. Consider the following:

    // The data for which we wish to calculate statistical properties:
    std::vector< double > data( /* stuff */ );

    // The accumulator set which will calculate the properties for us:    
    accumulator_set< double, features< tag::min, tag::mean > > acc;

    // Use std::for_each to accumulate the statistical properties:
    acc = std::for_each( data.begin(), data.end(), acc );

Notice how we must assign the return value of `std::for_each` back to the _accumulator_set_.
This works, but some accumulators are not cheap to copy. For
example, the _tail_ and _tail_variate_ accumulators must store a `std::vector<>`, so copying
these accumulators involves a dynamic allocation. We might be better off in this
case passing the accumulator by reference, with the help of `boost::bind()` and
`boost::ref()`. See below:

    // The data for which we wish to calculate statistical properties:
    std::vector< double > data( /* stuff */ );

    // The accumulator set which will calculate the properties for us:
    accumulator_set< double, features< tag::tail<left> > > acc( 
        tag::tail<left>::cache_size = 4 );

    // Use std::for_each to accumulate the statistical properties:
    std::for_each( data.begin(), data.end(), bind<void>( ref(acc), _1 ) );

Notice now that we don't care about the return value of `std::for_each()` anymore because
`std::for_each()` is modifying `acc` directly.

[note To use `boost::bind()` and `boost::ref()`, you must `#include` [^<boost/bind.hpp>]
and [^<boost/ref.hpp>]]

[endsect]

[section Extracting Results]

Once we have declared an _accumulator_set_ and pushed data into it, we need to be able
to extract results from it. For each feature we can add to an _accumulator_set_, there
is a corresponding extractor for fetching its result. Usually, the extractor has the
same name as the feature, but in a different namespace. For example, if we accumulate
the `tag::min` and `tag::max` features, we can extract the results with the `min` and `max`
extractors, as follows:

    // Calculate the minimum and maximum for a sequence of integers.
    accumulator_set< int, features< tag::min, tag::max > > acc;
    acc( 2 );
    acc( -1 );
    acc( 1 );

    // This displays "(-1, 2)"
    std::cout << '(' << min( acc ) << ", " << max( acc ) << ")\n";

The extractors are all declared in the `boost::accumulators::extract` namespace, but they
are brought into the `boost::accumulators` namespace with a `using` declaration.

[tip On the Windows platform, `min` and `max` are preprocessor macros defined in [^WinDef.h].
 To use the `min` and `max` extractors, you should either compile with `NOMINMAX` defined, or
 you should invoke the extractors like: `(min)( acc )` and `(max)( acc )`. The parentheses
 keep the macro from being invoked.]

Another way to extract a result from an _accumulator_set_ is with the
`extract_result()` function. This can be more convenient if there isn't an extractor
object handy for a certain feature. The line above which displays results could 
equally be written as:

    // This displays "(-1, 2)"
    std::cout << '('  << extract_result< tag::min >( acc )
              << ", " << extract_result< tag::max >( acc ) << ")\n";

Finally, we can define our own extractor using the _extractor_ class template. For
instance, another way to avoid the `min` / `max` macro business would be to define 
extractors with names that don't conflict with the macros, like this:

    extractor< tag::min > min_;
    extractor< tag::min > max_;

    // This displays "(-1, 2)"
    std::cout << '(' << min_( acc ) << ", " << max_( acc ) << ")\n";

[endsect]

[section Passing Optional Parameters]

Some accumulators need initialization parameters. In addition, perhaps some auxiliary
information needs to be passed into the _accumulator_set_ along with each sample.
Boost.Accumulators handles these cases with named parameters from the _parameter_
library.

For example, consider the _tail_ and _tail_variate_ features. _tail_ keeps
an ordered list of the largest [^['N]] samples, where [^['N]] can be specified at
construction time. Also, the _tail_variate_ feature, which depends on _tail_, keeps
track of some data that is covariate with the [^['N]] samples tracked by _tail_. The
code below shows how this all works, and is described in more detail below.

    // Define a feature for tracking covariate data
    typedef tag::tail_variate< int, tag::covariate1, left > my_tail_variate_tag;

    // This will calculate the left tail and my_tail_variate_tag for N == 2
    // using the tag::tail<left>::cache_size named parameter
    accumulator_set< double, features< my_tail_variate_tag > > acc( 
        tag::tail<left>::cache_size = 2 );

    // push in some samples and some covariates by using 
    // the covariate1 named parameter
    acc( 1.2, covariate1 =  12 );
    acc( 2.3, covariate1 = -23 );
    acc( 3.4, covariate1 =  34 );
    acc( 4.5, covariate1 = -45 );

    // Define an extractor for the my_tail_variate_tag feature
    extractor< my_tail_variate_tag > my_tail_variate;
    
    // Write the tail statistic to std::cout. This will print "4.5, 3.4, "
    std::ostream_iterator< double > dout( std::cout, ", " );
    std::copy( tail( acc ).begin(), tail( acc ).end(), dout );
    
    // Write the tail_variate statistic to std::cout. This will print "-45, 34, "
    std::ostream_iterator< int > iout( std::cout, ", " );
    std::copy( my_tail_variate( acc ).begin(), my_tail_variate( acc ).end(), iout );

There are several things to note about the code above. First, notice that we didn't have
to request that the _tail_ feature be calculated. That is implicit because the _tail_variate_
feature depends on the _tail_ feature. Next, notice how the `acc` object
is initialized: `acc( tag::tail<left>::cache_size = 2 )`. Here, `cache_size` is a named parameter.
It is used to tell the _tail_ and _tail_variate_ accumulators how many samples and
covariates to store. Conceptually, every construction parameter is made available to
every accumulator in an accumulator set. 

We also use a named parameter to pass covariate data into the accumulator set along with
the samples. As with the constructor parameters, all parameters to the accumulate function
are made available to all the accumulators in the set. In this case, only the accumulator
for the `my_tail_variate` feature would be interested in the value of the `covariate1` named
parameter.

We can make one final observation about the example above. Since _tail_ and _tail_variate_
are multi-valued features, the result we extract for them is represented as an iterator
range. That is why we can say `tail( acc ).begin()` and `tail( acc ).end()`.

Even the extractors can accept named parameters. In a bit, we'll see a situation where that
is useful.

[endsect]

[section Weighted Samples]

Some accumulators, statistical accumulators in particular, deal with data that are
['weighted]. Each sample pushed into the accumulator has an associated weight, by which
the sample is conceptually multiplied. The Statistical Accumulators Library provides an 
assortment of these weighted statistical accumulators. And many unweighted statistical
accumulators have weighted variants. For instance, the weighted variant of the `sum`
accumulator is called `weighted_sum`, and is calculated by accumulating all the
samples multiplied by their weights.

To declare an _accumulator_set_ that accepts weighted samples, you must specify the
type of the weight parameter as the 3rd template parameter, as follows:

    // 3rd template parameter 'int' means this is a weighted
    // accumulator set where the weights have type 'int'
    accumulator_set< int, features< tag::sum >, int > acc;

When you specify a weight, all the accumulators in the set are replaced with
their weighted equivalents. For example, the above _accumulator_set_ declaration
is equivalent to the following:

    // Since we specified a weight, tag::sum becomes tag::weighted_sum
    accumulator_set< int, features< tag::weighted_sum >, int > acc;

When passing samples to the accumulator set, you must also specify the 
weight of each sample. You can do that with the `weight` named parameter,
as follows:

    acc(1, weight = 2); //   1 * 2
    acc(2, weight = 4); //   2 * 4
    acc(3, weight = 6); // + 3 * 6
                        // -------
                        // =    28

You can then extract the result with the `sum()` extractor, as follows:

    // This prints "28"
    std::cout << sum(acc) << std::endl;

[note When working with weighted statistical accumulators from the Statistical
Accumulators Library, be sure to include the appropriate header. For instance, 
`weighted_sum` is defined in `<boost/accumulators/statistics/weighted_sum.hpp>`.]

[endsect]

[section Numeric Operators Sub-Library]

This section describes the function objects in the `boost::numeric` namespace, which
is a sub-library that provides function objects and meta-functions corresponding
to the infix operators in C++.

In the `boost::numeric::operators` namespace are additional operator overloads for
some useful operations not provided by the standard library, such as multiplication 
of a `std::complex<>` with a scalar.

In the `boost::numeric::functional` namespace are function object equivalents of 
the infix operators. These function object types are heterogeneous, and so are more 
general than the standard ones found in the [^<functional>] header. They use the
Boost.Typeof library to deduce the return types of the infix expressions they
evaluate. In addition, they look within the `boost::numeric::operators` namespace
to consider any additional overloads that might be defined there.

In the `boost::numeric` namespace are global polymorphic function objects 
corresponding to the function object types defined in the `boost::numeric::functional`
namespace. For example, `boost::numeric::plus(a, b)` is equivalent to
`boost::numeric::functional::plus<A, B>()(a, b)`, and both are equivalent to
`using namespace boost::numeric::operators; a + b;`.

The Numeric Operators Sub-Library also gives several ways to sub-class and
a way to sub-class and specialize operations. One way uses tag dispatching on
the types of the operands. The other way is based on the compile-time
properties of the operands.

[endsect]

[section Extending the Accumulators Framework]

This section describes how to extend the Accumulators Framework by defining new accumulators,
features and extractors. Also covered are how to control the dependency resolution of
features within an accumulator set.

[section Defining a New Accumulator]

All new accumulators must satisfy the [link
accumulators.user_s_guide.the_accumulators_framework.concepts.accumulator_concept Accumulator
Concept]. Below is a sample class that satisfies the accumulator concept, which simply sums
the values of all samples passed into it.

    #include <boost/accumulators/framework/accumulator_base.hpp>
    #include <boost/accumulators/framework/parameters/sample.hpp>
    
    namespace boost {                           // Putting your accumulators in the
    namespace accumulators {                    // impl namespace has some
    namespace impl {                            // advantages. See below.
    
    template<typename Sample>
    struct sum_accumulator                      // All accumulators should inherit from
      : accumulator_base                        // accumulator_base.
    {
        typedef Sample result_type;             // The type returned by result() below.
        
        template<typename Args>                 // The constructor takes an argument pack.
        sum_accumulator(Args const & args)
          : sum(args[sample | Sample()])        // Maybe there is an initial value in the
        {                                       // argument pack. ('sample' is defined in
        }                                       // sample.hpp, included above.)
        
        template<typename Args>                 // The accumulate function is the function
        void operator ()(Args const & args)     // call operator, and it also accepts an
        {                                       // argument pack.
            this->sum += args[sample];
        }
        
        result_type result(dont_care) const     // The result function will also be passed
        {                                       // an argument pack, but we don't use it here,
            return this->sum;                   // so we use "dont_care" as the argument type.
        }
    private:
        Sample sum;
    };
    
    }}}

Much of the above should be pretty self-explanitory, except for the use of argument packs
which may be confusing if you have never used the _parameter_ library before. An argument
pack is a cluster of values, each of which can be accessed with a key. So `args[sample]`
extracts from the pack the value associated with the `sample` key. And the cryptic
`args[sample | Sample()]` evaluates to the value associated with the `sample` key if it
exists, or a default-constructed `Sample` if it doesn't.

The example above demonstrates the most common attributes of an accumulator. There are
other optional member functions that have special meaning. In particular:

[variablelist Optional Accumulator Member Functions
[[[^on_drop(Args)]]         [Defines an action to be taken when this accumulator is
                             dropped. See the section on
                             [link accumulators.user_s_guide.the_accumulators_framework.extending_the_accumulators_framework.defining_a_new_accumulator.droppable_accumulators
                              Droppable Accumulators].]]
]

[h3 Accessing Other Accumulators in the Set]

Some accumulators depend on other accumulators within the same accumulator set. In those
cases, it is necessary to be able to access those other accumulators. To make this possible,
the _accumulator_set_ passes a reference to itself when invoking the member functions of
its contained accumulators. It can be accessed by using the special `accumulator` key with
the argument pack. Consider how we might implement `mean_accumulator`:

    // Mean == (Sum / Count)
    template<typename Sample>
    struct mean_accumulator : accumulator_base
    {
        typedef Sample result_type;
        mean_accumulator(dont_care) {}

        template<typename Args>
        result_type result(Args const &args) const
        {
            return sum(args[accumulator]) / count(args[accumulator]);
        }
    };

`mean` depends on the `sum` and `count` accumulators. (We'll see in the next section how
to specify these dependencies.) The result of the mean accumulator is merely the result of 
the sum accumulator divided by the result of the count accumulator. Consider how we write
that: `sum(args[accumulator]) / count(args[accumulator])`. The expression `args[accumulator]`
evaluates to a reference to the _accumulator_set_ that contains this `mean_accumulator`. It
also contains the `sum` and `count` accumulators, and we can access their results with the
extractors defined for those features: `sum` and `count`.

[note Accumulators that inherit from _accumulator_base_ get an empty `operator ()`, so
 accumulators like `mean_accumulator` above need not define one.]

All the member functions that accept an argument pack have access to the enclosing 
_accumulator_set_ via the `accumulator` key, including the constructor. The
accumulators within the set are constructed in an order determined by their interdependencies.
As a result, it is safe for an accumulator to access one on which it depends during construction.

[h3 Infix Notation and the Numeric Operators Sub-Library]

Although not necessary, it can be a good idea to put your accumulator implementations in 
the `boost::accumulators::impl` namespace. This namespace pulls in any operators defined
in the `boost::numeric::operators` namespace with a using directive. The Numeric Operators 
Sub-Library defines some additional overloads that will make your accumulators work with 
all sorts of data types.

Consider `mean_accumulator` defined above. It divides the sum of the samples by the count.
The type of the count is `std::size_t`. What if the sample type doesn't define division by
`std::size_t`? That's the case for `std::complex<>`. You might think that if the sample type
is `std::complex<>`, the code would not work, but in fact it does. That's because
Numeric Operators Sub-Library defines an overloaded `operator/` for `std::complex<>` 
and `std::size_t`. This operator is defined in the `boost::numeric::operators` namespace and
will be found within the `boost::accumulators::impl` namespace. That's why it's a good idea
to put your accumulators there.

[h3 Droppable Accumulators]

The term "droppable" refers to an accumulator that can be removed from the _accumulator_set_.
You can request that an accumulator be made droppable by using the _droppable_ class template.

    // calculate sum and count, make sum droppable:
    accumulator_set< double, features< tag::count, droppable<tag::sum> > > acc;

    // add some data
    acc(3.0);
    acc(2.0);

    // drop the sum (sum is 5 here)
    acc.drop<tag::sum>();
    
    // add more data
    acc(1.0);
    
    // This will display "3" and "5"
    std::cout << count(acc) << ' ' << sum(acc);    

Any accumulators that get added to an accumulator set in order to satisfy
dependencies on droppable accumulators are themselves droppable. Consider
the following accumulator:

    // Sum is not droppable. Mean is droppable. Count, brought in to 
    // satisfy mean's dependencies, is implicitly droppable, too.
    accumulator_set< double, features< tag::sum, droppable<tag::mean> > > acc;

`mean` depends on `sum` and `count`. Since `mean` is droppable, so too is `count`.
However, we have explictitly requested that `sum` be not droppable, so it isn't. Had
we left `tag::sum` out of the above declaration, the `sum` accumulator would have
been implicitly droppable.

A droppable accumulator is reference counted, and is only really dropped after all the
accumulators that depend on it have been dropped. This can lead to some surprising
behavior in some situations. 
    
    // calculate sum and mean, make mean droppable. 
    accumulator_set< double, features< tag::sum, droppable<tag::mean> > > acc;
    
    // add some data
    acc(1.0);
    acc(2.0);
    
    // drop the mean. mean's reference count
    // drops to 0, so it's really dropped. So
    // too, count's reference count drops to 0
    // and is really dropped.
    acc.drop<tag::mean>();
    
    // add more data. Sum continues to accumulate!
    acc(3.0);
    
    // This will display "6 2 3"
    std::cout << sum(acc) << ' '
              << count(acc) << ' '    
              << mean(acc);

Note that at the point at which `mean` is dropped, `sum` is 3, `count` is 2, and
therefore `mean` is 1.5. But since `sum` continues to accumulate even after `mean`
has been dropped, the value of `mean` continues to change. If you want to remember
the value of `mean` at the point it is dropped, you should save its value into
a local variable.

The following rules more precisely specify how droppable and non-droppable
accumulators behave within an accumulator set.

* There are two types of accumulators: droppable and non-droppable.
  The default is non-droppable.
* For any feature `X`, both `X` and `droppable<X>` satisfy the `X` dependency.
* If feature `X` depends on `Y` and `Z`, then `droppable<X>` depends on
  `droppable<Y>` and `droppable<Z>`.
* All accumulators have `add_ref()` and `drop()` member functions.
* For non-droppable accumulators, `drop()` is a no-op, and `add_ref()`
  invokes `add_ref()` on all accumulators corresponding to the features
  upon which the current accumulator depends.
* Droppable accumulators have a reference count and define `add_ref()`
  and `drop()` to manipulate the reference count.
* For droppable accumulators, `add_ref()` increments the accumulator's
  reference count, and also `add_ref()`'s the accumulators corresponding
  to the features upon which the current accumulator depends.
* For droppable accumulators, `drop()` decrements the accumulator's
  reference count, and also `drop()`'s the accumulators corresponding to
  the features upon which the current accumulator depends.
* The accumulator_set constructor walks the list of *user-specified*
  features and `add_ref()`'s the accumulator that corresponds to each of
  them. (Note: that means that an accumulator that is not user-specified
  but in the set merely to satisfy a dependency will be dropped as soon
  as all its dependencies have been dropped. Ones that have been user
  specified are not dropped until their dependencies have been
  dropped *and* the user has explicitly dropped the accumulator.)
* Droppable accumulators check their reference count in their
  accumulate member function. If the reference count is 0, the function
  is a no-op.
* Users are not allowed to drop a feature that is not user-specified and
  marked as droppable.

