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User's Guide

The Accumulators Framework
The Statistical Accumulators Library

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: " << accumulators::moment<2>(acc) << std::endl;

    return 0;
}

This program displays the following:

Mean:   2.85
Moment: 9.635

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 caching 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, propagating 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.


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] 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] Note

To use boost::bind() and boost::ref(), you must #include <boost/bind.hpp> and <boost/ref.hpp>

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] 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";

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.

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] 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>.

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.

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.

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-explanatory, 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] 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 explicitly 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.

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 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.

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 " << accumulators::moment<2>(acc) << std::endl;

In the expression accumulators::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.

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 have 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_sets and get the correct accumulator.

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::fdiv(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::fdiv() do something sensible with a std::vector<>, for instance, we'll need to partially specialize the boost::numeric::functional::fdiv<> 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 fdiv;
}}}

If you have some user-defined type MyDouble for which you would like to customize the behavior of numeric::fdiv(), you would specialize numeric::functional::fdiv<> 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 fdiv<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::fdiv() will use your specialization of numeric::functional::fdiv<> 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.

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. Feature Requirements

Expression

Return type

Assertion / Note / Pre- / Post-condition

F::dependencies

unspecified

An MPL sequence of other features on 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 defines accumulators for incremental statistical computations. It is built on top of The Accumulator Framework.

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

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

Headers

#include <boost/accumulators/statistics/covariance.hpp>
#include <boost/accumulators/statistics/variates/covariate.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

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

Note

Results from the density accumulator can only be extracted after the number of samples meets or exceeds the cache size.

See also

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

Result Type

numeric::functional::fdiv<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

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

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::fdiv<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

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::fdiv<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( accumulators::moment<2>(acc), 159./5. );
BOOST_CHECK_EQUAL( accumulators::moment<3>(acc), 1171./5. );
BOOST_CHECK_EQUAL( accumulators::moment<4>(acc), 1863 );
BOOST_CHECK_CLOSE( kurtosis(acc), -1.39965397924, 1e-6 );

See also

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

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::fdiv<sample-type, std::size_t>::result_type
For weights, numeric::functional::fdiv<weight-type, std::size_t>::result_type
For variates, numeric::functional::fdiv<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., (accumulators::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, (accumulators::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., (accumulators::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., (accumulators::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, (accumulators::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., (accumulators::mean_of_variates<int, tag::covariate1>(acc2)), 1e-5);

See also

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::fdiv<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

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

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::fdiv<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., accumulators::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., accumulators::moment<5>(acc2), 1e-5);

See also

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::fdiv<sample-type, std::size_t>::result_type
          , numeric::functional::fdiv<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_cumul_dist.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

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

Result Type

numeric::functional::fdiv<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

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::fdiv<sample-type, std::size_t>::result_type // u_bar
  , numeric::functional::fdiv<sample-type, std::size_t>::result_type // beta_bar
  , numeric::functional::fdiv<sample-type, std::size_t>::result_type // xi_hat
>

Depends On

count
In addition, 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

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::fdiv<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

Estimation of the (coherent) tail mean based on the peaks over threshold method (for both left and right tails). For implementation 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::fdiv<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

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

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

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

Lazy or iterative calculation of the mean over the last N samples. The lazy calculation is associated with the tag::lazy_rolling_mean feature, and the iterative calculation (which is the default) with the tag::immediate_rolling_mean feature. Both can be extracted using the tag::rolling_mean() extractor. For more implementation details, see lazy_rolling_mean_impl and immediate_rolling_mean_impl

Result Type

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

Depends On

lazy_rolling_mean depends on rolling_sum and rolling_count
immediate_rolling_mean depends on rolling_count

Variants

lazy_rolling_mean (a.k.a. rolling_mean(lazy))
immediate_rolling_mean (a.k.a. rolling_mean(immediate))

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_moment<M> calculates the M-th moment of the samples, which is defined as the sum of the M-th power of the samples over the count of samples, over the last N samples.

Result Type

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

Depends On

none

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_moment.hpp>

Example

accumulator_set<int, stats<tag::rolling_moment<2> > > acc(tag::rolling_window::window_size = 3);

acc(2);
acc(4);

BOOST_CHECK_CLOSE( rolling_moment<2>(acc), (4.0 + 16.0)/2, 1e-5 );

acc(5);
acc(6);

BOOST_CHECK_CLOSE( rolling_moment<2>(acc), (16.0 + 25.0 + 36.0)/3, 1e-5 );

See also

Lazy or iterative calculation of the variance over the last N samples. The lazy calculation is associated with the tag::lazy_rolling_variance feature, and the iterative calculation with the tag::immediate_rolling_variance feature. Both can be extracted using the tag::rolling_variance() extractor. For more implementation details, see lazy_rolling_variance_impl and immediate_rolling_variance_impl

Result Type

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

Depends On

lazy_rolling_variance depends on rolling_moment<2>, rolling_count and rolling_mean
immediate_rolling_variance depends on rolling_count and immediate_rolling_mean

