    // Copyright (C) 2008  Davis E. King (davis@dlib.net), Steve Taylor
    // License: Boost Software License   See LICENSE.txt for the full license.
    #undef DLIB_STATISTICs_ABSTRACT_
    #ifdef DLIB_STATISTICs_ABSTRACT_

    #include <limits>
    #include <cmath>
    #include "../matrix/matrix_abstract.h"
    #include "../svm/sparse_vector_abstract.h"

    namespace dlib
    {

    // ----------------------------------------------------------------------------------------

        template <
            typename T,
            typename alloc
            >
        double mean_sign_agreement (
            const std::vector<T,alloc>& a,
            const std::vector<T,alloc>& b
        );
        /*!
            requires
                - a.size() == b.size()
            ensures
                - returns the number of times a[i] has the same sign as b[i] divided by
                  a.size().  So we return the probability that elements of a and b have
                  the same sign.
        !*/

    // ----------------------------------------------------------------------------------------

        template <
            typename T,
            typename alloc
            >
        double correlation (
            const std::vector<T,alloc>& a,
            const std::vector<T,alloc>& b
        );
        /*!
            requires
                - a.size() == b.size()
                - a.size() > 1
            ensures
                - returns the correlation coefficient between all the elements of a and b.
                  (i.e. how correlated is a(i) with b(i))
        !*/

    // ----------------------------------------------------------------------------------------

        template <
            typename T,
            typename alloc
            >
        double covariance (
            const std::vector<T,alloc>& a,
            const std::vector<T,alloc>& b
        );
        /*!
            requires
                - a.size() == b.size()
                - a.size() > 1
            ensures
                - returns the covariance between all the elements of a and b.
                  (i.e. how does a(i) vary with b(i))
        !*/

    // ----------------------------------------------------------------------------------------

        template <
            typename T,
            typename alloc
            >
        double r_squared (
            const std::vector<T,alloc>& a,
            const std::vector<T,alloc>& b
        );
        /*!
            requires
                - a.size() == b.size()
                - a.size() > 1
            ensures
                - returns the R^2 coefficient of determination between all the elements of a and b.
                  This value is just the square of correlation(a,b).
        !*/

    // ----------------------------------------------------------------------------------------

        template <
            typename T,
            typename alloc
            >
        double mean_squared_error (
            const std::vector<T,alloc>& a,
            const std::vector<T,alloc>& b
        );
        /*!
            requires
                - a.size() == b.size()
            ensures
                - returns the mean squared error between all the elements of a and b.
                  (i.e. mean(squared(mat(a)-mat(b))))
        !*/

    // ----------------------------------------------------------------------------------------

        double binomial_random_vars_are_different (
            double k1,
            double n1,
            double k2,
            double n2
        );
        /*!
            requires
                - k1 <= n1
                - k2 <= n2
            ensures
                - Given two binomially distributed random variables, X1 and X2, we want to know
                  if these variables have the same parameter (i.e. the chance of "success").
                  So assume that:
                    - You observed X1 to give k1 successes out of n1 trials.
                    - You observed X2 to give k2 successes out of n2 trials.
                - This function performs a simple likelihood ratio test to determine if X1 and
                  X2 have the same parameter.  The return value of this function will be:
                    - Close to 0 if they are probably the same.
                    - Larger than 0 if X1 probably has a higher "success" rate than X2.
                    - Smaller than 0 if X2 probably has a higher "success" rate than X1.
                  Moreover, the larger the absolute magnitude of the return value the more
                  likely it is that X1 and X2 have different distributions.
                - For a discussion of the technique and applications see:
                      Dunning, Ted. "Accurate methods for the statistics of surprise and
                      coincidence." Computational linguistics 19.1 (1993): 61-74.
        !*/

