=================
Zero-Mean Scaling
=================
:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/MultiDimensionalAlgorithm/index`

*Zero-Mean Scaling* algorithm is a algorithm of normalization which uses the mean and standard deviation of the data set to normalize each data point. The basic formulas are stated as follows:

.. math::

    \hat{Y}_i = \frac{(Y_i - \mu)}{\sigma}

for a one-dimensional data vector,

.. math::

   \hat{Y}_{i,j} = \frac{(Y_{i,j} - \mu)}{\sigma}

for a two-dimensional data matrix,

.. math::

   \hat{Y}_{i,j,k} = \frac{(Y_{i,j,k} - \mu)}{\sigma}

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. :math:`\mu` is the mean, and :math:`\sigma` is the standard deviation of the input data.

.. 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:`\mu`                | :math:`\mu \in \mathbb{R}`                                                                                                                                         |                                                        | Mean value of :math:`Y`                  |         |
+----------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------+------------------------------------------+---------+
| :math:`\sigma`             | :math:`\sigma \in \mathbb{R}`                                                                                                                                      |                                                        | Standard deviation of :math:`Y`          |         |
+----------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------+------------------------------------------+---------+

.. rubric:: Output Parameters

+----------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------+-------------+---------+
| Parameter                  | Type                                                                                                                                                           | Constraint                                             | Description | Remarks |
+============================+================================================================================================================================================================+========================================================+=============+=========+
| :math:`\hat{Y}`            | :math:`\hat{Y} \in (-1,1)^{N_1}, (-1,1)^{N_1 \times N_2}, \text{ or } (-1,1)^{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/DataScalingWithZeroMeanScaling/index`

.. rubric:: References

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