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## 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 Reference section.

### 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:

    Mean:   2.85
    Moment: 9.635

### 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.

### Terminology

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

Sample

  A datum that is pushed into an `accumulator_set<>`. The type of the sample is the sample type.

Weight

  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<>`.

### Overview

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

Table 1.1. 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.

#### 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:

`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>`

#### 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";

#### 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 Boost.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.

#### 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>`.

#### 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.

#### 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.

##### Defining a New Accumulator

All new accumulators must satisfy the 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 Boost.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:

Optional Accumulator Member Functions

`on_drop(Args)`

  Defines an action to be taken when this accumulator is dropped. See the section on Droppable Accumulators.

#### 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.

#### 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.

#### 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.

##### 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 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 accumulators::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 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.

##### 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 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.

##### 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 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.

##### 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 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 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.

#### Concepts

### 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 Boost.Parameter library.

Table 1.2. 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` |

### Feature Concept

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

Table 1.3. 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.

### The Statistical Accumulators Library

The Statistical Accumulators Library defines accumulators for incremental statistial computations. It is built on top of The Accumulator Framework.

#### count

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

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

    #include <boost/accumulators/statistics/count.hpp>

Example

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

See also

- `count_impl`

#### covariance

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

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`
  `mean`
  `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

    #include <boost/accumulators/statistics/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

- `covariance_impl`
- `count`
- `mean`

#### density

The `tag::density` feature returns a histogram of the sample distribution. For more implementation details, see `density_impl`.

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`
  `min`
  `max`

Variants

  none

Initialization Parameters

  `density::cache_size`
  `density::num_bins`

Accumulator Parameters

  none

Extractor Parameters

  none

Accumulator Complexity

  TODO

Extractor Complexity

  O(N), when N is `density::num_bins`

Header

    #include <boost/accumulators/statistics/density.hpp>

See also

- `density_impl`
- `count`
- `min`
- `max`

#### error_of<mean>

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

Result Type

      numeric::functional::average<sample-type, std::size_t>::result_type

Depends On

  `count`
  `variance`

Variants

  `error_of<immediate_mean>`

Initialization Parameters

  none

Accumulator Parameters

  none

Extractor Parameters

  none

Accumulator Complexity

  TODO

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/error_of.hpp>
    #include <boost/accumulators/statistics/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

- `error_of_mean_impl`
- `count`
- `variance`

#### extended_p_square

Multiple quantile estimation with the extended `P^2` algorithm. For further details, see `extended_p_square_impl`.

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

    #include <boost/accumulators/statistics/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

- `extended_p_square_impl`
- `count`

#### 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 `extended_p_square_quantile_impl`.

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

Result Type

      numeric::functional::average<sample-type, std::size_t>::result_type

Depends On

  weighted variants depend on `weighted_extended_p_square`
  unweighted variants depend on `extended_p_square`

Variants

  `extended_p_square_quantile_quadratic`
  `weighted_extended_p_square_quantile`
  `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

    #include <boost/accumulators/statistics/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

- `extended_p_square_quantile_impl`
- `extended_p_square`
- `weighted_extended_p_square`

#### 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 `kurtosis_impl`

Result Type

      numeric::functional::average<sample-type, sample-type>::result_type

Depends On

  `mean`
  `moment<2>`
  `moment<3>`
  `moment<4>`

Variants

  none

Initialization Parameters

  none

Accumulator Parameters

  none

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/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

- `kurtosis_impl`
- `mean`
- `moment`

#### max

Calculates the maximum value of all the samples.

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

    #include <boost/accumulators/statistics/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

- `max_impl`

#### 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 `mean_impl` or. `immediate_mean_impl`

Result Type

  For samples, `numeric::functional::average<`sample-type`, std::size_t>::result_type`
  For weights, `numeric::functional::average<`weight-type`, std::size_t>::result_type`
  For variates, `numeric::functional::average<`variate-type`, std::size_t>::result_type`

Depends On

  `count`
  The lazy mean of samples depends on `sum`
  The lazy mean of weights depends on `sum_of_weights`
  The lazy mean of variates depends on `sum_of_variates<>`

Variants

  `mean_of_weights`
  `mean_of_variates<`variate-type`, `variate-tag`>`
  `immediate_mean`
  `immediate_mean_of_weights`
  `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

    #include <boost/accumulators/statistics/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

- `mean_impl`
- `immediate_mean_impl`
- `count`
- `sum`

#### 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 `median_impl`, `with_density_median_impl`, and `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.