    // ----------------------------------------------------------------------------------------

        double event_correlation (
            double A_count,
            double B_count,
            double AB_count,
            double total_num_observations
        );
        /*!
            requires
                - AB_count <= A_count <= total_num_observations
                - AB_count <= B_count <= total_num_observations
                - A_count + B_count - AB_count <= total_num_observations
            ensures
                - This function does a statistical test to determine if two events co-occur in
                  a statistically significant way.  In particular, we assume you performed
                  total_num_observations measurements and during those measurements you:
                    - Observed event A to happen A_count times.
                    - Observed event B to happen B_count times.
                    - Observed AB_count co-occurrences of the events.  That is, AB_count is the
                      number of times the events happened together during the same measurement.
                - This function returns a number, COR, which can take any real value.  It has
                  the following interpretations:
                    - COR == 0: there is no evidence of correlation between the two events.
                      They appear to be unrelated.
                    - COR > 0: There is evidence that A and B co-occur together.  That is,
                      they happen at the same times more often than you would expect if they
                      were independent events.  The larger the magnitude of COR the more
                      evidence we have for the correlation.
                    - COR < 0: There is evidence that A and B are anti-correlated.  That is,
                      when A happens B is unlikely to happen and vice versa.  The larger the
                      magnitude of COR the more evidence we have for the anti-correlation.
                - This function implements the simple likelihood ratio test discussed in the
                  following paper:
                      Dunning, Ted. "Accurate methods for the statistics of surprise and
                      coincidence." Computational linguistics 19.1 (1993): 61-74.
                  So for an extended discussion of the method see the above paper.
        !*/

    // ----------------------------------------------------------------------------------------

        template <
            typename T
            >
        class running_stats
        {
            /*!
                REQUIREMENTS ON T
                    - T must be a float, double, or long double type

                INITIAL VALUE
                    - mean() == 0
                    - current_n() == 0

                WHAT THIS OBJECT REPRESENTS
                    This object represents something that can compute the running mean,
                    variance, skewness, and excess kurtosis of a stream of real numbers.
            !*/
        public:

            running_stats(
            );
            /*!
                ensures
                    - this object is properly initialized
            !*/

            void clear(
            );
            /*!
                ensures
                    - this object has its initial value
                    - clears all memory of any previous data points
            !*/

            T current_n (
            ) const;
            /*!
                ensures
                    - returns the number of points given to this object so far.
            !*/

            void add (
                const T& val
            );
            /*!
                ensures
                    - updates the sum, mean, variance, skewness, and kurtosis stored in this
                      object so that the new value is factored into them.
                    - #sum() == sum() + val
                    - #mean() == mean()*current_n()/(current_n()+1) + val/(current_n()+1).
                      (i.e. the updated mean value that takes the new value into account)
                    - #variance() == the updated variance that takes this new value into account.
                    - #skewness() == the updated skewness that takes this new value into account.
                    - #ex_kurtosis() == the updated kurtosis that takes this new value into account.
                    - #current_n() == current_n() + 1
            !*/

            T sum (
            ) const;
            /*!
                ensures
                    - returns the sum of all the values presented to this object
                      so far.
            !*/

            T mean (
            ) const;
            /*!
                ensures
                    - returns the mean of all the values presented to this object
                      so far.
            !*/

            T variance (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the unbiased sample variance of all the values presented to this
                      object so far.
            !*/

            T stddev (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the unbiased sampled standard deviation of all the values
                      presented to this object so far.
            !*/

            T skewness (
            ) const;
            /*!
                requires
                    - current_n() > 2
                ensures
                    - returns the unbiased sample skewness of all the values presented
                      to this object so far.
            !*/

            T ex_kurtosis(
            ) const;
            /*!
                requires
                    - current_n() > 3
                ensures
                    - returns the unbiased sample kurtosis of all the values presented
                      to this object so far.
            !*/

            T max (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the largest value presented to this object so far.
            !*/

            T min (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the smallest value presented to this object so far.
            !*/

            T scale (
                const T& val
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - return (val-mean())/stddev();
            !*/

            running_stats operator+ (
                const running_stats& rhs
            ) const;
            /*!
                ensures
                    - returns a new running_stats object that represents the combination of all
                      the values given to *this and rhs.  That is, this function returns a
                      running_stats object, R, that is equivalent to what you would obtain if
                      all calls to this->add() and rhs.add() had instead been done to R.
            !*/
        };

        template <typename T>
        void serialize (
            const running_stats<T>& item,
            std::ostream& out
        );
        /*!
            provides serialization support
        !*/

        template <typename T>
        void deserialize (
            running_stats<T>& item,
            std::istream& in
        );
        /*!
            provides serialization support
        !*/