And as an optimization:

* If the user specifies the non-droppable feature `X`, which depends on `Y`
  and `Z`, then the accumulators for `Y` and `Z` can be safely made
  non-droppable, as well as any accumulators on which they depend.

[endsect]

[section Defining a New Feature]

Once we have implemented an accumulator, we must define a feature for it so
that users can specify the feature when declaring an _accumulator_set_. We
typically put the features into a nested namespace, so that later we can
define an extractor of the same name. All features must satisfy the
[link accumulators.user_s_guide.the_accumulators_framework.concepts.feature_concept
Feature Concept]. Using _depends_on_ makes satisfying the concept simple.
Below is an example of a feature definition.

    namespace boost { namespace accumulators { namespace tag {
    
    struct mean                         // Features should inherit from
      : depends_on< count, sum >        // depends_on<> to specify dependencies
    {
        // Define a nested typedef called 'impl' that specifies which
        // accumulator implements this feature. 
        typedef impl::mean_accumulator< mpl::_1 > impl;
    };
    
    }}}

The only two things we must do to define the `mean` feature is to specify the
dependencies with _depends_on_ and define the nested `impl` typedef. Even features
that have no dependencies should inherit from _depends_on_. The nested `impl` type
must be an _mpl_lambda_expression_. The result of 
`mpl::apply< impl, _sample_type_, _weight_type_ >::type` must be
be the type of the accumulator that implements this feature. The use of _mpl_
placeholders like `mpl::_1` make it especially easy to make a template such
as `mean_accumulator<>` an _mpl_lambda_expression_. Here, `mpl::_1` will be
replaced with the sample type. Had we used `mpl::_2`, it would have been replaced
with the weight type.

What about accumulator types that are not templates? If you have a `foo_accumulator`
which is a plain struct and not a template, you could turn it into an
_mpl_lambda_expression_ using `mpl::always<>`, like this:

    // An MPL lambda expression that always evaluates to
    // foo_accumulator:
    typedef mpl::always< foo_accumulator > impl;

If you are ever unsure, or if you are not comfortable with MPL lambda expressions,
you could always define `impl` explicitly:

    // Same as 'typedef mpl::always< foo_accumulator > impl;'
    struct impl
    {
        template< typename Sample, typename Weight >
        struct apply
        {
            typedef foo_accumulator type;
        };
    };

Here, `impl` is a binary [@../../libs/mpl/doc/refmanual/metafunction-class.html
MPL Metafunction Class], which is a kind of _mpl_lambda_expression_. The nested
`apply<>` template is part of the metafunction class protocol and tells MPL how
to to build the accumulator type given the sample and weight types.

All features must also provide a nested `is_weight_accumulator` typedef. It must
be either `mpl::true_` or `mpl::false_`. _depends_on_ provides a default of 
`mpl::false_` for all features that inherit from it, but that can be overridden
(or hidden, technically speaking) in the derived type. When the feature represents
an accumulation of information about the weights instead of the samples, we
can mark this feature as such with `typedef mpl::true_ is_weight_accumulator;`.
The weight accumulators are made external if the weight type is specified using
the _external_ template.

[endsect]

[section Defining a New Extractor]

Now that we have an accumulator and a feature, the only thing lacking is a way
to get results from the accumulator set. The Accumulators Framework provides the
_extractor_ class template to make it simple to define an extractor for your
feature. Here's an extractor for the `mean` feature we defined above:

    namespace boost {
    namespace accumulators {                // By convention, we put extractors
    namespace extract {                     // in the 'extract' namespace
    
    extractor< tag::mean > const mean = {}; // Simply define our extractor with
                                            // our feature tag, like this.
    }
    using extract::mean;                    // Pull the extractor into the
                                            // enclosing namespace.
    }}                                            

Once defined, the `mean` extractor can be used to extract the result of the
`tag::mean` feature from an _accumulator_set_.

Parameterized features complicate this simple picture. Consider the `moment`
feature, for calculating the [^['N]]-th moment, where [^['N]] is specified as
a template parameter:

    // An accumulator set for calculating the N-th moment, for N == 2 ...
    accumulator_set< double, features< tag::moment<2> > > acc;
    
    // ... add some data ...
    
    // Display the 2nd moment ...
    std::cout << "2nd moment is " << moment<2>(acc) << std::endl;

In the expression `moment<2>(acc)`, what is `moment`? It cannot be an object --
the syntax of C++ will not allow it. Clearly, if we want to provide this syntax,
we must make `moment` a function template. Here's what the definition of the 
`moment` extractor looks like:

    namespace boost {
    namespace accumulators {                // By convention, we put extractors
    namespace extract {                     // in the 'extract' namespace
    
    template<int N, typename AccumulatorSet>
    typename mpl::apply<AccumulatorSet, tag::moment<N> >::type::result_type
    moment(AccumulatorSet const &acc)
    {
        return extract_result<tag::moment<N> >(acc);
    }
    
    }
    using extract::moment;                  // Pull the extractor into the
                                            // enclosing namespace.
    }}                                            

The return type deserves some explanation. Every _accumulator_set_ type
is actually a unary [@../../libs/mpl/doc/refmanual/metafunction-class.html
MPL Metafunction Class]. When you `mpl::apply<>` an _accumulator_set_ and
a feature, the result is the type of the accumulator within the set that
implements that feature. And every accumulator provides a nested `result_type`
typedef that tells what its return type is. The extractor simply delegates
its work to the _extract_result_ function.

[endsect]

[section Controlling Dependencies]

The feature-based dependency resolution of the Accumulators Framework is
designed to allow multiple different implementation strategies for each
feature. For instance, two different accumulators may calculate the same
quantity with different rounding modes, or using different algorithms with
different size/speed tradeoffs. Other accumulators that depend on that
quantity shouldn't care how it's calculated. The Accumulators Framework
handles this by allowing several different accumulators satisfy the same
feature.

[*Aliasing feature dependencies with [^feature_of<>]]

Imagine that you would like to implement the hypothetical ['fubar] statistic,
and that you know two ways to calculate fubar on a bunch of samples: an
accurate but slow calculation and an approximate but fast calculation. You
might opt to make the accurate calculation the default, so you implement
two accumulators and call them `impl::fubar_impl` and `impl::fast_fubar_impl`.
You would also define the `tag::fubar` and `tag::fast_fubar` features as described
[link accumulators.user_s_guide.the_accumulators_framework.extending_the_accumulators_framework.defining_a_new_feature above].
Now, you would like to inform the Accumulators Framework that these two features
are the same from the point of view of dependency resolution. You can do that
with _feature_of_, as follows:

    namespace boost { namespace accumulators
    {
        // For the purposes of feature-based dependency resolution,
        // fast_fubar provides the same feature as fubar
        template<>
        struct feature_of<tag::fast_fubar>
          : feature_of<tag::fubar>
        {
        };
    }}

The above code instructs the Accumulators Framework that, if another accumulator
in the set depends on the `tag::fubar` feature, the `tag::fast_fubar` feature
is an acceptable substitute.

[*Registering feature variants with [^as_feature<>]]

You may hve noticed that some feature variants in the Accumulators Framework can be
specified with a nicer syntax. For instance, instead of `tag::mean` and `tag::immediate_mean`
you can specify them with `tag::mean(lazy)` and `tag::mean(immediate)` respectively.
These are merely aliases, but the syntax makes the relationship between the two clearer.
You can create these feature aliases with the _as_feature_ trait. Given the fubar example
above, you might decide to alias `tag::fubar(accurate)` with `tag::fubar` and
`tag::fubar(fast)` with `tag::fast_fubar`. You would do that as follows:

    namespace boost { namespace accumulators
    {
        struct fast {};     // OK to leave these tags empty
        struct accurate {};
        
        template<>
        struct as_feature<tag::fubar(accurate)>
        {
            typedef tag::fubar type;
        };
        
        template<>
        struct as_feature<tag::fubar(fast)>
        {
            typedef tag::fast_fubar type;
        };
    }}

Once you have done this, users of your fubar accumulator can request the `tag::fubar(fast)` 
and `tag::fubar(accurate)` features when defining their `accumulator_set`s and get the correct
accumulator.

[endsect]

[section:operators_ex Specializing Numeric Operators]

This section describes how to adapt third-party numeric types to work with the Accumulator
Framework.

Rather than relying on the built-in operators, the Accumulators Framework relies on functions
and operator overloads defined in the
[link accumulators.user_s_guide.the_accumulators_framework.numeric_operators_sub_library 
Numeric Operators Sub-Library] for many of its numeric operations. This is so that it
is possible to assign non-standard meanings to arithmetic operations. For instance, when
calculating an average by dividing two integers, the standard integer division behavior
would be mathematically incorrect for most statistical quantities. So rather than use `x / y`,
the Accumulators Framework uses `numeric::average(x, y)`, which does floating-point division
even if both `x` and `y` are integers.

Another example where the Numeric Operators Sub-Library is useful is when a type does not
define the operator overloads required to use it for some statistical calculations. For instance, 
`std::vector<>` does not overload any arithmetic operators, yet it may be useful to use
`std::vector<>` as a sample or variate type. The Numeric Operators Sub-Library
defines the necessary operator overloads in the `boost::numeric::operators` namespace,
which is brought into scope by the Accumulators Framework with a using directive. 

[*Numeric Function Objects and Tag Dispatching]

How are the numeric function object defined by the Numeric Operators Sub-Library made
to work with types such as `std::vector<>`? The free functions in the `boost::numeric` namespace
are implemented in terms of the function objects in the `boost::numeric::functional` namespace,
so to make `boost::numeric::average()` do something sensible with a `std::vector<>`, for instance,
we'll need to partially specialize the `boost::numeric::functional::average<>` function object.

The functional objects make use of a technique known as 
[@http://www.boost.org/community/generic_programming.html#tag_dispatching ['tag dispatching]] to
select the proper implementation for the given operands. It works as follows:

    namespace boost { namespace numeric { namespace functional
    {
        // Metafunction for looking up the tag associated with
        // a given numeric type T.
        template<typename T>
        struct tag
        {
            // by default, all types have void as a tag type
            typedef void type;
        };
    
        // Forward declaration looks up the tag types of each operand
        template<
            typename Left
          , typename Right
          , typename LeftTag = typename tag<Left>::type
          , typename RightTag = typename tag<Right>::type
        >
        struct average;
    }}}
    
If you have some user-defined type `MyDouble` for which you would like to customimze the behavior
of `numeric::average()`, you would specialize `numeric::functional::average<>` by
first defining a tag type, as shown below:

    namespace boost { namespace numeric { namespace functional
    {
        // Tag type for MyDouble
        struct MyDoubleTag {};
        
        // Specialize tag<> for MyDouble.
        // This only needs to be done once.
        template<>
        struct tag<MyDouble>
        {
            typedef MyDoubleTag type;
        };

        // Specify how to divide a MyDouble by an integral count
        template<typename Left, typename Right>
        struct average<Left, Right, MyDoubleTag, void>
        {
            // Define the type of the result
            typedef ... result_type;
            
            result_type operator()(Left & left, Right & right) const
            {
                return ...;
            }
        };
    }}}
    
Once you have done this, `numeric::average()` will use your specialization 
of `numeric::functional::average<>` when the first argument is a `MyDouble`
object. All of the function objects in the Numeric Operators Sub-Library can
be customized in a similar fashion.

[endsect]

[endsect]

[section Concepts]

[h2 Accumulator Concept]

In the following table, `Acc` is the type of an accumulator, `acc` and `acc2` are objects of type
`Acc`, and `args` is the name of an argument pack from the _parameter_ library.

[table Accumulator Requirements
    [[[*Expression]]                 [[*Return type]]       [[*Assertion / Note /
                                                               Pre- / Post-condition]]]
    [[`Acc::result_type`]            [['implementation
                                       defined]]            [The type returned by `Acc::result()`.]]
    [[`Acc acc(args)`]               [none]                 [Construct from an argument pack.]]
    [[`Acc acc(acc2)`]               [none]                 [Post: `acc.result(args)` is equivalent
                                                             to `acc2.result(args)`]]
    [[`acc(args)`]                   [['unspecified]]       []]
    [[`acc.on_drop(args)`]           [['unspecified]]       []]
    [[`acc.result(args)`]            [`Acc::result_type`]   []]
]

[h2 Feature Concept]

In the following table, `F` is the type of a feature and `S` is some scalar type.

[table Featue Requirements
    [[[*Expression]]                 [[*Return type]]       [[*Assertion / Note /
                                                               Pre- / Post-condition]]]
    [[`F::dependencies`]             [['unspecified]]       [An MPL sequence of other features on
                                                             which which `F` depends.]]
    [[`F::is_weight_accumulator`]    [`mpl::true_` or
                                      `mpl::false_`]        [`mpl::true_` if the accumulator for
                                                             this feature should be made external
                                                             when the weight type for the accumulator
                                                             set is `external<S>`, `mpl::false_`
                                                             otherwise.]]
    [[`F::impl`]                     [['unspecified]]       [An _mpl_lambda_expression_ that
                                                             returns the type of the accumulator that
                                                             implements this feature when passed a
                                                             sample type and a weight type.]]
]

[endsect]

[endsect]

[section The Statistical Accumulators Library]

The Statistical Accumulators Library defines accumulators for incremental statistial
computations. It is built on top of [link accumulators.user_s_guide.the_accumulators_framework
The Accumulator Framework].

[section:count count]

The `count` feature is a simple counter that tracks the
number of samples pushed into the accumulator set.

[variablelist
    [[Result Type] [``
                    std::size_t
                    ``]]
    [[Depends On] [['none]]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _COUNT_HPP_ [headerref boost/accumulators/statistics/count.hpp]]

    #include <_COUNT_HPP_>

[*Example]

    accumulator_set<int, features<tag::count> > acc;
    acc(0);
    acc(0);
    acc(0);
    assert(3 == count(acc));

[*See also]

* [classref boost::accumulators::impl::count_impl `count_impl`]

[endsect]

[section:covariance covariance]

The `covariance` feature is an iterative Monte Carlo estimator for the covariance.
It is specified as `tag::covariance<_variate_type_, _variate_tag_>`.

[variablelist
    [[Result Type] [``
                    numeric::functional::outer_product<
                        numeric::functional::average<_sample_type_, std::size_t>::result_type
                      , numeric::functional::average<_variate_type_, std::size_t>::result_type
                    >::result_type
                    ``]]
    [[Depends On] [`count` \n `mean` \n `mean_of_variates<_variate_type_, _variate_tag_>`]]
    [[Variants] [`abstract_covariance`]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [[~variate-tag]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _COVARIANCE_HPP_ [headerref boost/accumulators/statistics/covariance.hpp]]

    #include <_COVARIANCE_HPP_>

[*Example]

    accumulator_set<double, stats<tag::covariance<double, tag::covariate1> > > acc;
    acc(1., covariate1 = 2.);
    acc(1., covariate1 = 4.);
    acc(2., covariate1 = 3.);
    acc(6., covariate1 = 1.);
    assert(covariance(acc) == -1.75);

[*See also]

* [classref boost::accumulators::impl::covariance_impl [^covariance_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.mean [^mean]]

[endsect]

[section:density density]

The `tag::density` feature returns a histogram of the sample distribution. For more
implementation details, see [classref boost::accumulators::impl::density_impl [^density_impl]].

[variablelist
    [[Result Type] [``
                    iterator_range<
                        std::vector<
                            std::pair<
                                numeric::functional::average<_sample_type_, std::size_t>::result_type
                              , numeric::functional::average<_sample_type_, std::size_t>::result_type
                            >
                        >::iterator
                    >
                    ``]]
    [[Depends On] [`count` \n `min` \n `max`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [`density::cache_size` \n `density::num_bins`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(N), when N is `density::num_bins`]] 
]

[*Header]
[def _DENSITY_HPP_ [headerref boost/accumulators/statistics/density.hpp]]

    #include <_DENSITY_HPP_>

[/ TODO add example ]

[*See also]

* [classref boost::accumulators::impl::density_impl [^density_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.min [^min]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.max [^max]]

[endsect]

[section:error_of_mean error_of<mean>]

The `error_of<mean>` feature calculates the error of the mean feature. It is equal to
`sqrt(variance / (count - 1))`.

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`count` \n `variance`]]
    [[Variants] [`error_of<immediate_mean>`]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _ERROR_OF_HPP_      [headerref boost/accumulators/statistics/error_of.hpp]]
[def _ERROR_OF_MEAN_HPP_ [headerref boost/accumulators/statistics/error_of_mean.hpp]]

    #include <_ERROR_OF_HPP_>
    #include <_ERROR_OF_MEAN_HPP_>

[*Example]

    accumulator_set<double, stats<tag::error_of<tag::mean> > > acc;
    acc(1.1);
    acc(1.2);
    acc(1.3);
    assert(0.057735 == error_of<tag::mean>(acc));

[*See also]

* [classref boost::accumulators::impl::error_of_mean_impl [^error_of_mean_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.variance [^variance]]

[endsect]

[section:extended_p_square extended_p_square]

Multiple quantile estimation with the extended [^P^2] algorithm. For further 
details, see [classref boost::accumulators::impl::extended_p_square_impl [^extended_p_square_impl]].

