[/
  Copyright 2018 Nick Thompson

  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]

[section:bivariate_statistics Bivariate Statistics]

[heading Synopsis]

``
#include <boost/math/statistics/bivariate_statistics.hpp>

namespace boost{ namespace math{ namespace statistics {

    template<class Container>
    auto covariance(Container const & u, Container const & v);

    template<class Container>
    auto means_and_covariance(Container const & u, Container const & v);

    template<class Container>
    auto correlation_coefficient(Container const & u, Container const & v);

}}}
``

[heading Description]

This file provides functions for computing bivariate statistics.

[heading Covariance]

Computes the population covariance of two datasets:

    std::vector<double> u{1,2,3,4,5};
    std::vector<double> v{1,2,3,4,5};
    double cov_uv = boost::math::statistics::covariance(u, v);

The implementation follows [@https://doi.org/10.1109/CLUSTR.2009.5289161 Bennet et al].
The data is not modified. Requires a random-access container.
Works with real-valued inputs and does not work with complex-valued inputs.

The algorithm used herein simultaneously generates the mean values of the input data /u/ and /v/.
For certain applications, it might be useful to get them in a single pass through the data.
As such, we provide `means_and_covariance`:

    std::vector<double> u{1,2,3,4,5};
    std::vector<double> v{1,2,3,4,5};
    auto [mu_u, mu_v, cov_uv] = boost::math::statistics::means_and_covariance(u, v);

[heading Correlation Coefficient]

Computes the [@https://en.wikipedia.org/wiki/Pearson_correlation_coefficient Pearson correlation coefficient] of two datasets /u/ and /v/:

    std::vector<double> u{1,2,3,4,5};
    std::vector<double> v{1,2,3,4,5};
    double rho_uv = boost::math::statistics::correlation_coefficient(u, v);
    // rho_uv = 1.

The data must be random access and cannot be complex.

If one or both of the datasets is constant, the correlation coefficient is an indeterminant form (0/0) and definitions must be introduced to assign it a value.
We use the following: If both datasets are constant, then the correlation coefficient is 1.
If one dataset is constant, and the other is not, then the correlation coefficient is zero.


[heading References]

* Bennett, Janine, et al. ['Numerically stable, single-pass, parallel statistics algorithms.] Cluster Computing and Workshops, 2009. CLUSTER'09. IEEE International Conference on. IEEE, 2009.

[endsect]
[/section:bivariate_statistics Bivariate Statistics]