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.. highlight:: cpp
.. default-domain:: cpp

.. _alg_linear-regression:

=================
Linear Regression
=================

.. include::  ../../../includes/linear-regression/linear-regression-introduction.rst

------------------------
Mathematical Formulation
------------------------

.. _linreg_t_math:

Training
---------

Given a :math:`(x_{11}, \ldots, x_{1p}, y_1, \ldots, x_{n1}, \ldots, x_{np}, y_n)` set  of
training data, :math:`n \gg p`. The :math:`X` matrix of the :math:`n \times p` size contains
:math:`x_{ij}`, :math:`i=1, \ldots, n`, :math:`j = 1, \ldots, p` observations of independent
variables.

To estimate the coefficients :math:`(\beta_{0j}, \ldots, \beta_{pj})`,
you can use the Normal Equation method.

.. _linreg_p_math:

Partial Training
----------------

Given a :math:`(x_{11}, \ldots, x_{1p}, y_1, \ldots, x_{n1}, \ldots, x_{np}, y_n)` block of
training data, :math:`n \gg p`. The :math:`X` matrix of :math:`n \times p` size contains
:math:`x_{ij}`, :math:`i=1, \ldots, n`, :math:`j = 1, \ldots, p` observations of independent
variables.

To compute a cross product of Linear Regression, you can use the Normal Equation method.

.. _linreg_f_math:

Finalize Training
-----------------

Given a partial result with partial products.

To compute a cross product of Linear Regression, you can use the Normal Equation method.


.. _linreg_i_math:

Inference
---------
The linear regression-based prediction is made for the input vector
using the :math:`y_j = \beta_{0j} + \beta_{1j}x_1 + \ldots + \beta_{pj}x_p` equation
for each :math:`j=1, \ldots, k`.

.. _norm_eq:

Computation Method: *norm_eq*
-----------------------------
The method computes means, variance-covariance, or correlation matrix for the dense data.
This is the default and the only supported method.

---------------------
Programming Interface
---------------------

Refer to :ref:`API Reference: Linear Regression <api_linear-regression>`.

-----------
Online mode
-----------

The algorithm supports online mode.

----------------
Distributed mode
----------------

The algorithm supports distributed execution in SPMD mode (only on GPU).