[variablelist
    [[Result Type] [``
                    boost::iterator_range<
                        _implementation_defined_
                    >
                    ``]]
    [[Depends On] [`count`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [`tag::extended_p_square::probabilities`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _EXTENDED_P_SQUARE_HPP_ [headerref boost/accumulators/statistics/extended_p_square.hpp]]

    #include <_EXTENDED_P_SQUARE_HPP_>

[*Example]

    boost::array<double> probs = {0.001,0.01,0.1,0.25,0.5,0.75,0.9,0.99,0.999};
    accumulator_set<double, stats<tag::extended_p_square> >
        acc(tag::extended_p_square::probabilities = probs);

    boost::lagged_fibonacci607 rng; // a random number generator
    for (int i=0; i<10000; ++i)
        acc(rng());

    BOOST_CHECK_CLOSE(extended_p_square(acc)[0], probs[0], 25);
    BOOST_CHECK_CLOSE(extended_p_square(acc)[1], probs[1], 10);
    BOOST_CHECK_CLOSE(extended_p_square(acc)[2], probs[2], 5);
    
    for (std::size_t i=3; i < probs.size(); ++i)
    {
        BOOST_CHECK_CLOSE(extended_p_square(acc)[i], probs[i], 2);
    }

[*See also]

* [classref boost::accumulators::impl::extended_p_square_impl [^extended_p_square_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]

[endsect]

[section:extended_p_square_quantile extended_p_square_quantile ['and variants]]

Quantile estimation using the extended [^P^2] algorithm for weighted and unweighted samples.
By default, the calculation is linear and unweighted, but quadratic and weighted variants 
are also provided. For further implementation details, see 
[classref boost::accumulators::impl::extended_p_square_quantile_impl [^extended_p_square_quantile_impl]].

All the variants share the `tag::quantile` feature and can be extracted using the `quantile()`
extractor.

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [weighted variants depend on `weighted_extended_p_square` \n
                   unweighted variants depend on `extended_p_square`]]
    [[Variants] [`extended_p_square_quantile_quadratic` \n 
                 `weighted_extended_p_square_quantile` \n 
                 `weighted_extended_p_square_quantile_quadratic`]]
    [[Initialization Parameters] [`tag::extended_p_square::probabilities`]]
    [[Accumulator Parameters] [`weight` for the weighted variants]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(N) where N is the count of probabilities.]] 
]

[*Header]
[def _EXTENDED_P_SQUARE_QUANTILE_HPP_ [headerref boost/accumulators/statistics/extended_p_square_quantile.hpp]]

    #include <_EXTENDED_P_SQUARE_QUANTILE_HPP_>

[*Example]

    typedef accumulator_set<double, stats<tag::extended_p_square_quantile> >
        accumulator_t;
    typedef accumulator_set<double, stats<tag::weighted_extended_p_square_quantile>, double >
        accumulator_t_weighted;
    typedef accumulator_set<double, stats<tag::extended_p_square_quantile(quadratic)> >
        accumulator_t_quadratic;
    typedef accumulator_set<double, stats<tag::weighted_extended_p_square_quantile(quadratic)>, double >
        accumulator_t_weighted_quadratic;

    // tolerance
    double epsilon = 1;

    // a random number generator
    boost::lagged_fibonacci607 rng;

    boost::array<double> probs = { 0.990, 0.991, 0.992, 0.993, 0.994, 
                                   0.995, 0.996, 0.997, 0.998, 0.999 };
    accumulator_t acc(extended_p_square_probabilities = probs);
    accumulator_t_weighted acc_weighted(extended_p_square_probabilities = probs);
    accumulator_t_quadratic acc2(extended_p_square_probabilities = probs);
    accumulator_t_weighted_quadratic acc_weighted2(extended_p_square_probabilities = probs);
    
    for (int i=0; i<10000; ++i)
    {
        double sample = rng();
        acc(sample);
        acc2(sample);
        acc_weighted(sample, weight = 1.);
        acc_weighted2(sample, weight = 1.);
    }
    
    for (std::size_t i = 0; i < probs.size() - 1; ++i)
    {
        BOOST_CHECK_CLOSE(
            quantile(acc, quantile_probability = 0.99025 + i*0.001)
          , 0.99025 + i*0.001
          , epsilon
        );
        BOOST_CHECK_CLOSE(
            quantile(acc2, quantile_probability = 0.99025 + i*0.001)
          , 0.99025 + i*0.001
          , epsilon
        );
        BOOST_CHECK_CLOSE(
            quantile(acc_weighted, quantile_probability = 0.99025 + i*0.001)
          , 0.99025 + i*0.001
          , epsilon
        );
        BOOST_CHECK_CLOSE(
            quantile(acc_weighted2, quantile_probability = 0.99025 + i*0.001)
          , 0.99025 + i*0.001
          , epsilon
        );
    }

[*See also]

* [classref boost::accumulators::impl::extended_p_square_quantile_impl [^extended_p_square_quantile_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.extended_p_square [^extended_p_square]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_extended_p_square [^weighted_extended_p_square]]

[endsect]

[section:kurtosis kurtosis]

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 [^-3] is added
in order to ensure that the normal distribution has zero kurtosis. For more implementation 
details, see [classref boost::accumulators::impl::kurtosis_impl [^kurtosis_impl]]

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, _sample_type_>::result_type
                    ``]]
    [[Depends On] [`mean` \n `moment<2>` \n `moment<3>` \n `moment<4>`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _KURTOSIS_HPP_ [headerref boost/accumulators/statistics/kurtosis.hpp]]

    #include <_KURTOSIS_HPP_>

[*Example]

    accumulator_set<int, stats<tag::kurtosis > > acc;
        
    acc(2);
    acc(7);
    acc(4);
    acc(9);
    acc(3);
    
    BOOST_CHECK_EQUAL( mean(acc), 5 );
    BOOST_CHECK_EQUAL( moment<2>(acc), 159./5. );
    BOOST_CHECK_EQUAL( moment<3>(acc), 1171./5. );
    BOOST_CHECK_EQUAL( moment<4>(acc), 1863 );
    BOOST_CHECK_CLOSE( kurtosis(acc), -1.39965397924, 1e-6 );

[*See also]

* [classref boost::accumulators::impl::kurtosis_impl [^kurtosis_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.mean [^mean]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.moment [^moment]]

[endsect]

[section:max max]

Calculates the maximum value of all the samples.

[variablelist
    [[Result Type] [``
                    _sample_type_
                    ``]]
    [[Depends On] [['none]]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _MAX_HPP_ [headerref boost/accumulators/statistics/max.hpp]]

    #include <_MAX_HPP_>

[*Example]

    accumulator_set<int, stats<tag::max> > acc;

    acc(1);
    BOOST_CHECK_EQUAL(1, (max)(acc));

    acc(0);
    BOOST_CHECK_EQUAL(1, (max)(acc));

    acc(2);
    BOOST_CHECK_EQUAL(2, (max)(acc));

[*See also]

* [classref boost::accumulators::impl::max_impl [^max_impl]]

[endsect]

[section:mean mean ['and variants]]

Calculates the mean of samples, weights or variates. The calculation is either
lazy (in the result extractor), or immediate (in the accumulator). The lazy implementation
is the default. For more implementation details, see 
[classref boost::accumulators::impl::mean_impl [^mean_impl]] or.
[classref boost::accumulators::impl::immediate_mean_impl [^immediate_mean_impl]]

[variablelist
    [[Result Type] [For samples, `numeric::functional::average<_sample_type_, std::size_t>::result_type` \n
                    For weights, `numeric::functional::average<_weight_type_, std::size_t>::result_type` \n
                    For variates, `numeric::functional::average<_variate_type_, std::size_t>::result_type`]]
    [[Depends On] [`count` \n
                   The lazy mean of samples depends on `sum` \n
                   The lazy mean of weights depends on `sum_of_weights` \n
                   The lazy mean of variates depends on `sum_of_variates<>`]]
    [[Variants] [`mean_of_weights` \n
                 `mean_of_variates<_variate_type_, _variate_tag_>` \n
                 `immediate_mean` \n
                 `immediate_mean_of_weights` \n
                 `immediate_mean_of_variates<_variate_type_, _variate_tag_>`]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _MEAN_HPP_ [headerref boost/accumulators/statistics/mean.hpp]]

    #include <_MEAN_HPP_>

[*Example]

    accumulator_set<
        int
      , stats<
            tag::mean
          , tag::mean_of_weights
          , tag::mean_of_variates<int, tag::covariate1>
        >
      , int 
    > acc;

    acc(1, weight = 2, covariate1 = 3);
    BOOST_CHECK_CLOSE(1., mean(acc), 1e-5);
    BOOST_CHECK_EQUAL(1u, count(acc));
    BOOST_CHECK_EQUAL(2, sum(acc));
    BOOST_CHECK_CLOSE(2., mean_of_weights(acc), 1e-5);
    BOOST_CHECK_CLOSE(3., (mean_of_variates<int, tag::covariate1>(acc)), 1e-5);

    acc(0, weight = 4, covariate1 = 4);
    BOOST_CHECK_CLOSE(0.33333333333333333, mean(acc), 1e-5);
    BOOST_CHECK_EQUAL(2u, count(acc));
    BOOST_CHECK_EQUAL(2, sum(acc));
    BOOST_CHECK_CLOSE(3., mean_of_weights(acc), 1e-5);
    BOOST_CHECK_CLOSE(3.5, (mean_of_variates<int, tag::covariate1>(acc)), 1e-5);

    acc(2, weight = 9, covariate1 = 8);
    BOOST_CHECK_CLOSE(1.33333333333333333, mean(acc), 1e-5);
    BOOST_CHECK_EQUAL(3u, count(acc));
    BOOST_CHECK_EQUAL(20, sum(acc));
    BOOST_CHECK_CLOSE(5., mean_of_weights(acc), 1e-5);
    BOOST_CHECK_CLOSE(5., (mean_of_variates<int, tag::covariate1>(acc)), 1e-5);

    accumulator_set<
        int
      , stats<
            tag::mean(immediate)
          , tag::mean_of_weights(immediate)
          , tag::mean_of_variates<int, tag::covariate1>(immediate)
        >
      , int
    > acc2;

    acc2(1, weight = 2, covariate1 = 3);
    BOOST_CHECK_CLOSE(1., mean(acc2), 1e-5);
    BOOST_CHECK_EQUAL(1u, count(acc2));
    BOOST_CHECK_CLOSE(2., mean_of_weights(acc2), 1e-5);
    BOOST_CHECK_CLOSE(3., (mean_of_variates<int, tag::covariate1>(acc2)), 1e-5);
        
    acc2(0, weight = 4, covariate1 = 4);
    BOOST_CHECK_CLOSE(0.33333333333333333, mean(acc2), 1e-5);
    BOOST_CHECK_EQUAL(2u, count(acc2));
    BOOST_CHECK_CLOSE(3., mean_of_weights(acc2), 1e-5);
    BOOST_CHECK_CLOSE(3.5, (mean_of_variates<int, tag::covariate1>(acc2)), 1e-5);
    
    acc2(2, weight = 9, covariate1 = 8);
    BOOST_CHECK_CLOSE(1.33333333333333333, mean(acc2), 1e-5);
    BOOST_CHECK_EQUAL(3u, count(acc2));
    BOOST_CHECK_CLOSE(5., mean_of_weights(acc2), 1e-5);
    BOOST_CHECK_CLOSE(5., (mean_of_variates<int, tag::covariate1>(acc2)), 1e-5);

[*See also]

* [classref boost::accumulators::impl::mean_impl [^mean_impl]]
* [classref boost::accumulators::impl::immediate_mean_impl [^immediate_mean_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]

[endsect]

[section:median median ['and variants]]

Median estimation based on the [^P^2] quantile estimator, the density estimator, or
the [^P^2] cumulative distribution estimator. For more implementation details, see
[classref boost::accumulators::impl::median_impl [^median_impl]], 
[classref boost::accumulators::impl::with_density_median_impl [^with_density_median_impl]], 
and [classref boost::accumulators::impl::with_p_square_cumulative_distribution_median_impl [^with_p_square_cumulative_distribution_median_impl]].

The three median accumulators all satisfy the `tag::median` feature, and can all be
extracted with the `median()` extractor.

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`median` depends on `p_square_quantile_for_median` \n
                   `with_density_median` depends on `count` and `density` \n
                   `with_p_square_cumulative_distribution_median` depends on `p_square_cumulative_distribution`]]
    [[Variants] [`with_density_median` \n
                 `with_p_square_cumulative_distribution_median`]]
    [[Initialization Parameters] [`with_density_median` requires `tag::density::cache_size` and `tag::density::num_bins` \n
                                  `with_p_square_cumulative_distribution_median` requires `tag::p_square_cumulative_distribution::num_cells`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [TODO]] 
]

[*Header]
[def _MEDIAN_HPP_ [headerref boost/accumulators/statistics/median.hpp]]

    #include <_MEDIAN_HPP_>

[*Example]

    // two random number generators
    double mu = 1.;
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma(mu,1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> >
        normal(rng, mean_sigma);
    
    accumulator_set<double, stats<tag::median(with_p_square_quantile) > > acc;
    accumulator_set<double, stats<tag::median(with_density) > > 
        acc_dens( density_cache_size = 10000, density_num_bins = 1000 );
    accumulator_set<double, stats<tag::median(with_p_square_cumulative_distribution) > > 
        acc_cdist( p_square_cumulative_distribution_num_cells = 100 );
        
    for (std::size_t i=0; i<100000; ++i)
    {
        double sample = normal();
        acc(sample);
        acc_dens(sample);
        acc_cdist(sample);
    }
    
    BOOST_CHECK_CLOSE(1., median(acc), 1.);
    BOOST_CHECK_CLOSE(1., median(acc_dens), 1.);
    BOOST_CHECK_CLOSE(1., median(acc_cdist), 3.);

[*See also]

* [classref boost::accumulators::impl::median_impl [^median_impl]]
* [classref boost::accumulators::impl::with_density_median_impl [^with_density_median_impl]]
* [classref boost::accumulators::impl::with_p_square_cumulative_distribution_median_impl [^with_p_square_cumulative_distribution_median_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.p_square_quantile [^p_square_quantile]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.p_square_cumulative_distribution [^p_square_cumulative_distribution]]

[endsect]

[section:min min]

Calculates the minimum value of all the samples.

[variablelist
    [[Result Type] [``
                    _sample_type_
                    ``]]
    [[Depends On] [['none]]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _MIN_HPP_ [headerref boost/accumulators/statistics/min.hpp]]

    #include <_MIN_HPP_>

[*Example]

    accumulator_set<int, stats<tag::min> > acc;

    acc(1);
    BOOST_CHECK_EQUAL(1, (min)(acc));

    acc(0);
    BOOST_CHECK_EQUAL(0, (min)(acc));

    acc(2);
    BOOST_CHECK_EQUAL(0, (min)(acc));

[*See also]

* [classref boost::accumulators::impl::min_impl [^min_impl]]

[endsect]

[section:moment moment]

Calculates the N-th moment of the samples, which is defined as the sum of the N-th power of the
samples over the count of samples.

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`count`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _MOMENT_HPP_ [headerref boost/accumulators/statistics/moment.hpp]]

    #include <_MOMENT_HPP_>

[*Example]

    accumulator_set<int, stats<tag::moment<2> > > acc1;

    acc1(2); //    4
    acc1(4); //   16
    acc1(5); // + 25
             // = 45 / 3 = 15

    BOOST_CHECK_CLOSE(15., moment<2>(acc1), 1e-5);

    accumulator_set<int, stats<tag::moment<5> > > acc2;

    acc2(2); //     32
    acc2(3); //    243
    acc2(4); //   1024
    acc2(5); // + 3125
             // = 4424 / 4 = 1106

    BOOST_CHECK_CLOSE(1106., moment<5>(acc2), 1e-5);

[*See also]

* [classref boost::accumulators::impl::moment_impl [^moment_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]

[endsect]

[section:p_square_cumulative_distribution p_square_cumulative_distribution]

Histogram calculation of the cumulative distribution with the [^P^2] algorithm.
For more implementation details, see 
[classref boost::accumulators::impl::p_square_cumulative_distribution_impl [^p_square_cumulative_distribution_impl]]

[variablelist
    [[Result Type] [``
                    iterator_range<
                        std::vector<
                            std::pair<
                                numeric::functional::average<_sample_type_, std::size_t>::result_type
                              , numeric::functional::average<_sample_type_, std::size_t>::result_type
                            >
                        >::iterator
                    >
                    ``]]
    [[Depends On] [`count`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [`tag::p_square_cumulative_distribution::num_cells`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(N) where N is `num_cells`]] 
]

[*Header]
[def _P_SQUARE_CUMULATIVE_DISTRIBUTION_HPP_ [headerref boost/accumulators/statistics/p_square_cumulative_distribution.hpp]]

    #include <_P_SQUARE_CUMULATIVE_DISTRIBUTION_HPP_>

[*Example]

    // tolerance in %
    double epsilon = 3;

    typedef accumulator_set<double, stats<tag::p_square_cumulative_distribution> > accumulator_t;
        
    accumulator_t acc(tag::p_square_cumulative_distribution::num_cells = 100);
    
    // two random number generators
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma(0,1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal(rng, mean_sigma);
    
    for (std::size_t i=0; i<100000; ++i)
    {
        acc(normal());
    }
    
    typedef iterator_range<std::vector<std::pair<double, double> >::iterator > histogram_type;
    histogram_type histogram = p_square_cumulative_distribution(acc);
    
    for (std::size_t i = 0; i < histogram.size(); ++i)
    {   
        // problem with small results: epsilon is relative (in percent), not absolute!
        if ( histogram[i].second > 0.001 )    
            BOOST_CHECK_CLOSE( 0.5 * (1.0 + erf( histogram[i].first / sqrt(2.0) )), histogram[i].second, epsilon );
    }        

[*See also]

* [classref boost::accumulators::impl::p_square_cumulative_distribution_impl [^p_square_cumulative_distribution_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]