    // ----------------------------------------------------------------------------------------

        template <
            typename T
            >
        class running_scalar_covariance
        {
            /*!
                REQUIREMENTS ON T
                    - T must be a float, double, or long double type

                INITIAL VALUE
                    - mean_x() == 0
                    - mean_y() == 0
                    - current_n() == 0

                WHAT THIS OBJECT REPRESENTS
                    This object represents something that can compute the running covariance
                    of a stream of real number pairs.
            !*/

        public:

            running_scalar_covariance(
            );
            /*!
                ensures
                    - this object is properly initialized
            !*/

            void clear(
            );
            /*!
                ensures
                    - this object has its initial value
                    - clears all memory of any previous data points
            !*/

            void add (
                const T& x,
                const T& y
            );
            /*!
                ensures
                    - updates the statistics stored in this object so that
                      the new pair (x,y) is factored into them.
                    - #current_n() == current_n() + 1
            !*/

            T current_n (
            ) const;
            /*!
                ensures
                    - returns the number of points given to this object so far.
            !*/

            T mean_x (
            ) const;
            /*!
                ensures
                    - returns the mean value of all x samples presented to this object
                      via add().
            !*/

            T mean_y (
            ) const;
            /*!
                ensures
                    - returns the mean value of all y samples presented to this object
                      via add().
            !*/

            T covariance (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the covariance between all the x and y samples presented
                      to this object via add()
            !*/

            T correlation (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the correlation coefficient between all the x and y samples
                      presented to this object via add()
            !*/

            T variance_x (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the unbiased sample variance value of all x samples presented
                      to this object via add().
            !*/

            T variance_y (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the unbiased sample variance value of all y samples presented
                      to this object via add().
            !*/

            T stddev_x (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the unbiased sample standard deviation of all x samples
                      presented to this object via add().
            !*/

            T stddev_y (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the unbiased sample standard deviation of all y samples
                      presented to this object via add().
            !*/

            running_scalar_covariance operator+ (
                const running_covariance& rhs
            ) const;
            /*!
                ensures
                    - returns a new running_scalar_covariance object that represents the
                      combination of all the values given to *this and rhs.  That is, this
                      function returns a running_scalar_covariance object, R, that is
                      equivalent to what you would obtain if all calls to this->add() and
                      rhs.add() had instead been done to R.
            !*/
        };

    // ----------------------------------------------------------------------------------------

        template <
            typename T
            >
        class running_scalar_covariance_decayed
        {
            /*!
                REQUIREMENTS ON T
                    - T must be a float, double, or long double type

                INITIAL VALUE
                    - mean_x() == 0
                    - mean_y() == 0
                    - current_n() == 0

                WHAT THIS OBJECT REPRESENTS
                    This object represents something that can compute the running covariance of
                    a stream of real number pairs.  It is essentially the same as
                    running_scalar_covariance except that it forgets about data it has seen
                    after a certain period of time.  It does this by exponentially decaying old
                    statistics.
            !*/

        public:

            running_scalar_covariance_decayed(
                T decay_halflife = 1000
            );
            /*!
                requires
                    - decay_halflife > 0
                ensures
                    - #forget_factor() == std::pow(0.5, 1/decay_halflife);
                      (i.e. after decay_halflife calls to add() the data given to the first add
                      will be down weighted by 0.5 in the statistics stored in this object).
            !*/

            T forget_factor (
            ) const;
            /*!
                ensures
                    - returns the exponential forget factor used to forget old statistics when
                      add() is called.
            !*/

            void add (
                const T& x,
                const T& y
            );
            /*!
                ensures
                    - updates the statistics stored in this object so that
                      the new pair (x,y) is factored into them.
                    - #current_n() == current_n()*forget_factor() + forget_factor()
                    - Down weights old statistics by a factor of forget_factor().
            !*/

            T current_n (
            ) const;
            /*!
                ensures
                    - returns the effective number of points given to this object.   As add()
                      is called this value will converge to a constant, the value of which is
                      based on the decay_halflife supplied to the constructor.
            !*/

