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Min-max
=======

Min-max normalization is an algorithm to linearly scale the observations by each feature (column) into the range :math:`[a, b]`.

Problem Statement
*****************

Given a set :math:`X` of :math:`n` feature vectors :math:`x_1 = (x_{11}, \ldots, x_{1p}), \ldots, x_n = (x_{n1}, \ldots, x_{np})`
of dimension :math:`p`, the problem is to compute the matrix :math:`Y = (y_{ij})_{n \times p}` where the :math:`j`-th column
:math:`(Y)_j = (y_{ij})_{i = 1, \ldots, n}` is obtained as a result of normalizing the column
:math:`(X)_j = (x_{ij})_{i = 1, \ldots, n}`  of the original matrix as:

.. math::

    y_{ij} = a + \frac {x_{ij} - \min(j)}{\max(j) - \min(j)} (b-a),

where:

.. math::
    \min(j) = \min _{i = 1, \ldots, n} x_{ij},

.. math::
    \max(j) = \max _{i = 1, \ldots, n} x_{ij},

:math:`a` and :math:`b` are the parameters of the algorithm.

Batch Processing
****************

Algorithm Input
---------------

The min-max normalization algorithm accepts the input described below.
Pass the ``Input ID`` as a parameter to the methods that provide input for your algorithm.
For more details, see :ref:`algorithms`.

.. tabularcolumns::  |\Y{0.2}|\Y{0.8}|

.. list-table:: Algorithm Input for Min-max (Batch Processing)
   :widths: 10 60
   :header-rows: 1

   * - Input ID
     - Input
   * - ``data``
     - Pointer to the numeric table of size :math:`n \times p`.

       .. note:: This table can be an object of any class derived from ``NumericTable``.

Algorithm Parameters
--------------------

The min-max normalization algorithm has the following parameters:

.. tabularcolumns::  |\Y{0.15}|\Y{0.15}|\Y{0.7}|

.. list-table:: Algorithm Parameters for Min-max (Batch Processing)
   :header-rows: 1
   :widths: 10 10 60
   :align: left
   :class: longtable

   * - Parameter
     - Default Value
     - Description
   * - ``algorithmFPType``
     - ``float``
     - The floating-point type that the algorithm uses for intermediate computations. Can be ``float`` or ``double``.
   * - ``method``
     - ``defaultDense``
     - Performance-oriented computation method, the only method supported by the algorithm.
   * - ``lowerBound``
     - :math:`0.0`
     - The lower bound of the range to which the normalization scales values of the features.
   * - ``upperBound``
     - :math:`1.0`
     - The upper bound of the range to which the normalization scales values of the features.
   * - ``moments``
     - `SharedPtr<low_order_moments::Batch<algorithmFPType, low_order_moments::defaultDense> >`
     - Pointer to the low order moments algorithm that computes minimums and maximums
       to be used for min-max normalization with the defaultDense method.
       For more details, see :ref:`Batch Processing for Moments of Low Order <moments_batch>`.

Algorithm Output
----------------

The min-max normalization algorithm calculates the result described below.
Pass the ``Result ID`` as a parameter to the methods that access the results of your algorithm.
For more details, see ``Algorithms``.

.. tabularcolumns::  |\Y{0.2}|\Y{0.8}|

.. list-table:: Algorithm Output for Min-max (Batch Processing)
   :widths: 10 60
   :header-rows: 1

   * - Result ID
     - Result
   * - ``normalizedData``
     - Pointer to the :math:`n \times p` numeric table that stores the result of normalization.

       .. note::

          By default, the result is an object of the ``HomogenNumericTable`` class,
          but you can define the result as an object of any class derived from ``NumericTable``
          except ``PackedTriangularMatrix``, ``PackedSymmetricMatrix``, and ``CSRNumericTable``.

Examples
********

.. tabs::

  .. tab:: C++ (CPU)

    Batch Processing:

    - :cpp_example:`minmax_dense_batch.cpp <normalization/minmax_dense_batch.cpp>`

  .. tab:: Python*

    Batch Processing:

    - :daal4py_example:`normalization_minmax.py`