[endsect]

[section:p_square_quantile p_square_quantile ['and variants]]

Single quantile estimation with the [^P^2] algorithm. For more implementation details, see
[classref boost::accumulators::impl::p_square_quantile_impl [^p_square_quantile_impl]]

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`count`]]
    [[Variants] [`p_square_quantile_for_median`]]
    [[Initialization Parameters] [`quantile_probability`, which defaults to `0.5`.
                                  (Note: for `p_square_quantile_for_median`, the `quantile_probability`
                                  parameter is ignored and is always `0.5`.)]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _P_SQUARE_QUANTILE_HPP_ [headerref boost/accumulators/statistics/p_square_quantile.hpp]]

    #include <_P_SQUARE_QUANTILE_HPP_>

[*Example]

    typedef accumulator_set<double, stats<tag::p_square_quantile> > accumulator_t;

    // tolerance in %
    double epsilon = 1;
    
    // a random number generator
    boost::lagged_fibonacci607 rng;
    
    accumulator_t acc0(quantile_probability = 0.001);
    accumulator_t acc1(quantile_probability = 0.01 );
    accumulator_t acc2(quantile_probability = 0.1  );
    accumulator_t acc3(quantile_probability = 0.25 );
    accumulator_t acc4(quantile_probability = 0.5  );
    accumulator_t acc5(quantile_probability = 0.75 );
    accumulator_t acc6(quantile_probability = 0.9  );
    accumulator_t acc7(quantile_probability = 0.99 );
    accumulator_t acc8(quantile_probability = 0.999);
    
    for (int i=0; i<100000; ++i)
    {
        double sample = rng();
        acc0(sample);
        acc1(sample);
        acc2(sample);
        acc3(sample);
        acc4(sample);
        acc5(sample);
        acc6(sample);
        acc7(sample);
        acc8(sample);
    }
    
    BOOST_CHECK_CLOSE( p_square_quantile(acc0), 0.001, 15*epsilon );
    BOOST_CHECK_CLOSE( p_square_quantile(acc1), 0.01 , 5*epsilon );
    BOOST_CHECK_CLOSE( p_square_quantile(acc2), 0.1  , epsilon );
    BOOST_CHECK_CLOSE( p_square_quantile(acc3), 0.25 , epsilon );
    BOOST_CHECK_CLOSE( p_square_quantile(acc4), 0.5  , epsilon );
    BOOST_CHECK_CLOSE( p_square_quantile(acc5), 0.75 , epsilon );
    BOOST_CHECK_CLOSE( p_square_quantile(acc6), 0.9  , epsilon );
    BOOST_CHECK_CLOSE( p_square_quantile(acc7), 0.99 , epsilon );
    BOOST_CHECK_CLOSE( p_square_quantile(acc8), 0.999, epsilon );    

[*See also]

* [classref boost::accumulators::impl::p_square_quantile_impl [^p_square_quantile_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]

[endsect]

[section:peaks_over_threshold peaks_over_threshold ['and variants]]

Peaks Over Threshold method for quantile and tail mean estimation. For implementation
details, see [classref boost::accumulators::impl::peaks_over_threshold_impl [^peaks_over_threshold_impl]]
and [classref boost::accumulators::impl::peaks_over_threshold_prob_impl [^peaks_over_threshold_prob_impl]].

Both `tag::peaks_over_threshold` and `tag::peaks_over_threshold_prob<>` satisfy the `tag::abstract_peaks_over_threshold`
feature, and can be extracted with the `peaks_over_threshold()` extractor. The result is a 3-tuple representing
the fit parameters `u_bar`, `beta_bar` and `xi_hat`.

[variablelist
    [[Result Type] [``
                    boost::tuple<
                        numeric::functional::average<_sample_type_, std::size_t>::result_type // u_bar
                      , numeric::functional::average<_sample_type_, std::size_t>::result_type // beta_bar
                      , numeric::functional::average<_sample_type_, std::size_t>::result_type // xi_hat
                    >
                    ``]]
    [[Depends On] [`count` \n
                   In addtion, `tag::peaks_over_threshold_prob<>` depends on `tail<_left_or_right_>`]]
    [[Variants] [`peaks_over_threshold_prob<_left_or_right_>`]]
    [[Initialization Parameters] [ `tag::peaks_over_threshold::threshold_value` \n
                                   `tag::peaks_over_threshold_prob::threshold_probability` \n
                                   `tag::tail<_left_or_right_>::cache_size` ]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [TODO]] 
]

[*Header]
[def _PEAKS_OVER_THRESHOLD_HPP_ [headerref boost/accumulators/statistics/peaks_over_threshold.hpp]]

    #include <_PEAKS_OVER_THRESHOLD_HPP_>

[*Example]

See example for [link accumulators.user_s_guide.the_statistical_accumulators_library.pot_quantile [^pot_quantile]].

[*See also]

* [classref boost::accumulators::impl::peaks_over_threshold_impl [^peaks_over_threshold_impl]]
* [classref boost::accumulators::impl::peaks_over_threshold_prob_impl [^peaks_over_threshold_prob_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail [^tail]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.pot_quantile [^pot_quantile]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.pot_tail_mean [^pot_tail_mean]]

[endsect]

[section:pot_quantile pot_quantile ['and variants]]

Quantile estimation based on Peaks over Threshold method (for both left and right tails). For 
implementation details, see [classref boost::accumulators::impl::pot_quantile_impl [^pot_quantile_impl]].

Both `tag::pot_quantile<_left_or_right_>` and `tag::pot_quantile_prob<_left_or_right_>` satisfy the
`tag::quantile` feature and can be extracted using the `quantile()` extractor.

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`pot_quantile<_left_or_right_>` depends on `peaks_over_threshold<_left_or_right_>` \n
                   `pot_quantile_prob<_left_or_right_>` depends on `peaks_over_threshold_prob<_left_or_right_>` ]]
    [[Variants] [`pot_quantile_prob<_left_or_right_>`]]
    [[Initialization Parameters] [ `tag::peaks_over_threshold::threshold_value` \n
                                   `tag::peaks_over_threshold_prob::threshold_probability` \n
                                   `tag::tail<_left_or_right_>::cache_size` ]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [TODO]] 
]

[*Header]
[def _POT_QUANTILE_HPP_ [headerref boost/accumulators/statistics/pot_quantile.hpp]]

    #include <_POT_QUANTILE_HPP_>

[*Example]

    // tolerance in %
    double epsilon = 1.;

    double alpha = 0.999;
    double threshold_probability = 0.99;
    double threshold = 3.;
    
    // two random number generators
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma(0,1);
    boost::exponential_distribution<> lambda(1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal(rng, mean_sigma);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::exponential_distribution<> > exponential(rng, lambda);
    
    accumulator_set<double, stats<tag::pot_quantile<right>(with_threshold_value)> > acc1(
        tag::peaks_over_threshold::threshold_value = threshold
    );
    accumulator_set<double, stats<tag::pot_quantile<right>(with_threshold_probability)> > acc2(
        tag::tail<right>::cache_size = 2000
      , tag::peaks_over_threshold_prob::threshold_probability = threshold_probability
    );
    
    threshold_probability = 0.995;
    threshold = 5.;
    
    accumulator_set<double, stats<tag::pot_quantile<right>(with_threshold_value)> > acc3(
        tag::peaks_over_threshold::threshold_value = threshold
    );
    accumulator_set<double, stats<tag::pot_quantile<right>(with_threshold_probability)> > acc4(
        tag::tail<right>::cache_size = 2000
      , tag::peaks_over_threshold_prob::threshold_probability = threshold_probability
    );
    
    for (std::size_t i = 0; i < 100000; ++i)
    {
        double sample = normal();
        acc1(sample);
        acc2(sample);
    }
    
    for (std::size_t i = 0; i < 100000; ++i)
    {
        double sample = exponential();
        acc3(sample);
        acc4(sample);
    }
    
    BOOST_CHECK_CLOSE( quantile(acc1, quantile_probability = alpha), 3.090232, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc2, quantile_probability = alpha), 3.090232, epsilon );  
    
    BOOST_CHECK_CLOSE( quantile(acc3, quantile_probability = alpha), 6.908, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc4, quantile_probability = alpha), 6.908, epsilon );  

[*See also]

* [classref boost::accumulators::impl::pot_quantile_impl [^pot_quantile_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.peaks_over_threshold [^peaks_over_threshold]]

[endsect]

[section:pot_tail_mean pot_tail_mean]

Estimation of the (coherent) tail mean based on the peaks over threshold method (for both left and right tails).
For inplementation details, see [classref boost::accumulators::impl::pot_tail_mean_impl [^pot_tail_mean_impl]].

Both `tag::pot_tail_mean<_left_or_right_>` and `tag::pot_tail_mean_prob<_left_or_right_>` satisfy the
`tag::tail_mean` feature and can be extracted using the `tail_mean()` extractor.

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`pot_tail_mean<_left_or_right_>` depends on `peaks_over_threshold<_left_or_right_>`
                         and `pot_quantile<_left_or_right_>` \n
                   `pot_tail_mean_prob<_left_or_right_>` depends on `peaks_over_threshold_prob<_left_or_right_>`
                         and `pot_quantile_prob<_left_or_right_>` ]]
    [[Variants] [`pot_tail_mean_prob<_left_or_right_>`]]
    [[Initialization Parameters] [ `tag::peaks_over_threshold::threshold_value` \n
                                   `tag::peaks_over_threshold_prob::threshold_probability` \n
                                   `tag::tail<_left_or_right_>::cache_size` ]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [TODO]] 
]

[*Header]
[def _POT_TAIL_MEAN_HPP_ [headerref boost/accumulators/statistics/pot_tail_mean.hpp]]

    #include <_POT_TAIL_MEAN_HPP_>

[*Example]

    // TODO

[*See also]

* [classref boost::accumulators::impl::pot_tail_mean_impl [^pot_tail_mean_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.peaks_over_threshold [^peaks_over_threshold]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.pot_quantile [^pot_quantile]]

[endsect]

[section:skewness skewness]

The skewness of a sample distribution is defined as the ratio of the 3rd central moment and the [^3/2]-th power 
of the 2nd central moment (the variance) of the sampless 3. For implementation details, see
[classref boost::accumulators::impl::skewness_impl [^skewness_impl]].

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, _sample_type_>::result_type
                    ``]]
    [[Depends On] [`mean` \n `moment<2>` \n `moment<3>`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _SKEWNESS_HPP_ [headerref boost/accumulators/statistics/skewness.hpp]]

    #include <_SKEWNESS_HPP_>

[*Example]

    accumulator_set<int, stats<tag::skewness > > acc2;
    
    acc2(2);
    acc2(7);
    acc2(4);
    acc2(9);
    acc2(3);
    
    BOOST_CHECK_EQUAL( mean(acc2), 5 );
    BOOST_CHECK_EQUAL( moment<2>(acc2), 159./5. );
    BOOST_CHECK_EQUAL( moment<3>(acc2), 1171./5. );
    BOOST_CHECK_CLOSE( skewness(acc2), 0.406040288214, 1e-6 );

[*See also]

* [classref boost::accumulators::impl::skewness_impl [^skewness_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.mean [^mean]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.moment [^moment]]

[endsect]

[section:sum sum ['and variants]]

For summing the samples, weights or variates. 

[variablelist
    [[Result Type] [`_sample_type_` for summing samples \n
                    `_weight_type_` for summing weights \n
                    `_variate_type_` for summing variates]]
    [[Depends On] [['none]]]
    [[Variants] [`tag::sum` \n
                 `tag::sum_of_weights` \n
                 `tag::sum_of_variates<_variate_type_, _variate_tag_>`]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [`weight` for summing weights \n
                               `_variate_tag_` for summing variates]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _SUM_HPP_ [headerref boost/accumulators/statistics/sum.hpp]]

    #include <_SUM_HPP_>

[*Example]

    accumulator_set<
        int
      , stats<
            tag::sum
          , tag::sum_of_weights
          , tag::sum_of_variates<int, tag::covariate1>
        >
      , int
    > acc;

    acc(1, weight = 2, covariate1 = 3);
    BOOST_CHECK_EQUAL(2, sum(acc));  // weighted sample = 1 * 2
    BOOST_CHECK_EQUAL(2, sum_of_weights(acc));
    BOOST_CHECK_EQUAL(3, sum_of_variates(acc));

    acc(2, weight = 4, covariate1 = 6);
    BOOST_CHECK_EQUAL(10, sum(acc)); // weighted sample = 2 * 4
    BOOST_CHECK_EQUAL(6, sum_of_weights(acc));
    BOOST_CHECK_EQUAL(9, sum_of_variates(acc));

    acc(3, weight = 6, covariate1 = 9);
    BOOST_CHECK_EQUAL(28, sum(acc)); // weighted sample = 3 * 6
    BOOST_CHECK_EQUAL(12, sum_of_weights(acc));
    BOOST_CHECK_EQUAL(18, sum_of_variates(acc));

[*See also]

* [classref boost::accumulators::impl::sum_impl [^sum_impl]]

[endsect]

[section:tail tail]

Tracks the largest or smallest [^N] values. `tag::tail<right>` tracks the largest [^N],
and `tag::tail<left>` tracks the smallest. The parameter [^N] is specified with the
`tag::tail<_left_or_right_>::cache_size` initialization parameter. For implementation details, see
[classref boost::accumulators::impl::tail_impl [^tail_impl]].

Both `tag::tail<left>` and `tag::tail<right>` satisfy the `tag::abstract_tail` feature and 
can be extracted with the `tail()` extractor.

[variablelist
    [[Result Type] [``
                    boost::iterator_range<
                        boost::reverse_iterator<
                            boost::permutation_iterator<
                                std::vector<_sample_type_>::const_iterator  // samples
                              , std::vector<std::size_t>::iterator          // indices
                            >
                        >
                    >
                    ``]]
    [[Depends On] [['none]]]
    [[Variants] [`abstract_tail`]]
    [[Initialization Parameters] [`tag::tail<_left_or_right_>::cache_size`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(log N), where N is the cache size]] 
    [[Extractor Complexity] [O(N log N), where N is the cache size]] 
]

[*Header]
[def _TAIL_HPP_ [headerref boost/accumulators/statistics/tail.hpp]]

    #include <_TAIL_HPP_>

[*Example]

See the Example for [link accumulators.user_s_guide.the_statistical_accumulators_library.tail_variate [^tail_variate]].

[*See also]

* [classref boost::accumulators::impl::tail_impl [^tail_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail_variate [^tail_variate]]

[endsect]

[section:coherent_tail_mean coherent_tail_mean]

Estimation of the coherent tail mean based on order statistics (for both left and right tails).
The left coherent tail mean feature is `tag::coherent_tail_mean<left>`, and the right choherent
tail mean feature is `tag::coherent_tail_mean<right>`. They both share the `tag::tail_mean` feature
and can be extracted with the `tail_mean()` extractor. For more implementation details, see
[classref boost::accumulators::impl::coherent_tail_mean_impl [^coherent_tail_mean_impl]]

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`count` \n `quantile` \n `non_coherent_tail_mean<_left_or_right_>`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [`tag::tail<_left_or_right_>::cache_size`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [O(log N), where N is the cache size]] 
    [[Extractor Complexity] [O(N log N), where N is the cache size]] 
]

[*Header]
[def _TAIL_MEAN_HPP_ [headerref boost/accumulators/statistics/tail_mean.hpp]]

    #include <_TAIL_MEAN_HPP_>

[*Example]

See the example for 
[link accumulators.user_s_guide.the_statistical_accumulators_library.non_coherent_tail_mean [^non_coherent_tail_mean]].