            T mean_x (
            ) const;
            /*!
                ensures
                    - returns the mean value of all x samples presented to this object
                      via add().
            !*/

            T mean_y (
            ) const;
            /*!
                ensures
                    - returns the mean value of all y samples presented to this object
                      via add().
            !*/

            T covariance (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the covariance between all the x and y samples presented
                      to this object via add()
            !*/

            T correlation (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the correlation coefficient between all the x and y samples
                      presented to this object via add()
            !*/

            T variance_x (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the sample variance value of all x samples presented
                      to this object via add().
            !*/

            T variance_y (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the sample variance value of all y samples presented
                      to this object via add().
            !*/

            T stddev_x (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the sample standard deviation of all x samples
                      presented to this object via add().
            !*/

            T stddev_y (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the sample standard deviation of all y samples
                      presented to this object via add().
            !*/
        };

    // ----------------------------------------------------------------------------------------

        template <
            typename T
            >
        class running_stats_decayed
        {
            /*!
                REQUIREMENTS ON T
                    - T must be a float, double, or long double type

                INITIAL VALUE
                    - mean() == 0
                    - current_n() == 0

                WHAT THIS OBJECT REPRESENTS
                    This object represents something that can compute the running mean and
                    variance of a stream of real numbers.  It is similar to running_stats
                    except that it forgets about data it has seen after a certain period of
                    time.  It does this by exponentially decaying old statistics.
            !*/

        public:

            running_stats_decayed(
                T decay_halflife = 1000
            );
            /*!
                requires
                    - decay_halflife > 0
                ensures
                    - #forget_factor() == std::pow(0.5, 1/decay_halflife);
                      (i.e. after decay_halflife calls to add() the data given to the first add
                      will be down weighted by 0.5 in the statistics stored in this object).
            !*/

            T forget_factor (
            ) const;
            /*!
                ensures
                    - returns the exponential forget factor used to forget old statistics when
                      add() is called.
            !*/

            void add (
                const T& x
            );
            /*!
                ensures
                    - updates the statistics stored in this object so that x is factored into
                      them.
                    - #current_n() == current_n()*forget_factor() + forget_factor()
                    - Down weights old statistics by a factor of forget_factor().
            !*/

            T current_n (
            ) const;
            /*!
                ensures
                    - returns the effective number of points given to this object.   As add()
                      is called this value will converge to a constant, the value of which is
                      based on the decay_halflife supplied to the constructor.
            !*/

            T mean (
            ) const;
            /*!
                ensures
                    - returns the mean value of all x samples presented to this object
                      via add().
            !*/

            T variance (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the sample variance value of all x samples presented to this
                      object via add().
            !*/

            T stddev (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the sample standard deviation of all x samples presented to this
                      object via add().
            !*/

        };

        template <typename T>
        void serialize (
            const running_stats_decayed<T>& item,
            std::ostream& out
        );
        /*!
            provides serialization support
        !*/

        template <typename T>
        void deserialize (
            running_stats_decayed<T>& item,
            std::istream& in
        );
        /*!
            provides serialization support
        !*/

    // ----------------------------------------------------------------------------------------

        template <
            typename matrix_type
            >
        class running_covariance
        {
            /*!
                REQUIREMENTS ON matrix_type
                    Must be some type of dlib::matrix.

                INITIAL VALUE
                    - in_vector_size() == 0
                    - current_n() == 0

                WHAT THIS OBJECT REPRESENTS
                    This object is a simple tool for computing the mean and
                    covariance of a sequence of vectors.
            !*/
        public:

            typedef typename matrix_type::mem_manager_type mem_manager_type;
            typedef typename matrix_type::type scalar_type;
            typedef typename matrix_type::layout_type layout_type;
            typedef matrix<scalar_type,0,0,mem_manager_type,layout_type> general_matrix;
            typedef matrix<scalar_type,0,1,mem_manager_type,layout_type> column_matrix;

            running_covariance(
            );
            /*!
                ensures
                    - this object is properly initialized
            !*/