Result Type

      numeric::functional::average<sample-type, std::size_t>::result_type

Depends On

  `median` depends on `p_square_quantile_for_median`
  `with_density_median` depends on `count` and `density`
  `with_p_square_cumulative_distribution_median` depends on `p_square_cumulative_distribution`

Variants

  `with_density_median`
  `with_p_square_cumulative_distribution_median`

Initialization Parameters

  `with_density_median` requires `tag::density::cache_size` and `tag::density::num_bins`
  `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

    #include <boost/accumulators/statistics/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

- `median_impl`
- `with_density_median_impl`
- `with_p_square_cumulative_distribution_median_impl`
- `count`
- `p_square_quantile`
- `p_square_cumulative_distribution`

#### min

Calculates the minimum value of all the samples.

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

    #include <boost/accumulators/statistics/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

- `min_impl`

#### 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.

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

    #include <boost/accumulators/statistics/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

- `moment_impl`
- `count`

#### p_square_cumulative_distribution

Histogram calculation of the cumulative distribution with the `P^2` algorithm. For more implementation details, see `p_square_cumulative_distribution_impl`

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

    #include <boost/accumulators/statistics/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

- `p_square_cumulative_distribution_impl`
- `count`

#### p_square_quantile and variants

Single quantile estimation with the `P^2` algorithm. For more implementation details, see `p_square_quantile_impl`

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

    #include <boost/accumulators/statistics/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

- `p_square_quantile_impl`
- `count`

#### peaks_over_threshold and variants

Peaks Over Threshold method for quantile and tail mean estimation. For implementation details, see `peaks_over_threshold_impl` and `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`.

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`
  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`
  `tag::peaks_over_threshold_prob::threshold_probability`
  `tag::tail<`left-or-right`>::cache_size`

Accumulator Parameters

  none

Extractor Parameters

  none

Accumulator Complexity

  TODO

Extractor Complexity

  TODO

Header

    #include <boost/accumulators/statistics/peaks_over_threshold.hpp>

Example

See example for `pot_quantile`.

See also

- `peaks_over_threshold_impl`
- `peaks_over_threshold_prob_impl`
- `count`
- `tail`
- `pot_quantile`
- `pot_tail_mean`

#### pot_quantile and variants

Quantile estimation based on Peaks over Threshold method (for both left and right tails). For implementation details, see `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.

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`>`
  `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`
  `tag::peaks_over_threshold_prob::threshold_probability`
  `tag::tail<`left-or-right`>::cache_size`

Accumulator Parameters

  none

Extractor Parameters

  `quantile_probability`

Accumulator Complexity

  TODO

Extractor Complexity

  TODO

Header

    #include <boost/accumulators/statistics/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

- `pot_quantile_impl`
- `peaks_over_threshold`

#### 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 `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.

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`>`
  `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`
  `tag::peaks_over_threshold_prob::threshold_probability`
  `tag::tail<`left-or-right`>::cache_size`

Accumulator Parameters

  none

Extractor Parameters

  `quantile_probability`

Accumulator Complexity

  TODO

Extractor Complexity

  TODO

Header

    #include <boost/accumulators/statistics/pot_tail_mean.hpp>

Example

    // TODO

See also

- `pot_tail_mean_impl`
- `peaks_over_threshold`
- `pot_quantile`

#### rolling_count

The rolling count is the current number of elements in the rolling window.

Result Type

      std::size_t

Depends On

  `rolling_window_plus1`

Variants

  none

Initialization Parameters

  `tag::rolling_window::window_size`

Accumulator Parameters

  none

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/rolling_count.hpp>

Example

    accumulator_set<int, stats<tag::rolling_count> > acc(tag::rolling_window::window_size = 3);

    BOOST_CHECK_EQUAL(0u, rolling_count(acc));

    acc(1);
    BOOST_CHECK_EQUAL(1u, rolling_count(acc));

    acc(1);
    BOOST_CHECK_EQUAL(2u, rolling_count(acc));

    acc(1);
    BOOST_CHECK_EQUAL(3u, rolling_count(acc));

    acc(1);
    BOOST_CHECK_EQUAL(3u, rolling_count(acc));

    acc(1);
    BOOST_CHECK_EQUAL(3u, rolling_count(acc));

See also

- `rolling_count_impl`

#### rolling_sum

The rolling sum is the sum of the last N samples.