[*See also]

* [classref boost::accumulators::impl::coherent_tail_mean_impl [^coherent_tail_mean_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.extended_p_square_quantile [^extended_p_square_quantile]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.pot_quantile [^pot_quantile]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail_quantile [^tail_quantile]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.non_coherent_tail_mean [^non_coherent_tail_mean]]

[endsect]

[section:non_coherent_tail_mean non_coherent_tail_mean]

Estimation of the (non-coherent) tail mean based on order statistics (for both left and right tails).
The left non-coherent tail mean feature is `tag::non_coherent_tail_mean<left>`, and the right non-choherent
tail mean feature is `tag::non_coherent_tail_mean<right>`. They both share the `tag::abstract_non_coherent_tail_mean`
feature and can be extracted with the `non_coherent_tail_mean()` extractor. For more implementation details, see
[classref boost::accumulators::impl::non_coherent_tail_mean_impl [^non_coherent_tail_mean_impl]]

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`count` \n `tail<_left_or_right_>`]]
    [[Variants] [`abstract_non_coherent_tail_mean`]]
    [[Initialization Parameters] [`tag::tail<_left_or_right_>::cache_size`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [O(log N), where N is the cache size]] 
    [[Extractor Complexity] [O(N log N), where N is the cache size]] 
]

[*Header]
[def _TAIL_MEAN_HPP_ [headerref boost/accumulators/statistics/tail_mean.hpp]]

    #include <_TAIL_MEAN_HPP_>

[*Example]

    // tolerance in %
    double epsilon = 1;
    
    std::size_t n = 100000; // number of MC steps
    std::size_t c =  10000; // cache size
    
    typedef accumulator_set<double, stats<tag::non_coherent_tail_mean<right>, tag::tail_quantile<right> > > accumulator_t_right1;
    typedef accumulator_set<double, stats<tag::non_coherent_tail_mean<left>, tag::tail_quantile<left> > > accumulator_t_left1;
    typedef accumulator_set<double, stats<tag::coherent_tail_mean<right>, tag::tail_quantile<right> > > accumulator_t_right2;
    typedef accumulator_set<double, stats<tag::coherent_tail_mean<left>, tag::tail_quantile<left> > > accumulator_t_left2;
    
    accumulator_t_right1 acc0( right_tail_cache_size = c );
    accumulator_t_left1 acc1( left_tail_cache_size = c );
    accumulator_t_right2 acc2( right_tail_cache_size = c );
    accumulator_t_left2 acc3( left_tail_cache_size = c );
    
    // a random number generator
    boost::lagged_fibonacci607 rng;
    
    for (std::size_t i = 0; i < n; ++i)
    {
        double sample = rng();
        acc0(sample);
        acc1(sample);
        acc2(sample);
        acc3(sample);
    }
    
    // check uniform distribution
    BOOST_CHECK_CLOSE( non_coherent_tail_mean(acc0, quantile_probability = 0.95), 0.975, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_tail_mean(acc0, quantile_probability = 0.975), 0.9875, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_tail_mean(acc0, quantile_probability = 0.99), 0.995, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_tail_mean(acc0, quantile_probability = 0.999), 0.9995, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_tail_mean(acc1, quantile_probability = 0.05), 0.025, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_tail_mean(acc1, quantile_probability = 0.025), 0.0125, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_tail_mean(acc1, quantile_probability = 0.01), 0.005, 5 );
    BOOST_CHECK_CLOSE( non_coherent_tail_mean(acc1, quantile_probability = 0.001), 0.0005, 10 );
    BOOST_CHECK_CLOSE( tail_mean(acc2, quantile_probability = 0.95), 0.975, epsilon );
    BOOST_CHECK_CLOSE( tail_mean(acc2, quantile_probability = 0.975), 0.9875, epsilon );
    BOOST_CHECK_CLOSE( tail_mean(acc2, quantile_probability = 0.99), 0.995, epsilon );
    BOOST_CHECK_CLOSE( tail_mean(acc2, quantile_probability = 0.999), 0.9995, epsilon );
    BOOST_CHECK_CLOSE( tail_mean(acc3, quantile_probability = 0.05), 0.025, epsilon );
    BOOST_CHECK_CLOSE( tail_mean(acc3, quantile_probability = 0.025), 0.0125, epsilon );
    BOOST_CHECK_CLOSE( tail_mean(acc3, quantile_probability = 0.01), 0.005, 5 );
    BOOST_CHECK_CLOSE( tail_mean(acc3, quantile_probability = 0.001), 0.0005, 10 );

[*See also]

* [classref boost::accumulators::impl::non_coherent_tail_mean_impl [^non_coherent_tail_mean_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail [^tail]]

[endsect]

[section:tail_quantile tail_quantile]

Tail quantile estimation based on order statistics (for both left and right tails).
The left tail quantile feature is `tag::tail_quantile<left>`, and the right
tail quantile feature is `tag::tail_quantile<right>`. They both share the `tag::quantile`
feature and can be extracted with the `quantile()` extractor. For more implementation details, see
[classref boost::accumulators::impl::tail_quantile_impl [^tail_quantile_impl]]

[variablelist
    [[Result Type] [``
                    _sample_type_
                    ``]]
    [[Depends On] [`count` \n `tail<_left_or_right_>`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [`tag::tail<_left_or_right_>::cache_size`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [O(log N), where N is the cache size]] 
    [[Extractor Complexity] [O(N log N), where N is the cache size]] 
]

[*Header]
[def _TAIL_QUANTILE_HPP_ [headerref boost/accumulators/statistics/tail_quantile.hpp]]

    #include <_TAIL_QUANTILE_HPP_>

[*Example]

    // tolerance in %
    double epsilon = 1;
    
    std::size_t n = 100000; // number of MC steps
    std::size_t c =  10000; // cache size
        
    typedef accumulator_set<double, stats<tag::tail_quantile<right> > > accumulator_t_right;
    typedef accumulator_set<double, stats<tag::tail_quantile<left> > > accumulator_t_left;
    
    accumulator_t_right acc0( tag::tail<right>::cache_size = c );
    accumulator_t_right acc1( tag::tail<right>::cache_size = c );
    accumulator_t_left  acc2( tag::tail<left>::cache_size = c );
    accumulator_t_left  acc3( tag::tail<left>::cache_size = c );
    
    // two random number generators
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma(0,1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal(rng, mean_sigma);
                    
    for (std::size_t i = 0; i < n; ++i)
    {
        double sample1 = rng();
        double sample2 = normal();
        acc0(sample1);
        acc1(sample2);
        acc2(sample1);
        acc3(sample2);
    }
    
    // check uniform distribution
    BOOST_CHECK_CLOSE( quantile(acc0, quantile_probability = 0.95 ), 0.95,  epsilon );
    BOOST_CHECK_CLOSE( quantile(acc0, quantile_probability = 0.975), 0.975, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc0, quantile_probability = 0.99 ), 0.99,  epsilon );
    BOOST_CHECK_CLOSE( quantile(acc0, quantile_probability = 0.999), 0.999, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc2, quantile_probability  = 0.05 ), 0.05,  2 );
    BOOST_CHECK_CLOSE( quantile(acc2, quantile_probability  = 0.025), 0.025, 2 );
    BOOST_CHECK_CLOSE( quantile(acc2, quantile_probability  = 0.01 ), 0.01,  3 );
    BOOST_CHECK_CLOSE( quantile(acc2, quantile_probability  = 0.001), 0.001, 20 );
    
    // check standard normal distribution
    BOOST_CHECK_CLOSE( quantile(acc1, quantile_probability = 0.975),  1.959963, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc1, quantile_probability = 0.999),  3.090232, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc3, quantile_probability  = 0.025), -1.959963, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc3, quantile_probability  = 0.001), -3.090232, epsilon );

[*See also]

* [classref boost::accumulators::impl::tail_quantile_impl [^tail_quantile_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail [^tail]]

[endsect]

[section:tail_variate tail_variate]

Tracks the covariates of largest or smallest [^N] samples. 
`tag::tail_variate<_variate_type_, _variate_tag_, right>` tracks the covariate associated with 
_variate_tag_ for the largest [^N], and `tag::tail_variate<_variate_type_, _variate_tag_, left>`
for the smallest. The parameter [^N] is specified with the `tag::tail<_left_or_right_>::cache_size` 
initialization parameter. For implementation details, see
[classref boost::accumulators::impl::tail_variate_impl [^tail_variate_impl]].

Both `tag::tail_variate<_variate_type_, _variate_tag_, right>` and 
`tag::tail_variate<_variate_type_, _variate_tag_, left>` satisfy the `tag::abstract_tail_variate` feature
and can be extracted with the `tail_variate()` extractor.

[variablelist
    [[Result Type] [``
                    boost::iterator_range<
                        boost::reverse_iterator<
                            boost::permutation_iterator<
                                std::vector<_variate_type_>::const_iterator // variates
                              , std::vector<std::size_t>::iterator          // indices
                            >
                        >
                    >
                    ``]]
    [[Depends On] [`tail<_left_or_right_>`]]
    [[Variants] [`abstract_tail_variate`]]
    [[Initialization Parameters] [`tag::tail<_left_or_right_>::cache_size`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(log N), where N is the cache size]] 
    [[Extractor Complexity] [O(N log N), where N is the cache size]] 
]

[*Header]
[def _TAIL_VARIATE_HPP_ [headerref boost/accumulators/statistics/tail_variate.hpp]]

    #include <_TAIL_VARIATE_HPP_>

[*Example]

    accumulator_set<int, stats<tag::tail_variate<int, tag::covariate1, right> > > acc(
        tag::tail<right>::cache_size = 4
    );
    
    acc(8, covariate1 = 3);
    CHECK_RANGE_EQUAL(tail(acc), {8});
    CHECK_RANGE_EQUAL(tail_variate(acc), {3});
    
    acc(16, covariate1 = 1);
    CHECK_RANGE_EQUAL(tail(acc), {16, 8});
    CHECK_RANGE_EQUAL(tail_variate(acc), {1, 3});

    acc(12, covariate1 = 4);
    CHECK_RANGE_EQUAL(tail(acc), {16, 12, 8});
    CHECK_RANGE_EQUAL(tail_variate(acc), {1, 4, 3});

    acc(24, covariate1 = 5);
    CHECK_RANGE_EQUAL(tail(acc), {24, 16, 12, 8});
    CHECK_RANGE_EQUAL(tail_variate(acc), {5, 1, 4, 3});

    acc(1, covariate1 = 9);
    CHECK_RANGE_EQUAL(tail(acc), {24, 16, 12, 8});
    CHECK_RANGE_EQUAL(tail_variate(acc), {5, 1, 4, 3});

    acc(9, covariate1 = 7);
    CHECK_RANGE_EQUAL(tail(acc), {24,  16, 12, 9});
    CHECK_RANGE_EQUAL(tail_variate(acc), {5, 1, 4, 7});

[*See also]

* [classref boost::accumulators::impl::tail_variate_impl [^tail_variate_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail [^tail]]

[endsect]

[section:tail_variate_means tail_variate_means ['and variants]]

Estimation of the absolute and relative tail variate means (for both left and right tails).
The absolute tail variate means has the feature 
`tag::absolute_tail_variate_means<_left_or_right_, _variate_type_, _variate_tag_>`
and the relative tail variate mean has the feature
`tag::relative_tail_variate_means<_left_or_right_, _variate_type_, _variate_tag_>`. All
absolute tail variate mean features share the `tag::abstract_absolute_tail_variate_means`
feature and can be extracted with the `tail_variate_means()` extractor. All the
relative tail variate mean features share the `tag::abstract_relative_tail_variate_means`
feature and can be extracted with the `relative_tail_variate_means()` extractor.

For more implementation details, see
[classref boost::accumulators::impl::tail_variate_means_impl [^tail_variate_means_impl]]

[variablelist
    [[Result Type] [``
                    boost::iterator_range<
                        std::vector<
                            numeric::functional::average<_sample_type_, std::size_t>::result_type
                        >::iterator
                    >
                    ``]]
    [[Depends On] [`non_coherent_tail_mean<_left_or_right_>` \n 
                   `tail_variate<_variate_type_, _variate_tag_, _left_or_right_>`]]
    [[Variants] [`tag::absolute_tail_variate_means<_left_or_right_, _variate_type_, _variate_tag_>` \n
                 `tag::relative_tail_variate_means<_left_or_right_, _variate_type_, _variate_tag_>`]]
    [[Initialization Parameters] [`tag::tail<_left_or_right_>::cache_size`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [O(log N), where N is the cache size]] 
    [[Extractor Complexity] [O(N log N), where N is the cache size]] 
]

[*Header]
[def _TAIL_VARIATE_MEANS_HPP_ [headerref boost/accumulators/statistics/tail_variate_means.hpp]]

    #include <_TAIL_VARIATE_MEANS_HPP_>

[*Example]

    std::size_t c = 5; // cache size
    
    typedef double variate_type;
    typedef std::vector<variate_type> variate_set_type;
    
    typedef accumulator_set<double, stats<
        tag::tail_variate_means<right, variate_set_type, tag::covariate1>(relative)>, tag::tail<right> >
    accumulator_t1;
    
    typedef accumulator_set<double, stats<
        tag::tail_variate_means<right, variate_set_type, tag::covariate1>(absolute)>, tag::tail<right> >
    accumulator_t2;
    
    typedef accumulator_set<double, stats<
        tag::tail_variate_means<left, variate_set_type, tag::covariate1>(relative)>, tag::tail<left> >
    accumulator_t3;
    
    typedef accumulator_set<double, stats<
        tag::tail_variate_means<left, variate_set_type, tag::covariate1>(absolute)>, tag::tail<left> >
    accumulator_t4;
    
    accumulator_t1 acc1( right_tail_cache_size = c );
    accumulator_t2 acc2( right_tail_cache_size = c );
    accumulator_t3 acc3( left_tail_cache_size = c );
    accumulator_t4 acc4( left_tail_cache_size = c );
    
    variate_set_type cov1, cov2, cov3, cov4, cov5;
    double c1[] = { 10., 20., 30., 40. }; // 100
    double c2[] = { 26.,  4., 17.,  3. }; // 50
    double c3[] = { 46., 64., 40., 50. }; // 200
    double c4[] = {  1.,  3., 70.,  6. }; // 80
    double c5[] = {  2.,  2.,  2., 14. }; // 20
    cov1.assign(c1, c1 + sizeof(c1)/sizeof(variate_type));
    cov2.assign(c2, c2 + sizeof(c2)/sizeof(variate_type));
    cov3.assign(c3, c3 + sizeof(c3)/sizeof(variate_type));
    cov4.assign(c4, c4 + sizeof(c4)/sizeof(variate_type));
    cov5.assign(c5, c5 + sizeof(c5)/sizeof(variate_type));            
    
    acc1(100., covariate1 = cov1);
    acc1( 50., covariate1 = cov2);
    acc1(200., covariate1 = cov3);
    acc1( 80., covariate1 = cov4);
    acc1( 20., covariate1 = cov5);
    
    acc2(100., covariate1 = cov1);
    acc2( 50., covariate1 = cov2);
    acc2(200., covariate1 = cov3);
    acc2( 80., covariate1 = cov4);
    acc2( 20., covariate1 = cov5);
    
    acc3(100., covariate1 = cov1);
    acc3( 50., covariate1 = cov2);
    acc3(200., covariate1 = cov3);
    acc3( 80., covariate1 = cov4);
    acc3( 20., covariate1 = cov5);
    
    acc4(100., covariate1 = cov1);
    acc4( 50., covariate1 = cov2);
    acc4(200., covariate1 = cov3);
    acc4( 80., covariate1 = cov4);
    acc4( 20., covariate1 = cov5);
   
    // check relative risk contributions
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc1, quantile_probability = 0.7).begin()     ), 14./75. ); // (10 + 46) / 300 = 14/75
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc1, quantile_probability = 0.7).begin() + 1),  7./25. ); // (20 + 64) / 300 =  7/25
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc1, quantile_probability = 0.7).begin() + 2),  7./30. ); // (30 + 40) / 300 =  7/30
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc1, quantile_probability = 0.7).begin() + 3),  3./10. ); // (40 + 50) / 300 =  3/10
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc3, quantile_probability = 0.3).begin()    ), 14./35. ); // (26 +  2) /  70 = 14/35
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc3, quantile_probability = 0.3).begin() + 1),  3./35. ); // ( 4 +  2) /  70 =  3/35
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc3, quantile_probability = 0.3).begin() + 2), 19./70. ); // (17 +  2) /  70 = 19/70
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc3, quantile_probability = 0.3).begin() + 3), 17./70. ); // ( 3 + 14) /  70 = 17/70
    
    // check absolute risk contributions
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc2, quantile_probability = 0.7).begin()    ), 28 ); // (10 + 46) / 2 = 28
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc2, quantile_probability = 0.7).begin() + 1), 42 ); // (20 + 64) / 2 = 42
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc2, quantile_probability = 0.7).begin() + 2), 35 ); // (30 + 40) / 2 = 35
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc2, quantile_probability = 0.7).begin() + 3), 45 ); // (40 + 50) / 2 = 45
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc4, quantile_probability = 0.3).begin()    ), 14 ); // (26 +  2) / 2 = 14
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc4, quantile_probability = 0.3).begin() + 1),  3 ); // ( 4 +  2) / 2 =  3
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc4, quantile_probability = 0.3).begin() + 2),9.5 ); // (17 +  2) / 2 =  9.5
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc4, quantile_probability = 0.3).begin() + 3),8.5 ); // ( 3 + 14) / 2 =  8.5
    
    // check relative risk contributions
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc1, quantile_probability = 0.9).begin()    ), 23./100. ); // 46/200 = 23/100
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc1, quantile_probability = 0.9).begin() + 1),  8./25.  ); // 64/200 =  8/25
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc1, quantile_probability = 0.9).begin() + 2),  1./5.   ); // 40/200 =  1/5
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc1, quantile_probability = 0.9).begin() + 3),  1./4.   ); // 50/200 =  1/4
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc3, quantile_probability = 0.1).begin()    ),  1./10.  ); //  2/ 20 =  1/10
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc3, quantile_probability = 0.1).begin() + 1),  1./10.  ); //  2/ 20 =  1/10
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc3, quantile_probability = 0.1).begin() + 2),  1./10.  ); //  2/ 20 =  1/10
    BOOST_CHECK_EQUAL( *(relative_tail_variate_means(acc3, quantile_probability = 0.1).begin() + 3),  7./10.  ); // 14/ 20 =  7/10
    
    // check absolute risk contributions
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc2, quantile_probability = 0.9).begin()    ), 46 ); // 46
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc2, quantile_probability = 0.9).begin() + 1), 64 ); // 64
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc2, quantile_probability = 0.9).begin() + 2), 40 ); // 40
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc2, quantile_probability = 0.9).begin() + 3), 50 ); // 50
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc4, quantile_probability = 0.1).begin()    ),  2 ); //  2
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc4, quantile_probability = 0.1).begin() + 1),  2 ); //  2
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc4, quantile_probability = 0.1).begin() + 2),  2 ); //  2
    BOOST_CHECK_EQUAL( *(tail_variate_means(acc4, quantile_probability = 0.1).begin() + 3), 14 ); // 14

[*See also]

* [classref boost::accumulators::impl::tail_variate_means_impl [^tail_variate_means_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.non_coherent_tail_mean [^non_coherent_tail_mean]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail_variate [^tail_variate]]

[endsect]

[section:variance variance ['and variants]]

Lazy or iterative calculation of the variance. The lazy calculation is associated with the `tag::lazy_variance`
feature, and the iterative calculation with the `tag::variance` feature. Both can be extracted
using the `tag::variance()` extractor. For more implementation details, see
[classref boost::accumulators::impl::lazy_variance_impl [^lazy_variance_impl]] and
[classref boost::accumulators::impl::variance_impl [^variance_impl]]