            void clear(
            );
            /*!
                ensures
                    - this object has its initial value
                    - clears all memory of any previous data points
            !*/

            long current_n (
            ) const;
            /*!
                ensures
                    - returns the number of samples that have been presented to this object
            !*/

            long in_vector_size (
            ) const;
            /*!
                ensures
                    - if (this object has been presented with any input vectors or
                      set_dimension() has been called) then
                        - returns the dimension of the column vectors used with this object
                    - else
                        - returns 0
            !*/

            void set_dimension (
                long size
            );
            /*!
                requires
                    - size > 0
                ensures
                    - #in_vector_size() == size
                    - #current_n() == 0
            !*/

            template <typename T>
            void add (
                const T& val
            );
            /*!
                requires
                    - val must represent a column vector.  It can either be a dlib::matrix
                      object or some kind of unsorted sparse vector type.  See the top of
                      dlib/svm/sparse_vector_abstract.h for a definition of unsorted sparse vector.
                    - val must have a number of dimensions which is compatible with the current
                      setting of in_vector_size().  In particular, this means that the
                      following must hold:
                        - if (val is a dlib::matrix) then
                            - in_vector_size() == 0 || val.size() == val_vector_size()
                        - else
                            - max_index_plus_one(val) <= in_vector_size()
                            - in_vector_size() > 0
                              (i.e. you must call set_dimension() prior to calling add() if
                              you want to use sparse vectors.)
                ensures
                    - updates the mean and covariance stored in this object so that
                      the new value is factored into them.
                    - if (val is a dlib::matrix) then
                        - #in_vector_size() == val.size()
            !*/

            const column_matrix mean (
            ) const;
            /*!
                requires
                    - in_vector_size() != 0
                ensures
                    - returns the mean of all the vectors presented to this object
                      so far.
            !*/

            const general_matrix covariance (
            ) const;
            /*!
                requires
                    - in_vector_size() != 0
                    - current_n() > 1
                ensures
                    - returns the unbiased sample covariance matrix for all the vectors
                      presented to this object so far.
            !*/

            const running_covariance operator+ (
                const running_covariance& item
            ) const;
            /*!
                requires
                    - in_vector_size() == 0 || item.in_vector_size() == 0 || in_vector_size() == item.in_vector_size()
                      (i.e. the in_vector_size() of *this and item must match or one must be zero)
                ensures
                    - returns a new running_covariance object that represents the combination of all
                      the vectors given to *this and item.  That is, this function returns a
                      running_covariance object, R, that is equivalent to what you would obtain if all
                      calls to this->add() and item.add() had instead been done to R.
            !*/
        };

    // ----------------------------------------------------------------------------------------

        template <
            typename matrix_type
            >
        class running_cross_covariance
        {
            /*!
                REQUIREMENTS ON matrix_type
                    Must be some type of dlib::matrix.

                INITIAL VALUE
                    - x_vector_size() == 0
                    - y_vector_size() == 0
                    - current_n() == 0

                WHAT THIS OBJECT REPRESENTS
                    This object is a simple tool for computing the mean and cross-covariance
                    matrices of a sequence of pairs of vectors.
            !*/

        public:

            typedef typename matrix_type::mem_manager_type mem_manager_type;
            typedef typename matrix_type::type scalar_type;
            typedef typename matrix_type::layout_type layout_type;
            typedef matrix<scalar_type,0,0,mem_manager_type,layout_type> general_matrix;
            typedef matrix<scalar_type,0,1,mem_manager_type,layout_type> column_matrix;

            running_cross_covariance(
            );
            /*!
                ensures
                    - this object is properly initialized
            !*/

            void clear(
            );
            /*!
                ensures
                    - This object has its initial value.
                    - Clears all memory of any previous data points.
            !*/

            long x_vector_size (
            ) const;
            /*!
                ensures
                    - if (this object has been presented with any input vectors or
                      set_dimensions() has been called) then
                        - returns the dimension of the x vectors given to this object via add().
                    - else
                        - returns 0
            !*/

            long y_vector_size (
            ) const;
            /*!
                ensures
                    - if (this object has been presented with any input vectors or
                      set_dimensions() has been called) then
                        - returns the dimension of the y vectors given to this object via add().
                    - else
                        - returns 0
            !*/