Result Type

      sample-type

Depends On

  `rolling_window_plus1`

Variants

  none

Initialization Parameters

  `tag::rolling_window::window_size`

Accumulator Parameters

  none

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/rolling_sum.hpp>

Example

    accumulator_set<int, stats<tag::rolling_sum> > acc(tag::rolling_window::window_size = 3);

    BOOST_CHECK_EQUAL(0, rolling_sum(acc));

    acc(1);
    BOOST_CHECK_EQUAL(1, rolling_sum(acc));

    acc(2);
    BOOST_CHECK_EQUAL(3, rolling_sum(acc));

    acc(3);
    BOOST_CHECK_EQUAL(6, rolling_sum(acc));

    acc(4);
    BOOST_CHECK_EQUAL(9, rolling_sum(acc));

    acc(5);
    BOOST_CHECK_EQUAL(12, rolling_sum(acc));

See also

- `rolling_sum_impl`

#### rolling_mean

The rolling mean is the mean over the last N samples. It is computed by dividing the rolling sum by the rolling count.

Result Type

      numeric::functional::average<sample-type, std::size_t>::result_type

Depends On

  `rolling_sum`
  `rolling_count`

Variants

  none

Initialization Parameters

  `tag::rolling_window::window_size`

Accumulator Parameters

  none

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/rolling_mean.hpp>

Example

    accumulator_set<int, stats<tag::rolling_mean> > acc(tag::rolling_window::window_size = 5);

    acc(1);
    acc(2);
    acc(3);

    BOOST_CHECK_CLOSE( rolling_mean(acc), 2.0, 1e-6 );

    acc(4);
    acc(5);
    acc(6);
    acc(7);

    BOOST_CHECK_CLOSE( rolling_mean(acc), 5.0, 1e-6 );

See also

- `rolling_mean_impl`

#### 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 `skewness_impl`.

Result Type

      numeric::functional::average<sample-type, sample-type>::result_type

Depends On

  `mean`
  `moment<2>`
  `moment<3>`

Variants

  none

Initialization Parameters

  none

Accumulator Parameters

  none

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/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

- `skewness_impl`
- `mean`
- `moment`

#### sum and variants

For summing the samples, weights or variates.

Result Type

  ``sample-type`` for summing samples
  ``weight-type`` for summing weights
  ``variate-type`` for summing variates

Depends On

  none

Variants

  `tag::sum`
  `tag::sum_of_weights`
  `tag::sum_of_variates<`variate-type`, `variate-tag`>`

Initialization Parameters

  none

Accumulator Parameters

  `weight` for summing weights
  ``variate-tag`` for summing variates

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/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

- `sum_impl`

#### 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 `tail_impl`.

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

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

    #include <boost/accumulators/statistics/tail.hpp>

Example

See the Example for `tail_variate`.

See also

- `tail_impl`
- `tail_variate`

#### 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 `coherent_tail_mean_impl`

Result Type

      numeric::functional::average<sample-type, std::size_t>::result_type

Depends On

  `count`
  `quantile`
  `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

    #include <boost/accumulators/statistics/tail_mean.hpp>

Example

See the example for `non_coherent_tail_mean`.

See also

- `coherent_tail_mean_impl`
- `count`
- `extended_p_square_quantile`
- `pot_quantile`
- `tail_quantile`
- `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 `non_coherent_tail_mean_impl`

Result Type

      numeric::functional::average<sample-type, std::size_t>::result_type

Depends On

  `count`
  `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

    #include <boost/accumulators/statistics/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

- `non_coherent_tail_mean_impl`
- `count`
- `tail`

#### 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 `tail_quantile_impl`

Result Type

      sample-type

Depends On

  `count`
  `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

    #include <boost/accumulators/statistics/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

- `tail_quantile_impl`
- `count`
- `tail`

#### 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 `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.

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

    #include <boost/accumulators/statistics/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