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`tag::lazy_variance` depends on `tag::moment<2>` and `tag::mean` \n
                   `tag::variance` depends on `tag::count` and `tag::immediate_mean`]]
    [[Variants] [`tag::lazy_variance` (a.k.a. `tag::variance(lazy))` \n
                 `tag::variance` (a.k.a. `tag::variance(immediate)`)]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _VARIANCE_HPP_ [headerref boost/accumulators/statistics/variance.hpp]]

    #include <_VARIANCE_HPP_>

[*Example]

    // lazy variance
    accumulator_set<int, stats<tag::variance(lazy)> > acc1;

    acc1(1);
    acc1(2);
    acc1(3);
    acc1(4);
    acc1(5);

    BOOST_CHECK_EQUAL(5u, count(acc1));
    BOOST_CHECK_CLOSE(3., mean(acc1), 1e-5);
    BOOST_CHECK_CLOSE(11., moment<2>(acc1), 1e-5);
    BOOST_CHECK_CLOSE(2., variance(acc1), 1e-5);

    // immediate variance
    accumulator_set<int, stats<tag::variance> > acc2;

    acc2(1);
    acc2(2);
    acc2(3);
    acc2(4);
    acc2(5);

    BOOST_CHECK_EQUAL(5u, count(acc2));
    BOOST_CHECK_CLOSE(3., mean(acc2), 1e-5);
    BOOST_CHECK_CLOSE(2., variance(acc2), 1e-5);

[*See also]

* [classref boost::accumulators::impl::lazy_variance_impl [^lazy_variance_impl]]
* [classref boost::accumulators::impl::variance_impl [^variance_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.mean [^mean]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.moment [^moment]]

[endsect]

[section:weighted_covariance weighted_covariance]

An iterative Monte Carlo estimator for the weighted covariance. The feature is specified as
`tag::weighted_covariance<_variate_type_, _variate_tag_>` and is extracted with the `weighted_variate()`
extractor. For more implementation details, see 
[classref boost::accumulators::impl::weighted_covariance_impl [^weighted_covariance_impl]]

[variablelist
    [[Result Type] [``
                    numeric::functional::outer_product<
                        numeric::functional::multiplies<
                            _weight_type_
                          , numeric::functional::average<_sample_type_, std::size_t>::result_type
                        >::result_type
                      , numeric::functional::multiplies<
                            _weight_type_
                          , numeric::functional::average<_variate_type_, std::size_t>::result_type
                        >::result_type
                    >
                    ``]]
    [[Depends On] [`count` \n 
                   `sum_of_weights` \n 
                   `weighted_mean` \n 
                   `weighted_mean_of_variates<_variate_type_, _variate_tag_>`]]
    [[Variants] [`abstract_weighted_covariance`]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [`weight` \n
                               `_variate_tag_`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _WEIGHTED_COVARIANCE_HPP_ [headerref boost/accumulators/statistics/weighted_covariance.hpp]]

    #include <_WEIGHTED_COVARIANCE_HPP_>

[*Example]

    accumulator_set<double, stats<tag::weighted_covariance<double, tag::covariate1> >, double > acc;

    acc(1., weight = 1.1, covariate1 = 2.);
    acc(1., weight = 2.2, covariate1 = 4.);
    acc(2., weight = 3.3, covariate1 = 3.);
    acc(6., weight = 4.4, covariate1 = 1.);        

    double epsilon = 1e-6;
    BOOST_CHECK_CLOSE(weighted_covariance(acc), -2.39, epsilon);

[*See also]

* [classref boost::accumulators::impl::weighted_covariance_impl [^weighted_covariance_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_mean [^weighted_mean]]

[endsect]

[section:weighted_density weighted_density]

The `tag::weighted_density` feature returns a histogram of the weighted sample distribution. For more
implementation details, see [classref boost::accumulators::impl::weighted_density_impl [^weighted_density_impl]].

[variablelist
    [[Result Type] [``
                    iterator_range<
                        std::vector<
                            std::pair<
                                numeric::functional::average<_weight_type_, std::size_t>::result_type
                              , numeric::functional::average<_weight_type_, std::size_t>::result_type
                            >
                        >::iterator
                    >
                    ``]]
    [[Depends On] [`count` \n `sum_of_weights` \n `min` \n `max`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [`tag::weighted_density::cache_size` \n `tag::weighted_density::num_bins`]]
    [[Accumulator Parameters] [`weight`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(N), when N is `weighted_density::num_bins`]] 
]

[*Header]
[def _WEIGHTED_DENSITY_HPP_ [headerref boost/accumulators/statistics/weighted_density.hpp]]

    #include <_WEIGHTED_DENSITY_HPP_>

[/ TODO add example ]

[*See also]

* [classref boost::accumulators::impl::weighted_density_impl [^weighted_density_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.min [^min]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.max [^max]]

[endsect]

[section:weighted_extended_p_square weighted_extended_p_square]

Multiple quantile estimation with the extended [^P^2] algorithm for weighted samples. For further 
details, see [classref boost::accumulators::impl::weighted_extended_p_square_impl [^weighted_extended_p_square_impl]].

[variablelist
    [[Result Type] [``
                    boost::iterator_range<
                        _implementation_defined_
                    >
                    ``]]
    [[Depends On] [`count` \n `sum_of_weights`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [`tag::weighted_extended_p_square::probabilities`]]
    [[Accumulator Parameters] [`weight`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _WEIGHTED_EXTENDED_P_SQUARE_HPP_ [headerref boost/accumulators/statistics/weighted_extended_p_square.hpp]]

    #include <_WEIGHTED_EXTENDED_P_SQUARE_HPP_>

[*Example]

    typedef accumulator_set<double, stats<tag::weighted_extended_p_square>, double> accumulator_t;

    // tolerance in %
    double epsilon = 1;
    
    // some random number generators
    double mu1 = -1.0;
    double mu2 =  1.0;
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma1(mu1, 1);
    boost::normal_distribution<> mean_sigma2(mu2, 1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal1(rng, mean_sigma1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal2(rng, mean_sigma2);
    
    std::vector<double> probs_uniform, probs_normal1, probs_normal2, probs_normal_exact1, probs_normal_exact2;
    
    double p1[] = {/*0.001,*/ 0.01, 0.1, 0.5, 0.9, 0.99, 0.999};
    probs_uniform.assign(p1, p1 + sizeof(p1) / sizeof(double));
    
    double p2[] = {0.001, 0.025};
    double p3[] = {0.975, 0.999};
    probs_normal1.assign(p2, p2 + sizeof(p2) / sizeof(double));
    probs_normal2.assign(p3, p3 + sizeof(p3) / sizeof(double));
    
    double p4[] = {-3.090232, -1.959963};
    double p5[] = {1.959963, 3.090232};
    probs_normal_exact1.assign(p4, p4 + sizeof(p4) / sizeof(double));
    probs_normal_exact2.assign(p5, p5 + sizeof(p5) / sizeof(double));
    
    accumulator_t acc_uniform(tag::weighted_extended_p_square::probabilities = probs_uniform);
    accumulator_t acc_normal1(tag::weighted_extended_p_square::probabilities = probs_normal1);
    accumulator_t acc_normal2(tag::weighted_extended_p_square::probabilities = probs_normal2);
    
    for (std::size_t i = 0; i < 100000; ++i)
    {
        acc_uniform(rng(), weight = 1.);
        
        double sample1 = normal1();
        double sample2 = normal2();
        acc_normal1(sample1, weight = std::exp(-mu1 * (sample1 - 0.5 * mu1)));
        acc_normal2(sample2, weight = std::exp(-mu2 * (sample2 - 0.5 * mu2)));
    }
    
    // check for uniform distribution    
    for (std::size_t i = 0; i < probs_uniform.size(); ++i)
    {
        BOOST_CHECK_CLOSE(weighted_extended_p_square(acc_uniform)[i], probs_uniform[i], epsilon);
    }
    
    // check for standard normal distribution
    for (std::size_t i = 0; i < probs_normal1.size(); ++i)
    {
        BOOST_CHECK_CLOSE(weighted_extended_p_square(acc_normal1)[i], probs_normal_exact1[i], epsilon);
        BOOST_CHECK_CLOSE(weighted_extended_p_square(acc_normal2)[i], probs_normal_exact2[i], epsilon);
    }

[*See also]

* [classref boost::accumulators::impl::weighted_extended_p_square_impl [^weighted_extended_p_square_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]

[endsect]

[section:weighted_kurtosis weighted_kurtosis]

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 [^-3] is added
in order to ensure that the normal distribution has zero kurtosis. For more implementation 
details, see [classref boost::accumulators::impl::weighted_kurtosis_impl [^weighted_kurtosis_impl]]

[variablelist
    [[Result Type] [``
                    numeric::functional::average<
                        numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type
                      , numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type
                    >::result_type
                    ``]]
    [[Depends On] [`weighted_mean` \n `weighted_moment<2>` \n `weighted_moment<3>` \n `weighted_moment<4>`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _WEIGHTED_KURTOSIS_HPP_ [headerref boost/accumulators/statistics/weighted_kurtosis.hpp]]

    #include <_WEIGHTED_KURTOSIS_HPP_>

[*Example]

    accumulator_set<int, stats<tag::weighted_kurtosis>, int > acc2;
    
    acc2(2, weight = 4);
    acc2(7, weight = 1);
    acc2(4, weight = 3);
    acc2(9, weight = 1);
    acc2(3, weight = 2);
    
    BOOST_CHECK_EQUAL( weighted_mean(acc2), 42./11. );
    BOOST_CHECK_EQUAL( weighted_moment<2>(acc2), 212./11. );
    BOOST_CHECK_EQUAL( weighted_moment<3>(acc2), 1350./11. );
    BOOST_CHECK_EQUAL( weighted_moment<4>(acc2), 9956./11. );
    BOOST_CHECK_CLOSE( weighted_kurtosis(acc2), 0.58137026432, 1e-6 );

[*See also]

* [classref boost::accumulators::impl::weighted_kurtosis_impl [^weighted_kurtosis_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_mean [^weighted_mean]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_moment [^weighted_moment]]

[endsect]

[section:weighted_mean weighted_mean ['and variants]]

Calculates the weighted mean of samples or variates. The calculation is either
lazy (in the result extractor), or immediate (in the accumulator). The lazy implementation
is the default. For more implementation details, see 
[classref boost::accumulators::impl::weighted_mean_impl [^weighted_mean_impl]] or.
[classref boost::accumulators::impl::immediate_weighted_mean_impl [^immediate_weighted_mean_impl]]

[variablelist
    [[Result Type] [For samples, `numeric::functional::average<numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type, _weight_type_>::result_type` \n
                    For variates, `numeric::functional::average<numeric::functional::multiplies<_variate_type_, _weight_type_>::result_type, _weight_type_>::result_type`]]
    [[Depends On] [`sum_of_weights` \n
                   The lazy mean of samples depends on `weighted_sum` \n
                   The lazy mean of variates depends on `weighted_sum_of_variates<>`]]
    [[Variants] [`weighted_mean_of_variates<_variate_type_, _variate_tag_>` \n
                 `immediate_weighted_mean` \n
                 `immediate_weighted_mean_of_variates<_variate_type_, _variate_tag_>`]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _WEIGHTED_MEAN_HPP_ [headerref boost/accumulators/statistics/weighted_mean.hpp]]

    #include <_WEIGHTED_MEAN_HPP_>

[*Example]

    accumulator_set<
        int
      , stats<
            tag::weighted_mean
          , tag::weighted_mean_of_variates<int, tag::covariate1>
        >
      , int
    > acc;

    acc(10, weight = 2, covariate1 = 7);          //  20
    BOOST_CHECK_EQUAL(2, sum_of_weights(acc));    //
                                                  //
    acc(6, weight = 3, covariate1 = 8);           //  18
    BOOST_CHECK_EQUAL(5, sum_of_weights(acc));    //
                                                  //
    acc(4, weight = 4, covariate1 = 9);           //  16
    BOOST_CHECK_EQUAL(9, sum_of_weights(acc));    //
                                                  //
    acc(6, weight = 5, covariate1 = 6);           //+ 30
    BOOST_CHECK_EQUAL(14, sum_of_weights(acc));   //
                                                  //= 84  / 14 = 6

    BOOST_CHECK_EQUAL(6., weighted_mean(acc));
    BOOST_CHECK_EQUAL(52./7., (weighted_mean_of_variates<int, tag::covariate1>(acc)));

    accumulator_set<
        int
      , stats<
            tag::weighted_mean(immediate)
          , tag::weighted_mean_of_variates<int, tag::covariate1>(immediate)
        >
      , int
    > acc2;

    acc2(10, weight = 2, covariate1 = 7);         //  20
    BOOST_CHECK_EQUAL(2, sum_of_weights(acc2));   //
                                                  //
    acc2(6, weight = 3, covariate1 = 8);          //  18
    BOOST_CHECK_EQUAL(5, sum_of_weights(acc2));   //
                                                  //
    acc2(4, weight = 4, covariate1 = 9);          //  16
    BOOST_CHECK_EQUAL(9, sum_of_weights(acc2));   //
                                                  //
    acc2(6, weight = 5, covariate1 = 6);          //+ 30
    BOOST_CHECK_EQUAL(14, sum_of_weights(acc2));  //
                                                  //= 84  / 14 = 6

    BOOST_CHECK_EQUAL(6., weighted_mean(acc2));
    BOOST_CHECK_EQUAL(52./7., (weighted_mean_of_variates<int, tag::covariate1>(acc2)));

[*See also]

* [classref boost::accumulators::impl::weighted_mean_impl [^weighted_mean_impl]]
* [classref boost::accumulators::impl::immediate_weighted_mean_impl [^immediate_weighted_mean_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_sum [^weighted_sum]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]

[endsect]

[section:weighted_median weighted_median ['and variants]]

Median estimation for weighted samples based on the [^P^2] quantile estimator, the density estimator, or
the [^P^2] cumulative distribution estimator. For more implementation details, see
[classref boost::accumulators::impl::weighted_median_impl [^weighted_median_impl]], 
[classref boost::accumulators::impl::with_weighted_density_median_impl [^with_weighted_density_median_impl]], 
and [classref boost::accumulators::impl::with_weighted_p_square_cumulative_distribution_median_impl [^with_weighted_p_square_cumulative_distribution_median_impl]].

The three median accumulators all satisfy the `tag::weighted_median` feature, and can all be
extracted with the `weighted_median()` extractor.

[variablelist
    [[Result Type] [``
                    numeric::functional::average<_sample_type_, std::size_t>::result_type
                    ``]]
    [[Depends On] [`weighted_median` depends on `weighted_p_square_quantile_for_median` \n
                   `with_weighted_density_median` depends on `count` and `weighted_density` \n
                   `with_weighted_p_square_cumulative_distribution_median` depends on `weighted_p_square_cumulative_distribution`]]
    [[Variants] [`with_weighted_density_median` (a.k.a. `weighted_median(with_weighted_density)`) \n
                 `with_weighted_p_square_cumulative_distribution_median` (a.k.a. `weighted_median(with_weighted_p_square_cumulative_distribution)`)]]
    [[Initialization Parameters] [`with_weighted_density_median` requires `tag::weighted_density::cache_size` and `tag::weighted_density::num_bins` \n
                                  `with_weighted_p_square_cumulative_distribution_median` requires `tag::weighted_p_square_cumulative_distribution::num_cells`]]
    [[Accumulator Parameters] [`weight`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [TODO]] 
]

[*Header]
[def _WEIGHTED_MEDIAN_HPP_ [headerref boost/accumulators/statistics/weighted_median.hpp]]

    #include <_WEIGHTED_MEDIAN_HPP_>

[*Example]

    // Median estimation of normal distribution N(1,1) using samples from a narrow normal distribution N(1,0.01)
    // The weights equal to the likelihood ratio of the corresponding samples
    
    // two random number generators
    double mu = 1.;
    double sigma_narrow = 0.01;
    double sigma = 1.;
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma_narrow(mu,sigma_narrow);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal_narrow(rng, mean_sigma_narrow);
    
    accumulator_set<double, stats<tag::weighted_median(with_weighted_p_square_quantile) >, double > acc;
    accumulator_set<double, stats<tag::weighted_median(with_weighted_density) >, double > 
        acc_dens( tag::weighted_density::cache_size = 10000, tag::weighted_density::num_bins = 1000 );
    accumulator_set<double, stats<tag::weighted_median(with_weighted_p_square_cumulative_distribution) >, double > 
        acc_cdist( tag::weighted_p_square_cumulative_distribution::num_cells = 100 );
        
    for (std::size_t i=0; i<100000; ++i)
    {
        double sample = normal_narrow();
        acc(sample, weight = std::exp(0.5 * (sample - mu) * (sample - mu) * ( 1./sigma_narrow/sigma_narrow - 1./sigma/sigma )));
        acc_dens(sample, weight = std::exp(0.5 * (sample - mu) * (sample - mu) * ( 1./sigma_narrow/sigma_narrow - 1./sigma/sigma )));
        acc_cdist(sample, weight = std::exp(0.5 * (sample - mu) * (sample - mu) * ( 1./sigma_narrow/sigma_narrow - 1./sigma/sigma )));
    }
    
    BOOST_CHECK_CLOSE(1., weighted_median(acc), 1e-1);
    BOOST_CHECK_CLOSE(1., weighted_median(acc_dens), 1e-1);
    BOOST_CHECK_CLOSE(1., weighted_median(acc_cdist), 1e-1);

[*See also]

* [classref boost::accumulators::impl::weighted_median_impl [^weighted_median_impl]]
* [classref boost::accumulators::impl::with_weighted_density_median_impl [^with_weighted_density_median_impl]]
* [classref boost::accumulators::impl::with_weighted_p_square_cumulative_distribution_median_impl [^with_weighted_p_square_cumulative_distribution_median_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_p_square_quantile [^weighted_p_square_quantile]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_p_square_cumulative_distribution [^weighted_p_square_cumulative_distribution]]

[endsect]

[section:weighted_moment weighted_moment]

Calculates the N-th moment of the weighted samples, which is defined as the sum of the weighted N-th
power of the samples over the sum of the weights.