            void set_dimensions (
                long x_size,
                long y_size
            );
            /*!
                requires
                    - x_size > 0
                    - y_size > 0
                ensures
                    - #x_vector_size() == x_size
                    - #y_vector_size() == y_size
                    - #current_n() == 0
            !*/

            long current_n (
            ) const;
            /*!
                ensures
                    - returns the number of samples that have been presented to this object.
            !*/

            template <typename T, typename U>
            void add (
                const T& x,
                const U& y
            );
            /*!
                requires
                    - x and y must represent column vectors.  They can either be dlib::matrix
                      objects or some kind of unsorted sparse vector type.  See the top of
                      dlib/svm/sparse_vector_abstract.h for a definition of unsorted sparse vector.
                    - x and y must have a number of dimensions which is compatible with the
                      current setting of x_vector_size() and y_vector_size().  In particular,
                      this means that the following must hold:
                        - if (x or y is a sparse vector type) then
                            - x_vector_size() > 0 && y_vector_size() > 0
                              (i.e. you must call set_dimensions() prior to calling add() if
                              you want to use sparse vectors.)
                        - if (x is a dlib::matrix) then
                            - x_vector_size() == 0 || x.size() == x_vector_size()
                        - else
                            - max_index_plus_one(x) <= x_vector_size()
                        - if (y is a dlib::matrix) then
                            - y_vector_size() == 0 || y.size() == y_vector_size()
                        - else
                            - max_index_plus_one(y) <= y_vector_size()
                ensures
                    - updates the mean and cross-covariance matrices stored in this object so
                      that the new (x,y) vector pair is factored into them.
                    - if (x is a dlib::matrix) then
                        - #x_vector_size() == x.size()
                    - if (y is a dlib::matrix) then
                        - #y_vector_size() == y.size()
            !*/

            const column_matrix mean_x (
            ) const;
            /*!
                requires
                    - current_n() != 0
                ensures
                    - returns the mean of all the x vectors presented to this object so far.
                    - The returned vector will have x_vector_size() dimensions.
            !*/

            const column_matrix mean_y (
            ) const;
            /*!
                requires
                    - current_n() != 0
                ensures
                    - returns the mean of all the y vectors presented to this object so far.
                    - The returned vector will have y_vector_size() dimensions.
            !*/

            const general_matrix covariance_xy (
            ) const;
            /*!
                requires
                    - current_n() > 1
                ensures
                    - returns the unbiased sample cross-covariance matrix for all the vector
                      pairs presented to this object so far.  In particular, returns a matrix
                      M such that:
                        - M.nr() == x_vector_size()
                        - M.nc() == y_vector_size()
                        - M == the cross-covariance matrix of the data given to add().
            !*/

            const running_cross_covariance operator+ (
                const running_cross_covariance& item
            ) const;
            /*!
                requires
                    - x_vector_size() == 0 || item.x_vector_size() == 0 || x_vector_size() == item.x_vector_size()
                      (i.e. the x_vector_size() of *this and item must match or one must be zero)
                    - y_vector_size() == 0 || item.y_vector_size() == 0 || y_vector_size() == item.y_vector_size()
                      (i.e. the y_vector_size() of *this and item must match or one must be zero)
                ensures
                    - returns a new running_cross_covariance object that represents the
                      combination of all the vectors given to *this and item.  That is, this
                      function returns a running_cross_covariance object, R, that is equivalent
                      to what you would obtain if all calls to this->add() and item.add() had
                      instead been done to R.
            !*/
        };

    // ----------------------------------------------------------------------------------------

        template <
            typename matrix_type
            >
        class vector_normalizer
        {
            /*!
                REQUIREMENTS ON matrix_type
                    - must be a dlib::matrix object capable of representing column
                      vectors

                INITIAL VALUE
                    - in_vector_size() == 0
                    - out_vector_size() == 0
                    - means().size() == 0
                    - std_devs().size() == 0

                WHAT THIS OBJECT REPRESENTS
                    This object represents something that can learn to normalize a set
                    of column vectors.  In particular, normalized column vectors should
                    have zero mean and a variance of one.

