===============
Min-Max Scaling
===============
:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/MultiDimensionalAlgorithm/index`

The basic formula of *Min-Max Scaling* algorithm is stated as follows:

.. math::

    \hat{Y}_i = \frac{Y_i - min(Y)}{max(Y)-min(Y)} \cdot (\text{max}^{\text{new}} - \text{min}^{\text{new}}) + \text{min}^{\text{new}}

for a one-dimensional data vector,

.. math::

   \hat{Y}_{i,j} = \frac{Y_{i,j} - min(Y)}{max(Y)-min(Y)} \cdot (\text{max}^{\text{new}} - \text{min}^{\text{new}}) + \text{min}^{\text{new}}

for a two-dimensional data matrix, and

.. math::

   \hat{Y}_{i,j,k} = \frac{Y_{i,j,k} - min(Y)}{max(Y)-min(Y)} \cdot (\text{max}^{\text{new}} - \text{min}^{\text{new}}) + \text{min}^{\text{new}}

for a three-dimensional data matrix, where :math:`Y` represents the input data and :math:`i, j, k` represent the corresponding indices for the data entry considered.

.. rubric:: Input Parameters

+----------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+-------------+---------+
| Parameter                                          | Type                                                                                                                                                               | Constraint                                                                             | Description | Remarks |
+====================================================+====================================================================================================================================================================+========================================================================================+=============+=========+
| :math:`Y`                                          | :math:`Y \in \mathbb R^{N_1}, \mathbb R^{N_1 \times N_2}, \text{ or } \mathbb R^{N_1 \times N_2 \times N_3}, \ldots`                                               | :math:`N_1, N_2, N_3 \in \mathbb{N}`                                                   |             |         |
+----------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+-------------+---------+
| :math:`\text{min}^{\text{new}}`                    | :math:`\text{min}^{\text{new}} \in \mathbb R`                                                                                                                      | :math:`\text{min}^{\text{new}} \leq \text{max}^{\text{new}}`                           |             |         |
+----------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+-------------+---------+
| :math:`\text{max}^{\text{new}}`                    | :math:`\text{max}^{\text{new}} \in \mathbb R`                                                                                                                      | :math:`\text{min}^{\text{new}} \leq \text{max}^{\text{new}}`                           |             |         |
+----------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+-------------+---------+

.. rubric:: Output Parameters

+----------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------+-------------+---------+
| Parameter                  | Type                                                                                                                                                                                                                                                                                                                                           | Constraint                                             | Description | Remarks |
+============================+================================================================================================================================================================================================================================================================================================================================================+========================================================+=============+=========+
| :math:`\hat{Y}`            | :math:`\hat{Y} \in (\text{min}^{\text{new}}, \text{max}^{\text{new}})^{N_1}, (\text{min}^{\text{new}}, \text{max}^{\text{new}})^{N_1 \times N_2}, \text{ or } (\text{min}^{\text{new}}, \text{max}^{\text{new}})^{N_1 \times N_2 \times N_3}, \ldots`                                                                                          | :math:`N_1, N_2, N_3 \in \mathbb{N}`                   |             |         |
+----------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------+-------------+---------+

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/DataTransformation/DataScaling/DataScalingWithMinMaxScaling/index`

.. rubric:: References

- J.\  Han, M. Kamber and J. Pei, Data Mining - Concepts and Techniques, 3rd ed., Amsterdam: Morgan Kaufmann Publishers, 2012.