Variants

lazy_rolling_variance (a.k.a. rolling_variance(lazy))
immediate_rolling_variance (a.k.a. rolling_variance(immediate))

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_variance.hpp>

Example

accumulator_set<double, stats<tag::rolling_variance> > acc(tag::rolling_window::window_size = 4);

acc(1.2);

BOOST_CHECK_CLOSE( rolling_variance(acc), 0.0, 1e-10 ); // variance is not defined for a single sample

acc(2.3);
acc(3.4);

BOOST_CHECK_CLOSE( rolling_variance(acc), 1.21, 1e-10 ); // variance over samples 1-3

acc(4.5);
acc(0.4);
acc(2.2);
acc(7.1);

BOOST_CHECK_CLOSE( rolling_variance(acc), 8.41666666666667, 1e-10 ); // variance over samples 4-7

See also

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 samples 3. For implementation details, see skewness_impl.

Result Type

numeric::functional::fdiv<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( accumulators::moment<2>(acc2), 159./5. );
BOOST_CHECK_EQUAL( accumulators::moment<3>(acc2), 1171./5. );
BOOST_CHECK_CLOSE( skewness(acc2), 0.406040288214, 1e-6 );

See also

For summing the samples, weights or variates. The default implementation uses the standard sum operation, but variants using the Kahan summation algorithm are also provided.

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>
tag::sum_kahan (a.k.a. tag::sum(kahan))
tag::sum_of_weights_kahan (a.k.a. tag::sum_of_weights(kahan))
tag::sum_of_variates_kahan<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). Note that the Kahan sum performs four floating-point sum operations per accumulated value, whereas the naive sum performs only one.

Extractor Complexity

O(1)

Header

#include <boost/accumulators/statistics/sum.hpp>
#include <boost/accumulators/statistics/sum_kahan.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));

// demonstrate Kahan summation
accumulator_set<float, stats<tag::sum_kahan> > acc;
BOOST_CHECK_EQUAL(0.0f, sum_kahan(acc));
for (size_t i = 0; i < 1e6; ++i) {
  acc(1e-6f);
}
BOOST_CHECK_EQUAL(1.0f, sum_kahan(acc));

See also

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

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 coherent 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::fdiv<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

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::fdiv<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

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

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

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::fdiv<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

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::fdiv<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., accumulators::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

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

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::fdiv<weight-type, std::size_t>::result_type
          , numeric::functional::fdiv<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

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

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::fdiv<
    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( accumulators::weighted_moment<2>(acc2), 212./11. );
BOOST_CHECK_EQUAL( accumulators::weighted_moment<3>(acc2), 1350./11. );
BOOST_CHECK_EQUAL( accumulators::weighted_moment<4>(acc2), 9956./11. );
BOOST_CHECK_CLOSE( weighted_kurtosis(acc2), 0.58137026432, 1e-6 );

See also

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::fdiv<numeric::functional::multiplies<sample-type, weight-type>::result_type, weight-type>::result_type
For variates, numeric::functional::fdiv<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., (accumulators::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., (accumulators::weighted_mean_of_variates<int, tag::covariate1>(acc2)));

See also

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::fdiv<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

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::fdiv<
    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, accumulators::weighted_moment<2>(acc2), 1e-5);
BOOST_CHECK_CLOSE(548.54182, accumulators::weighted_moment<7>(acc7), 1e-5);

See also

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::fdiv<weighted_sample, std::size_t>::result_type
          , numeric::functional::fdiv<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_cumul_dist.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 applying 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

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

Result Type

numeric::functional::fdiv<
    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 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::fdiv<
    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

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 samples 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::fdiv<
    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( accumulators::weighted_moment<2>(acc2), 212./11. );
BOOST_CHECK_EQUAL( accumulators::weighted_moment<3>(acc2), 1350./11. );
BOOST_CHECK_CLOSE( weighted_skewness(acc2), 1.30708406282, 1e-6 );

See also

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. Variants that implement the Kahan summation algorithm are also provided.

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>
tag::weighted_sum_kahan (a.k.a. tag::weighted_sum(kahan))
tag::weighted_sum_of_variates_kahan<variate-type, variate-tag>

Initialization Parameters

none

Accumulator Parameters

weight
variate-tag for summing variates

Extractor Parameters

none

Accumulator Complexity

O(1). Note that the Kahan sum performs four floating-point sum operations per accumulated value, whereas the naive sum performs only one.

Extractor Complexity

O(1)

Header

#include <boost/accumulators/statistics/weighted_sum.hpp>
#include <boost/accumulators/statistics/weighted_sum_kahan.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));

// demonstrate weighted Kahan summation
accumulator_set<float, stats<tag::weighted_sum_kahan>, float > acc;
BOOST_CHECK_EQUAL(0.0f, weighted_sum_kahan(acc));
for (size_t i = 0; i < 1e6; ++i) {
  acc(1.0f, weight = 1e-6f);
}
BOOST_CHECK_EQUAL(1.0f, weighted_sum_kahan(acc));

See also

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::fdiv<
    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

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

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::fdiv<
        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

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::fdiv<
    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, accumulators::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


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