                THREAD SAFETY
                    Note that this object contains a cached matrix object it uses
                    to store intermediate results for normalization.  This avoids
                    needing to reallocate it every time this object performs normalization
                    but also makes it non-thread safe.  So make sure you don't share
                    instances of this object between threads.
            !*/

        public:
            typedef typename matrix_type::mem_manager_type mem_manager_type;
            typedef typename matrix_type::type scalar_type;
            typedef matrix_type result_type;

            template <typename vector_type>
            void train (
                const vector_type& samples
            );
            /*!
                requires
                    - samples.size() > 0
                    - samples == a column matrix or something convertible to a column
                      matrix via mat().  Also, x should contain
                      matrix_type objects that represent nonempty column vectors.
                    - samples does not contain any infinite or NaN values
                ensures
                    - #in_vector_size() == samples(0).nr()
                    - #out_vector_size() == samples(0).nr()
                    - This object has learned how to normalize vectors that look like
                      vectors in the given set of samples.
                    - #means() == mean(samples)
                    - #std_devs() == reciprocal(sqrt(variance(samples)));
            !*/

            long in_vector_size (
            ) const;
            /*!
                ensures
                    - returns the number of rows that input vectors are
                      required to contain if they are to be normalized by
                      this object.
            !*/

            long out_vector_size (
            ) const;
            /*!
                ensures
                    - returns the number of rows in the normalized vectors
                      that come out of this object.
            !*/

            const matrix_type& means (
            ) const;
            /*!
                ensures
                    - returns a matrix M such that:
                        - M.nc() == 1
                        - M.nr() == in_vector_size()
                        - M(i) == the mean of the ith input feature shown to train()
            !*/

            const matrix_type& std_devs (
            ) const;
            /*!
                ensures
                    - returns a matrix SD such that:
                        - SD.nc() == 1
                        - SD.nr() == in_vector_size()
                        - SD(i) == the reciprocal of the standard deviation of the ith
                          input feature shown to train()
            !*/

            const result_type& operator() (
                const matrix_type& x
            ) const;
            /*!
                requires
                    - x.nr() == in_vector_size()
                    - x.nc() == 1
                ensures
                    - returns a normalized version of x, call it Z, that has the
                      following properties:
                        - Z.nr() == out_vector_size()
                        - Z.nc() == 1
                        - the mean of each element of Z is 0
                        - the variance of each element of Z is 1
                        - Z == pointwise_multiply(x-means(), std_devs());
            !*/

            void swap (
                vector_normalizer& item
            );
            /*!
                ensures
                    - swaps *this and item
            !*/
        };

        template <
            typename matrix_type
            >
        inline void swap (
            vector_normalizer<matrix_type>& a,
            vector_normalizer<matrix_type>& b
        ) { a.swap(b); }
        /*!
            provides a global swap function
        !*/

        template <
            typename matrix_type,
            >
        void deserialize (
            vector_normalizer<matrix_type>& item,
            std::istream& in
        );
        /*!
            provides deserialization support
        !*/

        template <
            typename matrix_type,
            >
        void serialize (
            const vector_normalizer<matrix_type>& item,
            std::ostream& out
        );
        /*!
            provides serialization support
        !*/

    // ----------------------------------------------------------------------------------------

        template <
            typename matrix_type
            >
        class vector_normalizer_pca
        {
            /*!
                REQUIREMENTS ON matrix_type
                    - must be a dlib::matrix object capable of representing column
                      vectors

                INITIAL VALUE
                    - in_vector_size() == 0
                    - out_vector_size() == 0
                    - means().size() == 0
                    - std_devs().size() == 0
                    - pca_matrix().size() == 0

                WHAT THIS OBJECT REPRESENTS
                    This object represents something that can learn to normalize a set
                    of column vectors.  In particular, normalized column vectors should
                    have zero mean and a variance of one.

                    Also, this object uses principal component analysis for the purposes
                    of reducing the number of elements in a vector.