- `tail_variate_impl`
- `tail`

#### 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 `tail_variate_means_impl`

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`>`
  `tail_variate<`variate-type`, `variate-tag`, `left-or-right`>`

Variants

  `tag::absolute_tail_variate_means<`left-or-right`, `variate-type`, `variate-tag`>`
  `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

    #include <boost/accumulators/statistics/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

- `tail_variate_means_impl`
- `non_coherent_tail_mean`
- `tail_variate`

#### 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 `lazy_variance_impl` and `variance_impl`

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`
  `tag::variance` depends on `tag::count` and `tag::immediate_mean`

Variants

  `tag::lazy_variance` (a.k.a. `tag::variance(lazy))`
  `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

    #include <boost/accumulators/statistics/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

- `lazy_variance_impl`
- `variance_impl`
- `count`
- `mean`
- `moment`

#### 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 `weighted_covariance_impl`

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`
  `sum_of_weights`
  `weighted_mean`
  `weighted_mean_of_variates<`variate-type`, `variate-tag`>`

Variants

  `abstract_weighted_covariance`

Initialization Parameters

  none

Accumulator Parameters

  `weight`
  ``variate-tag``

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/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

- `weighted_covariance_impl`
- `count`
- `sum`
- `weighted_mean`

#### weighted_density

The `tag::weighted_density` feature returns a histogram of the weighted sample distribution. For more implementation details, see `weighted_density_impl`.

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`
  `sum_of_weights`
  `min`
  `max`

Variants

  none

Initialization Parameters

  `tag::weighted_density::cache_size`
  `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

    #include <boost/accumulators/statistics/weighted_density.hpp>

See also

- `weighted_density_impl`
- `count`
- `sum`
- `min`
- `max`

#### weighted_extended_p_square

Multiple quantile estimation with the extended `P^2` algorithm for weighted samples. For further details, see `weighted_extended_p_square_impl`.

Result Type

      boost::iterator_range<
          implementation-defined
      >

Depends On

  `count`
  `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

    #include <boost/accumulators/statistics/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

- `weighted_extended_p_square_impl`
- `count`
- `sum`

#### 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 `weighted_kurtosis_impl`

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`
  `weighted_moment<2>`
  `weighted_moment<3>`
  `weighted_moment<4>`

Variants

  none

Initialization Parameters

  none

Accumulator Parameters

  none

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/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

- `weighted_kurtosis_impl`
- `weighted_mean`
- `weighted_moment`

#### 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 `weighted_mean_impl` or. `immediate_weighted_mean_impl`

Result Type

  For samples, `numeric::functional::average<numeric::functional::multiplies<`sample-type`, `weight-type`>::result_type, `weight-type`>::result_type`
  For variates, `numeric::functional::average<numeric::functional::multiplies<`variate-type`, `weight-type`>::result_type, `weight-type`>::result_type`

Depends On

  `sum_of_weights`
  The lazy mean of samples depends on `weighted_sum`
  The lazy mean of variates depends on `weighted_sum_of_variates<>`

Variants

  `weighted_mean_of_variates<`variate-type`, `variate-tag`>`
  `immediate_weighted_mean`
  `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

    #include <boost/accumulators/statistics/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

- `weighted_mean_impl`
- `immediate_weighted_mean_impl`
- `weighted_sum`
- `sum`

#### 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 `weighted_median_impl`, `with_weighted_density_median_impl`, and `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.

Result Type

      numeric::functional::average<sample-type, std::size_t>::result_type

Depends On

  `weighted_median` depends on `weighted_p_square_quantile_for_median`
  `with_weighted_density_median` depends on `count` and `weighted_density`
  `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)`)
  `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`
  `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

    #include <boost/accumulators/statistics/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

- `weighted_median_impl`
- `with_weighted_density_median_impl`
- `with_weighted_p_square_cumulative_distribution_median_impl`
- `count`
- `weighted_p_square_quantile`
- `weighted_p_square_cumulative_distribution`

#### 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.

Result Type

      numeric::functional::average<
          numeric::functional::multiplies<sample-type, weight-type>::result_type
        , weight_type
      >::result_type

Depends On

  `count`
  `sum_of_weights`

Variants

  none

Initialization Parameters

  none

Accumulator Parameters

  `weight`

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/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

