```// Copyright (C) 2012  Davis E. King (davis@dlib.net)
#undef DLIB_RLs_ABSTRACT_Hh_
#ifdef DLIB_RLs_ABSTRACT_Hh_

#include "../matrix/matrix_abstract.h"
#include "function_abstract.h"

namespace dlib
{

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

class rls
{
/*!
WHAT THIS OBJECT REPRESENTS
This is an implementation of the linear version of the recursive least
squares algorithm.  It accepts training points incrementally and, at
each step, maintains the solution to the following optimization problem:
find w minimizing: 0.5*dot(w,w) + C*sum_i(y_i - trans(x_i)*w)^2
Where (x_i,y_i) are training pairs.  x_i is some vector and y_i is a target
scalar value.

This object can also be configured to use exponential forgetting.  This is
where each training example is weighted by pow(forget_factor, i), where i
indicates the sample's age.  So older samples are weighted less in the
least squares solution and therefore become forgotten after some time.
Therefore, with forgetting, this object solves the following optimization
problem at each step:
find w minimizing: 0.5*dot(w,w) + C*sum_i pow(forget_factor, i)*(y_i - trans(x_i)*w)^2
Where i starts at 0 and i==0 corresponds to the most recent training point.
!*/

public:

explicit rls(
double forget_factor,
double C = 1000,
bool apply_forget_factor_to_C = false
);
/*!
requires
- 0 < forget_factor <= 1
- 0 < C
ensures
- #get_w().size() == 0
- #get_c() == C
- #get_forget_factor() == forget_factor
- #should_apply_forget_factor_to_C() == apply_forget_factor_to_C
!*/

rls(
);
/*!
ensures
- #get_w().size() == 0
- #get_c() == 1000
- #get_forget_factor() == 1
- #should_apply_forget_factor_to_C() == false
!*/

double get_c(
) const;
/*!
ensures
- returns the regularization parameter.  It is the parameter
that determines the trade-off between trying to fit the training
data or allowing more errors but hopefully improving the generalization
of the resulting regression.  Larger values encourage exact fitting while
smaller values of C may encourage better generalization.
!*/

double get_forget_factor(
) const;
/*!
ensures
- returns the exponential forgetting factor.  A value of 1 disables forgetting
and results in normal least squares regression.  On the other hand, a smaller
value causes the regression to forget about old training examples and prefer
instead to fit more recent examples.  The closer the forget factor is to
zero the faster old examples are forgotten.
!*/

bool should_apply_forget_factor_to_C (
) const;
/*!
ensures
- If this function returns false then it means we are optimizing the
objective function discussed in the WHAT THIS OBJECT REPRESENTS section
above. However, if it returns true then we will allow the forget factor
(get_forget_factor()) to be applied to the C value which causes the
algorithm to slowly increase C and convert into a textbook version of RLS
without regularization.   The main reason you might want to do this is
because it can make the algorithm run significantly faster.
!*/

template <typename EXP>
void train (
const matrix_exp<EXP>& x,
double y
)
/*!
requires
- is_col_vector(x) == true
- if (get_w().size() != 0) then
- x.size() == get_w().size()
(i.e. all training examples must have the same
dimensionality)
ensures
- #get_w().size() == x.size()
- updates #get_w() such that it contains the solution to the least
squares problem of regressing the given x onto the given y as well
as all the previous training examples supplied to train().
!*/

const matrix<double,0,1>& get_w(
) const;
/*!
ensures
- returns the regression weights.  These are the values learned by the
least squares procedure.  If train() has not been called then this
function returns an empty vector.
!*/

template <typename EXP>
double operator() (
const matrix_exp<EXP>& x
) const;
/*!
requires
- is_col_vector(x) == true
- get_w().size() == x.size()
ensures
- returns dot(x, get_w())
!*/

decision_function<linear_kernel<matrix<double,0,1> > > get_decision_function (
) const;
/*!
requires
- get_w().size() != 0
ensures
- returns a decision function DF such that:
- DF(x) == dot(x, get_w())
!*/

};

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

void serialize (
const rls& item,
std::ostream& out
);
/*!
provides serialization support
!*/

void deserialize (
rls& item,
std::istream& in
);
/*!
provides deserialization support
!*/

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

}

#endif // DLIB_RLs_ABSTRACT_Hh_

```