                THREAD SAFETY
                    Note that this object contains a cached matrix object it uses
                    to store intermediate results for normalization.  This avoids
                    needing to reallocate it every time this object performs normalization
                    but also makes it non-thread safe.  So make sure you don't share
                    instances of this object between threads.
            !*/

        public:
            typedef typename matrix_type::mem_manager_type mem_manager_type;
            typedef typename matrix_type::type scalar_type;
            typedef matrix<scalar_type,0,1,mem_manager_type> result_type;

            template <typename vector_type>
            void train (
                const vector_type& samples,
                const double eps = 0.99
            );
            /*!
                requires
                    - 0 < eps <= 1
                    - samples.size() > 0
                    - samples == a column matrix or something convertible to a column
                      matrix via mat().  Also, x should contain
                      matrix_type objects that represent nonempty column vectors.
                    - samples does not contain any infinite or NaN values
                ensures
                    - This object has learned how to normalize vectors that look like
                      vectors in the given set of samples.
                    - Principal component analysis is performed to find a transform
                      that might reduce the number of output features.
                    - #in_vector_size() == samples(0).nr()
                    - 0 < #out_vector_size() <= samples(0).nr()
                    - eps is a number that controls how "lossy" the pca transform will be.
                      Large values of eps result in #out_vector_size() being larger and
                      smaller values of eps result in #out_vector_size() being smaller.
                    - #means() == mean(samples)
                    - #std_devs() == reciprocal(sqrt(variance(samples)));
                    - #pca_matrix() == the PCA transform matrix that is out_vector_size()
                      rows by in_vector_size() columns.
            !*/

            long in_vector_size (
            ) const;
            /*!
                ensures
                    - returns the number of rows that input vectors are
                      required to contain if they are to be normalized by
                      this object.
            !*/

            long out_vector_size (
            ) const;
            /*!
                ensures
                    - returns the number of rows in the normalized vectors
                      that come out of this object.
            !*/

            const matrix<scalar_type,0,1,mem_manager_type>& means (
            ) const;
            /*!
                ensures
                    - returns a matrix M such that:
                        - M.nc() == 1
                        - M.nr() == in_vector_size()
                        - M(i) == the mean of the ith input feature shown to train()
            !*/

            const matrix<scalar_type,0,1,mem_manager_type>& std_devs (
            ) const;
            /*!
                ensures
                    - returns a matrix SD such that:
                        - SD.nc() == 1
                        - SD.nr() == in_vector_size()
                        - SD(i) == the reciprocal of the standard deviation of the ith
                          input feature shown to train()
            !*/

            const matrix<scalar_type,0,0,mem_manager_type>& pca_matrix (
            ) const;
            /*!
                ensures
                    - returns a matrix PCA such that:
                        - PCA.nr() == out_vector_size()
                        - PCA.nc() == in_vector_size()
                        - PCA == the principal component analysis transformation
                          matrix
            !*/

            const result_type& operator() (
                const matrix_type& x
            ) const;
            /*!
                requires
                    - x.nr() == in_vector_size()
                    - x.nc() == 1
                ensures
                    - returns a normalized version of x, call it Z, that has the
                      following properties:
                        - Z.nr() == out_vector_size()
                        - Z.nc() == 1
                        - the mean of each element of Z is 0
                        - the variance of each element of Z is 1
                        - Z == pca_matrix()*pointwise_multiply(x-means(), std_devs());
            !*/

            void swap (
                vector_normalizer_pca& item
            );
            /*!
                ensures
                    - swaps *this and item
            !*/
        };

        template <
            typename matrix_type
            >
        inline void swap (
            vector_normalizer_pca<matrix_type>& a,
            vector_normalizer_pca<matrix_type>& b
        ) { a.swap(b); }
        /*!
            provides a global swap function
        !*/

        template <
            typename matrix_type,
            >
        void deserialize (
            vector_normalizer_pca<matrix_type>& item,
            std::istream& in
        );
        /*!
            provides deserialization support
        !*/

        template <
            typename matrix_type,
            >
        void serialize (
            const vector_normalizer_pca<matrix_type>& item,
            std::ostream& out
        );
        /*!
            provides serialization support
        !*/

    // ----------------------------------------------------------------------------------------

    }

    #endif // DLIB_STATISTICs_ABSTRACT_