- `weighted_moment_impl`
- `count`
- `sum`

#### weighted_p_square_cumulative_distribution

Histogram calculation of the cumulative distribution with the `P^2` algorithm for weighted samples. For more implementation details, see `weighted_p_square_cumulative_distribution_impl`

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`
  `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

    #include <boost/accumulators/statistics/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

- `weighted_p_square_cumulative_distribution_impl`
- `count`
- `sum`

#### weighted_p_square_quantile and variants

Single quantile estimation with the `P^2` algorithm. For more implementation details, see `weighted_p_square_quantile_impl`

Result Type

      numeric::functional::average<
          numeric::functional::multiplies<sample-type, weight-type>::result_type
        , std::size_t
      >::result_type

Depends On

  `count`
  `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

    #include <boost/accumulators/statistics/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

- `weighted_p_square_quantile_impl`
- `count`
- `sum`

#### weighted_peaks_over_threshold and variants

Weighted peaks over threshold method for weighted quantile and weighted tail mean estimation. For more implementation details, see `weighted_peaks_over_threshold_impl` and `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.

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`
  `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`
  `tag::peaks_over_threshold_prob::threshold_probability`
  `tag::tail<`left-or-right`>::cache_size`

Accumulator Parameters

  `weight`

Extractor Parameters

  none

Accumulator Complexity

  TODO

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/weighted_peaks_over_threshold.hpp>

See also

- `weighted_peaks_over_threshold_impl`
- `weighted_peaks_over_threshold_prob_impl`
- `sum`
- `tail`

#### 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 `weighted_skewness_impl`.

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`
  `weighted_moment<2>`
  `weighted_moment<3>`

Variants

  none

Initialization Parameters

  none

Accumulator Parameters

  `weight`

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/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

- `weighted_skewness_impl`
- `weighted_mean`
- `weighted_moment`

#### 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.

Result Type

  `numeric::functional::multiplies<`sample-type`, `weight-type`>::result_type` for summing weighted samples
  `numeric::functional::multiplies<`variate-type`, `weight-type`>::result_type` for summing weighted variates

Depends On

  none

Variants

  `tag::weighted_sum`
  `tag::weighted_sum_of_variates<`variate-type`, `variate-tag`>`

Initialization Parameters

  none

Accumulator Parameters

  `weight`
  ``variate-tag`` for summing variates

Extractor Parameters

  none

Accumulator Complexity

  O(1)

Extractor Complexity

  O(1)

Header

    #include <boost/accumulators/statistics/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

- `weighted_sum_impl`

#### 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 `non_coherent_weighted_tail_mean_impl`.

Result Type

      numeric::functional::average<
          numeric::functional::multiplies<sample-type, weight-type>::result_type
        , std::size_t
      >::result_type

Depends On

  `sum_of_weights`
  `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

    #include <boost/accumulators/statistics/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

- `non_coherent_weighted_tail_mean_impl`
- `sum`
- `tail`

#### 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 `weighted_tail_quantile_impl`

Result Type

      sample-type

Depends On

  `sum_of_weights`
  `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

    #include <boost/accumulators/statistics/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

- `weighted_tail_quantile_impl`
- `sum`
- `tail`

#### 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 `weighted_tail_variate_means_impl`

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`>`
  `tail_variate<`variate-type`, `variate-tag`, `left-or-right`>`
  `tail_weights<`left-or-right`>`

Variants

  `tag::absolute_weighted_tail_variate_means<`left-or-right`, `variate-type`, `variate-tag`>`
  `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

    #include <boost/accumulators/statistics/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

- `weighted_tail_variate_means_impl`
- `non_coherent_weighted_tail_mean`
- `tail_variate`
- `tail`

#### 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 `lazy_weighted_variance_impl` and `weighted_variance_impl`

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`
  `tag::weighted_variance` depends on `tag::count` and `tag::immediate_weighted_mean`

Variants

  `tag::lazy_weighted_variance` (a.k.a. `tag::weighted_variance(lazy))`
  `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

    #include <boost/accumulators/statistics/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

- `lazy_weighted_variance_impl`
- `weighted_variance_impl`
- `count`
- `weighted_mean`
- `weighted_moment`