[variablelist
    [[Result Type] [``
                    numeric::functional::average<
                        numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type
                      , weight_type
                    >::result_type
                    ``]]
    [[Depends On] [`count` \n `sum_of_weights`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [`weight`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _WEIGHTED_MOMENT_HPP_ [headerref boost/accumulators/statistics/weighted_moment.hpp]]

    #include <_WEIGHTED_MOMENT_HPP_>

[*Example]

    accumulator_set<double, stats<tag::weighted_moment<2> >, double> acc2;
    accumulator_set<double, stats<tag::weighted_moment<7> >, double> acc7;
    
    acc2(2.1, weight = 0.7);
    acc2(2.7, weight = 1.4);
    acc2(1.8, weight = 0.9);
    
    acc7(2.1, weight = 0.7);
    acc7(2.7, weight = 1.4);
    acc7(1.8, weight = 0.9);
    
    BOOST_CHECK_CLOSE(5.403, weighted_moment<2>(acc2), 1e-5);
    BOOST_CHECK_CLOSE(548.54182, weighted_moment<7>(acc7), 1e-5);

[*See also]

* [classref boost::accumulators::impl::weighted_moment_impl [^weighted_moment_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]

[endsect]

[section:weighted_p_square_cumulative_distribution weighted_p_square_cumulative_distribution]

Histogram calculation of the cumulative distribution with the [^P^2] algorithm for weighted samples.
For more implementation details, see 
[classref boost::accumulators::impl::weighted_p_square_cumulative_distribution_impl [^weighted_p_square_cumulative_distribution_impl]]

[variablelist
    [[Result Type] [``
                    iterator_range<
                        std::vector<
                            std::pair<
                                numeric::functional::average<weighted_sample, std::size_t>::result_type
                              , numeric::functional::average<weighted_sample, std::size_t>::result_type
                            >
                        >::iterator
                    >
                    ``
                    where `weighted_sample` is `numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type`]]
    [[Depends On] [`count` \n `sum_or_weights`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [`tag::weighted_p_square_cumulative_distribution::num_cells`]]
    [[Accumulator Parameters] [`weight`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(N) where N is `num_cells`]] 
]

[*Header]
[def _WEIGHTED_P_SQUARE_CUMULATIVE_DISTRIBUTION_HPP_ [headerref boost/accumulators/statistics/weighted_p_square_cumulative_distribution.hpp]]

    #include <_WEIGHTED_P_SQUARE_CUMULATIVE_DISTRIBUTION_HPP_>

[*Example]

    // tolerance in %
    double epsilon = 4;

    typedef accumulator_set<double, stats<tag::weighted_p_square_cumulative_distribution>, double > accumulator_t;
        
    accumulator_t acc_upper(tag::weighted_p_square_cumulative_distribution::num_cells = 100);
    accumulator_t acc_lower(tag::weighted_p_square_cumulative_distribution::num_cells = 100);
    
    // two random number generators
    double mu_upper = 1.0;
    double mu_lower = -1.0;
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma_upper(mu_upper,1);
    boost::normal_distribution<> mean_sigma_lower(mu_lower,1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal_upper(rng, mean_sigma_upper);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal_lower(rng, mean_sigma_lower);
    
    for (std::size_t i=0; i<100000; ++i)
    {
        double sample = normal_upper();
        acc_upper(sample, weight = std::exp(-mu_upper * (sample - 0.5 * mu_upper)));
    }
    
    for (std::size_t i=0; i<100000; ++i)
    {
        double sample = normal_lower();
        acc_lower(sample, weight = std::exp(-mu_lower * (sample - 0.5 * mu_lower)));
    }
    
    typedef iterator_range<std::vector<std::pair<double, double> >::iterator > histogram_type;
    histogram_type histogram_upper = weighted_p_square_cumulative_distribution(acc_upper);
    histogram_type histogram_lower = weighted_p_square_cumulative_distribution(acc_lower);
    
    // Note that applaying importance sampling results in a region of the distribution 
    // to be estimated more accurately and another region to be estimated less accurately
    // than without importance sampling, i.e., with unweighted samples
    
    for (std::size_t i = 0; i < histogram_upper.size(); ++i)
    {
        // problem with small results: epsilon is relative (in percent), not absolute!
        
        // check upper region of distribution
        if ( histogram_upper[i].second > 0.1 )    
            BOOST_CHECK_CLOSE( 0.5 * (1.0 + erf( histogram_upper[i].first / sqrt(2.0) )), histogram_upper[i].second, epsilon );
        // check lower region of distribution
        if ( histogram_lower[i].second < -0.1 )    
            BOOST_CHECK_CLOSE( 0.5 * (1.0 + erf( histogram_lower[i].first / sqrt(2.0) )), histogram_lower[i].second, epsilon );
    }

[*See also]

* [classref boost::accumulators::impl::weighted_p_square_cumulative_distribution_impl [^weighted_p_square_cumulative_distribution_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]

[endsect]

[section:weighted_p_square_quantile weighted_p_square_quantile ['and variants]]

Single quantile estimation with the [^P^2] algorithm. For more implementation details, see
[classref boost::accumulators::impl::weighted_p_square_quantile_impl [^weighted_p_square_quantile_impl]]

[variablelist
    [[Result Type] [``
                    numeric::functional::average<
                        numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type
                      , std::size_t
                    >::result_type
                    ``]]
    [[Depends On] [`count` \n `sum_of_weights`]]
    [[Variants] [`weighted_p_square_quantile_for_median`]]
    [[Initialization Parameters] [`quantile_probability`, which defaults to `0.5`.
                                  (Note: for `weighted_p_square_quantile_for_median`, the `quantile_probability`
                                  parameter is ignored and is always `0.5`.)]]
    [[Accumulator Parameters] [`weight`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _WEIGHTED_P_SQUARE_QUANTILE_HPP_ [headerref boost/accumulators/statistics/weighted_p_square_quantile.hpp]]

    #include <_WEIGHTED_P_SQUARE_QUANTILE_HPP_>

[*Example]

    typedef accumulator_set<double, stats<tag::weighted_p_square_quantile>, double> accumulator_t;

    // tolerance in %
    double epsilon = 1;
    
    // some random number generators
    double mu4 = -1.0;
    double mu5 = -1.0;
    double mu6 = 1.0;
    double mu7 = 1.0;
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma4(mu4, 1);
    boost::normal_distribution<> mean_sigma5(mu5, 1);
    boost::normal_distribution<> mean_sigma6(mu6, 1);
    boost::normal_distribution<> mean_sigma7(mu7, 1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal4(rng, mean_sigma4);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal5(rng, mean_sigma5);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal6(rng, mean_sigma6);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal7(rng, mean_sigma7);
        
    accumulator_t acc0(quantile_probability = 0.001);
    accumulator_t acc1(quantile_probability = 0.025);
    accumulator_t acc2(quantile_probability = 0.975);
    accumulator_t acc3(quantile_probability = 0.999);

    accumulator_t acc4(quantile_probability = 0.001);
    accumulator_t acc5(quantile_probability = 0.025);
    accumulator_t acc6(quantile_probability = 0.975);
    accumulator_t acc7(quantile_probability = 0.999);

        
    for (std::size_t i=0; i<100000; ++i)
    {
        double sample = rng();
        acc0(sample, weight = 1.);
        acc1(sample, weight = 1.);
        acc2(sample, weight = 1.);
        acc3(sample, weight = 1.);
        
        double sample4 = normal4();
        double sample5 = normal5();
        double sample6 = normal6();
        double sample7 = normal7();
        acc4(sample4, weight = std::exp(-mu4 * (sample4 - 0.5 * mu4)));
        acc5(sample5, weight = std::exp(-mu5 * (sample5 - 0.5 * mu5)));
        acc6(sample6, weight = std::exp(-mu6 * (sample6 - 0.5 * mu6)));
        acc7(sample7, weight = std::exp(-mu7 * (sample7 - 0.5 * mu7)));
    }
    
    // check for uniform distribution with weight = 1
    BOOST_CHECK_CLOSE( weighted_p_square_quantile(acc0), 0.001, 15 );
    BOOST_CHECK_CLOSE( weighted_p_square_quantile(acc1), 0.025, 5 );
    BOOST_CHECK_CLOSE( weighted_p_square_quantile(acc2), 0.975, epsilon );
    BOOST_CHECK_CLOSE( weighted_p_square_quantile(acc3), 0.999, epsilon );
    
    // check for shifted standard normal distribution ("importance sampling")
    BOOST_CHECK_CLOSE( weighted_p_square_quantile(acc4), -3.090232, epsilon );
    BOOST_CHECK_CLOSE( weighted_p_square_quantile(acc5), -1.959963, epsilon );
    BOOST_CHECK_CLOSE( weighted_p_square_quantile(acc6),  1.959963, epsilon );
    BOOST_CHECK_CLOSE( weighted_p_square_quantile(acc7),  3.090232, epsilon );

[*See also]

* [classref boost::accumulators::impl::weighted_p_square_quantile_impl [^weighted_p_square_quantile_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]

[endsect]

[section:weighted_peaks_over_threshold weighted_peaks_over_threshold ['and variants]]

Weighted peaks over threshold method for weighted quantile and weighted tail mean estimation.
For more implementation details,
see [classref boost::accumulators::impl::weighted_peaks_over_threshold_impl [^weighted_peaks_over_threshold_impl]]
and [classref boost::accumulators::impl::weighted_peaks_over_threshold_prob_impl [^weighted_peaks_over_threshold_prob_impl]].

Both `tag::weighted_peaks_over_threshold<_left_or_right_>` and
`tag::weighted_peaks_over_threshold_prob<_left_or_right_>` satisfy the
`tag::weighted_peaks_over_threshold<_left_or_right_>` feature and can be extracted using the
`weighted_peaks_over_threshold()` extractor.

[variablelist
    [[Result Type] [`tuple<float_type, float_type, float_type>` where `float_type` is
                    ``
                    numeric::functional::average<
                        numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type
                      , std::size_t
                    >::result_type
                    ``]]
    [[Depends On] [`weighted_peaks_over_threshold<_left_or_right_>` depends on `sum_of_weights` \n
                   `weighted_peaks_over_threshold_prob<_left_or_right_>` depends on `sum_of_weights` and `tail_weights<_left_or_right_>`]]
    [[Variants] [`weighted_peaks_over_threshold_prob`]]
    [[Initialization Parameters] [ `tag::peaks_over_threshold::threshold_value` \n
                                   `tag::peaks_over_threshold_prob::threshold_probability` \n
                                   `tag::tail<_left_or_right_>::cache_size` ]]
    [[Accumulator Parameters] [`weight`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [TODO]] 
    [[Extractor Complexity] [O(1)]]
]

[*Header]
[def _WEIGHTED_PEAKS_OVER_THRESHOLD_HPP_ [headerref boost/accumulators/statistics/weighted_peaks_over_threshold.hpp]]

    #include <_WEIGHTED_PEAKS_OVER_THRESHOLD_HPP_>

[/ TODO Add example]

[*See also]

* [classref boost::accumulators::impl::weighted_peaks_over_threshold_impl [^weighted_peaks_over_threshold_impl]]
* [classref boost::accumulators::impl::weighted_peaks_over_threshold_prob_impl [^weighted_peaks_over_threshold_prob_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail [^tail]]

[endsect]

[section:weighted_skewness weighted_skewness]

The skewness of a sample distribution is defined as the ratio of the 3rd central moment and the [^3/2]-th power 
of the 2nd central moment (the variance) of the sampless 3. The skewness 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.

For implementation details, see
[classref boost::accumulators::impl::weighted_skewness_impl [^weighted_skewness_impl]].

[variablelist
    [[Result Type] [``
                    numeric::functional::average<
                        numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type
                      , numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type
                    >::result_type
                    ``]]
    [[Depends On] [`weighted_mean` \n `weighted_moment<2>` \n `weighted_moment<3>`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [`weight`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _WEIGHTED_SKEWNESS_HPP_ [headerref boost/accumulators/statistics/weighted_skewness.hpp]]

    #include <_WEIGHTED_SKEWNESS_HPP_>

[*Example]

    accumulator_set<int, stats<tag::weighted_skewness>, int > acc2;
    
    acc2(2, weight = 4);
    acc2(7, weight = 1);
    acc2(4, weight = 3);
    acc2(9, weight = 1);
    acc2(3, weight = 2);
    
    BOOST_CHECK_EQUAL( weighted_mean(acc2), 42./11. );
    BOOST_CHECK_EQUAL( weighted_moment<2>(acc2), 212./11. );
    BOOST_CHECK_EQUAL( weighted_moment<3>(acc2), 1350./11. );
    BOOST_CHECK_CLOSE( weighted_skewness(acc2), 1.30708406282, 1e-6 );

[*See also]

* [classref boost::accumulators::impl::weighted_skewness_impl [^weighted_skewness_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_mean [^weighted_mean]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_moment [^weighted_moment]]

[endsect]

[section:weighted_sum weighted_sum ['and variants]]

For summing the weighted samples or variates. All of the `tag::weighted_sum_of_variates<>` features 
can be extracted with the `weighted_sum_of_variates()` extractor.

[variablelist
    [[Result Type] [`numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type` for summing weighted samples \n
                    `numeric::functional::multiplies<_variate_type_, _weight_type_>::result_type` for summing weighted variates]]
    [[Depends On] [['none]]]
    [[Variants] [`tag::weighted_sum` \n
                 `tag::weighted_sum_of_variates<_variate_type_, _variate_tag_>`]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [`weight` \n
                               `_variate_tag_` for summing variates]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _WEIGHTED_SUM_HPP_ [headerref boost/accumulators/statistics/weighted_sum.hpp]]

    #include <_WEIGHTED_SUM_HPP_>

[*Example]

    accumulator_set<int, stats<tag::weighted_sum, tag::weighted_sum_of_variates<int, tag::covariate1> >, int> acc;

    acc(1, weight = 2, covariate1 = 3);
    BOOST_CHECK_EQUAL(2, weighted_sum(acc));
    BOOST_CHECK_EQUAL(6, weighted_sum_of_variates(acc));

    acc(2, weight = 3, covariate1 = 6);
    BOOST_CHECK_EQUAL(8, weighted_sum(acc));
    BOOST_CHECK_EQUAL(24, weighted_sum_of_variates(acc));

    acc(4, weight = 6, covariate1 = 9);
    BOOST_CHECK_EQUAL(32, weighted_sum(acc));
    BOOST_CHECK_EQUAL(78, weighted_sum_of_variates(acc));

[*See also]

* [classref boost::accumulators::impl::weighted_sum_impl [^weighted_sum_impl]]

[endsect]

[section:non_coherent_weighted_tail_mean non_coherent_weighted_tail_mean]

Estimation of the (non-coherent) weighted tail mean based on order statistics (for both left and right tails).
The left non-coherent weighted tail mean feature is `tag::non_coherent_weighted_tail_mean<left>`, and the right
non-choherent weighted tail mean feature is `tag::non_coherent_weighted_tail_mean<right>`. They both share the
`tag::abstract_non_coherent_tail_mean` feature with the unweighted non-coherent tail mean accumulators and can
be extracted with either the `non_coherent_tail_mean()` or the `non_coherent_weighted_tail_mean()` extractors.
For more implementation details, see
[classref boost::accumulators::impl::non_coherent_weighted_tail_mean_impl [^non_coherent_weighted_tail_mean_impl]].

[variablelist
    [[Result Type] [``
                    numeric::functional::average<
                        numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type
                      , std::size_t
                    >::result_type
                    ``]]
    [[Depends On] [`sum_of_weights` \n `tail_weights<_left_or_right_>`]]
    [[Variants] [`abstract_non_coherent_tail_mean`]]
    [[Initialization Parameters] [`tag::tail<_left_or_right_>::cache_size`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [O(log N), where N is the cache size]] 
    [[Extractor Complexity] [O(N log N), where N is the cache size]] 
]

[*Header]
[def _WEIGHTED_TAIL_MEAN_HPP_ [headerref boost/accumulators/statistics/weighted_tail_mean.hpp]]

    #include <_WEIGHTED_TAIL_MEAN_HPP_>

[*Example]

    // tolerance in %
    double epsilon = 1;

    std::size_t n = 100000; // number of MC steps
    std::size_t c = 25000; // cache size

    accumulator_set<double, stats<tag::non_coherent_weighted_tail_mean<right> >, double >
        acc0( right_tail_cache_size = c );
    accumulator_set<double, stats<tag::non_coherent_weighted_tail_mean<left> >, double >
        acc1( left_tail_cache_size = c );

    // random number generators
    boost::lagged_fibonacci607 rng;

    for (std::size_t i = 0; i < n; ++i)
    {
        double smpl = std::sqrt(rng());
        acc0(smpl, weight = 1./smpl);
    }

    for (std::size_t i = 0; i < n; ++i)
    {
        double smpl = rng();
        acc1(smpl*smpl, weight = smpl);
    }

    // check uniform distribution
    BOOST_CHECK_CLOSE( non_coherent_weighted_tail_mean(acc0, quantile_probability = 0.95), 0.975, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_weighted_tail_mean(acc0, quantile_probability = 0.975), 0.9875, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_weighted_tail_mean(acc0, quantile_probability = 0.99), 0.995, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_weighted_tail_mean(acc0, quantile_probability = 0.999), 0.9995, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_weighted_tail_mean(acc1, quantile_probability = 0.05), 0.025, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_weighted_tail_mean(acc1, quantile_probability = 0.025), 0.0125, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_weighted_tail_mean(acc1, quantile_probability = 0.01), 0.005, epsilon );
    BOOST_CHECK_CLOSE( non_coherent_weighted_tail_mean(acc1, quantile_probability = 0.001), 0.0005, 5*epsilon );

[*See also]

* [classref boost::accumulators::impl::non_coherent_weighted_tail_mean_impl [^non_coherent_weighted_tail_mean_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail [^tail]]

[endsect]

[section:weighted_tail_quantile weighted_tail_quantile]

Tail quantile estimation based on order statistics of weighted samples (for both left
and right tails). The left weighted tail quantile feature is `tag::weighted_tail_quantile<left>`,
and the right weighted tail quantile feature is `tag::weighted_tail_quantile<right>`. They both
share the `tag::quantile` feature with the unweighted tail quantile accumulators and can be
extracted with either the `quantile()` or the `weighted_tail_quantile()` extractors. For more
implementation details, see
[classref boost::accumulators::impl::weighted_tail_quantile_impl [^weighted_tail_quantile_impl]]

[variablelist
    [[Result Type] [``
                    _sample_type_
                    ``]]
    [[Depends On] [`sum_of_weights` \n `tail_weights<_left_or_right_>`]]
    [[Variants] [['none]]]
    [[Initialization Parameters] [`tag::tail<_left_or_right_>::cache_size`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [O(log N), where N is the cache size]] 
    [[Extractor Complexity] [O(N log N), where N is the cache size]] 
]

[*Header]
[def _WEIGHTED_TAIL_QUANTILE_HPP_ [headerref boost/accumulators/statistics/weighted_tail_quantile.hpp]]

    #include <_WEIGHTED_TAIL_QUANTILE_HPP_>

[*Example]

    // tolerance in %
    double epsilon = 1;

    std::size_t n = 100000; // number of MC steps
    std::size_t c =  20000; // cache size

    double mu1 = 1.0;
    double mu2 = -1.0;
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma1(mu1,1);
    boost::normal_distribution<> mean_sigma2(mu2,1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal1(rng, mean_sigma1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal2(rng, mean_sigma2);

    accumulator_set<double, stats<tag::weighted_tail_quantile<right> >, double>
        acc1(right_tail_cache_size = c);

    accumulator_set<double, stats<tag::weighted_tail_quantile<left> >, double>
        acc2(left_tail_cache_size = c);

    for (std::size_t i = 0; i < n; ++i)
    {
        double sample1 = normal1();
        double sample2 = normal2();
        acc1(sample1, weight = std::exp(-mu1 * (sample1 - 0.5 * mu1)));
        acc2(sample2, weight = std::exp(-mu2 * (sample2 - 0.5 * mu2)));
    }

    // check standard normal distribution
    BOOST_CHECK_CLOSE( quantile(acc1, quantile_probability = 0.975),  1.959963, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc1, quantile_probability = 0.999),  3.090232, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc2, quantile_probability  = 0.025), -1.959963, epsilon );
    BOOST_CHECK_CLOSE( quantile(acc2, quantile_probability  = 0.001), -3.090232, epsilon );

[*See also]

* [classref boost::accumulators::impl::weighted_tail_quantile_impl [^weighted_tail_quantile_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.sum [^sum]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail [^tail]]

[endsect]

[section:weighted_tail_variate_means weighted_tail_variate_means ['and variants]]

Estimation of the absolute and relative weighted tail variate means (for both left and right tails)
The absolute weighted tail variate means has the feature 
`tag::absolute_weighted_tail_variate_means<_left_or_right_, _variate_type_, _variate_tag_>`
and the relative weighted tail variate mean has the feature
`tag::relative_weighted_tail_variate_means<_left_or_right_, _variate_type_, _variate_tag_>`. All
absolute weighted tail variate mean features share the `tag::abstract_absolute_tail_variate_means`
feature with their unweighted variants and can be extracted with the `tail_variate_means()` and
`weighted_tail_variate_means()` extractors. All the relative weighted tail variate mean features 
share the `tag::abstract_relative_tail_variate_means` feature with their unweighted variants
and can be extracted with either the `relative_tail_variate_means()` or 
`relative_weighted_tail_variate_means()` extractors.

For more implementation details, see
[classref boost::accumulators::impl::weighted_tail_variate_means_impl [^weighted_tail_variate_means_impl]]

[variablelist
    [[Result Type] [``
                    boost::iterator_range<
                        numeric::functional::average<
                            numeric::functional::multiplies<_variate_type_, _weight_type_>::result_type
                          , _weight_type_
                        >::result_type::iterator
                    >
                    ``]]
    [[Depends On] [`non_coherent_weighted_tail_mean<_left_or_right_>` \n 
                   `tail_variate<_variate_type_, _variate_tag_, _left_or_right_>` \n
                   `tail_weights<_left_or_right_>`]]
    [[Variants] [`tag::absolute_weighted_tail_variate_means<_left_or_right_, _variate_type_, _variate_tag_>` \n
                 `tag::relative_weighted_tail_variate_means<_left_or_right_, _variate_type_, _variate_tag_>`]]
    [[Initialization Parameters] [`tag::tail<_left_or_right_>::cache_size`]]
    [[Accumulator Parameters] [['none]]]
    [[Extractor Parameters] [`quantile_probability`]]
    [[Accumulator Complexity] [O(log N), where N is the cache size]] 
    [[Extractor Complexity] [O(N log N), where N is the cache size]] 
]

[*Header]
[def _WEIGHTED_TAIL_VARIATE_MEANS_HPP_ [headerref boost/accumulators/statistics/weighted_tail_variate_means.hpp]]

    #include <_WEIGHTED_TAIL_VARIATE_MEANS_HPP_>

[*Example]

    std::size_t c = 5; // cache size

    typedef double variate_type;
    typedef std::vector<variate_type> variate_set_type;

    accumulator_set<double, stats<tag::weighted_tail_variate_means<right, variate_set_type, tag::covariate1>(relative)>, double >
        acc1( right_tail_cache_size = c );
    accumulator_set<double, stats<tag::weighted_tail_variate_means<right, variate_set_type, tag::covariate1>(absolute)>, double >
        acc2( right_tail_cache_size = c );
    accumulator_set<double, stats<tag::weighted_tail_variate_means<left, variate_set_type, tag::covariate1>(relative)>, double >
        acc3( left_tail_cache_size = c );
    accumulator_set<double, stats<tag::weighted_tail_variate_means<left, variate_set_type, tag::covariate1>(absolute)>, double >
        acc4( left_tail_cache_size = c );

    variate_set_type cov1, cov2, cov3, cov4, cov5;
    double c1[] = { 10., 20., 30., 40. }; // 100
    double c2[] = { 26.,  4., 17.,  3. }; // 50
    double c3[] = { 46., 64., 40., 50. }; // 200
    double c4[] = {  1.,  3., 70.,  6. }; // 80
    double c5[] = {  2.,  2.,  2., 14. }; // 20
    cov1.assign(c1, c1 + sizeof(c1)/sizeof(variate_type));
    cov2.assign(c2, c2 + sizeof(c2)/sizeof(variate_type));
    cov3.assign(c3, c3 + sizeof(c3)/sizeof(variate_type));
    cov4.assign(c4, c4 + sizeof(c4)/sizeof(variate_type));
    cov5.assign(c5, c5 + sizeof(c5)/sizeof(variate_type));

    acc1(100., weight = 0.8, covariate1 = cov1);
    acc1( 50., weight = 0.9, covariate1 = cov2);
    acc1(200., weight = 1.0, covariate1 = cov3);
    acc1( 80., weight = 1.1, covariate1 = cov4);
    acc1( 20., weight = 1.2, covariate1 = cov5);

    acc2(100., weight = 0.8, covariate1 = cov1);
    acc2( 50., weight = 0.9, covariate1 = cov2);
    acc2(200., weight = 1.0, covariate1 = cov3);
    acc2( 80., weight = 1.1, covariate1 = cov4);
    acc2( 20., weight = 1.2, covariate1 = cov5);

    acc3(100., weight = 0.8, covariate1 = cov1);
    acc3( 50., weight = 0.9, covariate1 = cov2);
    acc3(200., weight = 1.0, covariate1 = cov3);
    acc3( 80., weight = 1.1, covariate1 = cov4);
    acc3( 20., weight = 1.2, covariate1 = cov5);

    acc4(100., weight = 0.8, covariate1 = cov1);
    acc4( 50., weight = 0.9, covariate1 = cov2);
    acc4(200., weight = 1.0, covariate1 = cov3);
    acc4( 80., weight = 1.1, covariate1 = cov4);
    acc4( 20., weight = 1.2, covariate1 = cov5);

    // check relative risk contributions
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc1, quantile_probability = 0.7).begin()    ), (0.8*10 + 1.0*46)/(0.8*100 + 1.0*200) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc1, quantile_probability = 0.7).begin() + 1), (0.8*20 + 1.0*64)/(0.8*100 + 1.0*200) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc1, quantile_probability = 0.7).begin() + 2), (0.8*30 + 1.0*40)/(0.8*100 + 1.0*200) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc1, quantile_probability = 0.7).begin() + 3), (0.8*40 + 1.0*50)/(0.8*100 + 1.0*200) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc3, quantile_probability = 0.3).begin()    ), (0.9*26 + 1.2*2)/(0.9*50 + 1.2*20) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc3, quantile_probability = 0.3).begin() + 1), (0.9*4 + 1.2*2)/(0.9*50 + 1.2*20) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc3, quantile_probability = 0.3).begin() + 2), (0.9*17 + 1.2*2)/(0.9*50 + 1.2*20) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc3, quantile_probability = 0.3).begin() + 3), (0.9*3 + 1.2*14)/(0.9*50 + 1.2*20) );

    // check absolute risk contributions
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc2, quantile_probability = 0.7).begin()    ), (0.8*10 + 1.0*46)/1.8 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc2, quantile_probability = 0.7).begin() + 1), (0.8*20 + 1.0*64)/1.8 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc2, quantile_probability = 0.7).begin() + 2), (0.8*30 + 1.0*40)/1.8 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc2, quantile_probability = 0.7).begin() + 3), (0.8*40 + 1.0*50)/1.8 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc4, quantile_probability = 0.3).begin()    ), (0.9*26 + 1.2*2)/2.1 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc4, quantile_probability = 0.3).begin() + 1), (0.9*4 + 1.2*2)/2.1 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc4, quantile_probability = 0.3).begin() + 2), (0.9*17 + 1.2*2)/2.1 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc4, quantile_probability = 0.3).begin() + 3), (0.9*3 + 1.2*14)/2.1 );

    // check relative risk contributions
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc1, quantile_probability = 0.9).begin()    ), 1.0*46/(1.0*200) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc1, quantile_probability = 0.9).begin() + 1), 1.0*64/(1.0*200) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc1, quantile_probability = 0.9).begin() + 2), 1.0*40/(1.0*200) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc1, quantile_probability = 0.9).begin() + 3), 1.0*50/(1.0*200) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc3, quantile_probability = 0.1).begin()    ), 1.2*2/(1.2*20) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc3, quantile_probability = 0.1).begin() + 1), 1.2*2/(1.2*20) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc3, quantile_probability = 0.1).begin() + 2), 1.2*2/(1.2*20) );
    BOOST_CHECK_EQUAL( *(relative_weighted_tail_variate_means(acc3, quantile_probability = 0.1).begin() + 3), 1.2*14/(1.2*20) );

    // check absolute risk contributions
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc2, quantile_probability = 0.9).begin()    ), 1.0*46/1.0 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc2, quantile_probability = 0.9).begin() + 1), 1.0*64/1.0 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc2, quantile_probability = 0.9).begin() + 2), 1.0*40/1.0 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc2, quantile_probability = 0.9).begin() + 3), 1.0*50/1.0 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc4, quantile_probability = 0.1).begin()    ), 1.2*2/1.2 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc4, quantile_probability = 0.1).begin() + 1), 1.2*2/1.2 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc4, quantile_probability = 0.1).begin() + 2), 1.2*2/1.2 );
    BOOST_CHECK_EQUAL( *(weighted_tail_variate_means(acc4, quantile_probability = 0.1).begin() + 3), 1.2*14/1.2 );

[*See also]

* [classref boost::accumulators::impl::weighted_tail_variate_means_impl [^weighted_tail_variate_means_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.non_coherent_weighted_tail_mean [^non_coherent_weighted_tail_mean]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail_variate [^tail_variate]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.tail [^tail]]

[endsect]

[section:weighted_variance weighted_variance ['and variants]]

Lazy or iterative calculation of the weighted variance. The lazy calculation is associated with the `tag::lazy_weighted_variance`
feature, and the iterative calculation with the `tag::weighted_variance` feature. Both can be extracted
using the `tag::weighted_variance()` extractor. For more implementation details, see
[classref boost::accumulators::impl::lazy_weighted_variance_impl [^lazy_weighted_variance_impl]] and
[classref boost::accumulators::impl::weighted_variance_impl [^weighted_variance_impl]]

[variablelist
    [[Result Type] [``
                    numeric::functional::average<
                        numeric::functional::multiplies<_sample_type_, _weight_type_>::result_type
                      , std::size_t
                    >::result_type
                    ``]]
    [[Depends On] [`tag::lazy_weighted_variance` depends on `tag::weighted_moment<2>` and `tag::weighted_mean` \n
                   `tag::weighted_variance` depends on `tag::count` and `tag::immediate_weighted_mean`]]
    [[Variants] [`tag::lazy_weighted_variance` (a.k.a. `tag::weighted_variance(lazy))` \n
                 `tag::weighted_variance` (a.k.a. `tag::weighted_variance(immediate)`)]]
    [[Initialization Parameters] [['none]]]
    [[Accumulator Parameters] [`weight`]]
    [[Extractor Parameters] [['none]]]
    [[Accumulator Complexity] [O(1)]] 
    [[Extractor Complexity] [O(1)]] 
]

[*Header]
[def _WEIGHTED_VARIANCE_HPP_ [headerref boost/accumulators/statistics/weighted_variance.hpp]]

    #include <_WEIGHTED_VARIANCE_HPP_>

[*Example]

    // lazy weighted_variance
    accumulator_set<int, stats<tag::weighted_variance(lazy)>, int> acc1;

    acc1(1, weight = 2);    //  2
    acc1(2, weight = 3);    //  6
    acc1(3, weight = 1);    //  3
    acc1(4, weight = 4);    // 16
    acc1(5, weight = 1);    //  5

    // weighted_mean = (2+6+3+16+5) / (2+3+1+4+1) = 32 / 11 = 2.9090909090909090909090909090909

    BOOST_CHECK_EQUAL(5u, count(acc1));
    BOOST_CHECK_CLOSE(2.9090909, weighted_mean(acc1), 1e-5);
    BOOST_CHECK_CLOSE(10.1818182, weighted_moment<2>(acc1), 1e-5);
    BOOST_CHECK_CLOSE(1.7190083, weighted_variance(acc1), 1e-5);

    // immediate weighted_variance
    accumulator_set<int, stats<tag::weighted_variance>, int> acc2;

    acc2(1, weight = 2);
    acc2(2, weight = 3);
    acc2(3, weight = 1);
    acc2(4, weight = 4);
    acc2(5, weight = 1);

    BOOST_CHECK_EQUAL(5u, count(acc2));
    BOOST_CHECK_CLOSE(2.9090909, weighted_mean(acc2), 1e-5);
    BOOST_CHECK_CLOSE(1.7190083, weighted_variance(acc2), 1e-5);
    
    // check lazy and immediate variance with random numbers
    
    // two random number generators
    boost::lagged_fibonacci607 rng;
    boost::normal_distribution<> mean_sigma(0,1);
    boost::variate_generator<boost::lagged_fibonacci607&, boost::normal_distribution<> > normal(rng, mean_sigma);
    
    accumulator_set<double, stats<tag::weighted_variance>, double > acc_lazy;    
    accumulator_set<double, stats<tag::weighted_variance(immediate)>, double > acc_immediate;
    
    for (std::size_t i=0; i<10000; ++i)
    {
        double value = normal();
        acc_lazy(value, weight = rng());
        acc_immediate(value, weight = rng());
    }
    
    BOOST_CHECK_CLOSE(1., weighted_variance(acc_lazy), 1.);
    BOOST_CHECK_CLOSE(1., weighted_variance(acc_immediate), 1.);

[*See also]

* [classref boost::accumulators::impl::lazy_weighted_variance_impl [^lazy_weighted_variance_impl]]
* [classref boost::accumulators::impl::weighted_variance_impl [^weighted_variance_impl]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.count [^count]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_mean [^weighted_mean]]
* [link accumulators.user_s_guide.the_statistical_accumulators_library.weighted_moment [^weighted_moment]]

[endsect]

[endsect]

[endsect]

[section Acknowledgements]

Boost.Accumulators represents the efforts of many individuals. I would like to thank 
Daniel Egloff of _ZKB_ for helping to conceive the library and realize its 
implementation. I would also like to thank David Abrahams and Matthias Troyer for 
their key contributions to the design of the library. Many thanks are due to Michael 
Gauckler and Olivier Gygi, who, along with Daniel Egloff, implemented many of the 
statistical accumulators.

Finally, I would like to thank _ZKB_ for sponsoring the work on Boost.Accumulators 
and graciously donating it to the community. 

[endsect]

[section Reference]

[xinclude accdoc.xml]

[xinclude statsdoc.xml]

[xinclude opdoc.xml]

[endsect]
