================
Chi-Squared Test
================
:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/MultiDimensionalAlgorithm/index`

*Chi-Squared Test* algorithm is used to analyze a correlation relationship between two attributes. The :math:`\chi^2`\ -value for attributes :math:`A` and :math:`B` is computed as:

.. math::

   \chi^2 = \sum_{i=1}^{N} \sum_{j=1}^{M} \frac{(o_{ij} - e_{ij})^2}{e_{ij}} \, \text{,}

where :math:`o_{ij}` is the observed frequency of the joint event of pair :math:`(A_i, B_j)` and :math:`e_{ij}` is the corresponding expected frequency.

.. rubric:: Input Parameters

+--------------------------------+--------------------------------------------------------------------+----------------------------------------+-------------+---------+
| Parameter                      | Type                                                               | Constraint                             | Description | Remarks |
+================================+====================================================================+========================================+=============+=========+
| :math:`A`                      | :math:`A = \{a_i\}, i = 1, 2, \ldots, N`                           | :math:`N \in \mathbb{N}`               |             |         |
+--------------------------------+--------------------------------------------------------------------+----------------------------------------+-------------+---------+
| :math:`B`                      | :math:`B = \{b_i\}, i = 1, 2, \ldots, M`                           | :math:`M \in \mathbb{N}`               |             |         |
+--------------------------------+--------------------------------------------------------------------+----------------------------------------+-------------+---------+
| :math:`(e_{ij})`               | :math:`(e_{ij}) \in \mathbb N^{N \times M}`                        |                                        |             |         |
+--------------------------------+--------------------------------------------------------------------+----------------------------------------+-------------+---------+

.. rubric:: Output Parameters

+----------------------------+------------------------------------------------+------------+-------------+---------+
| Parameter                  | Type                                           | Constraint | Description | Remarks |
+============================+================================================+============+=============+=========+
| :math:`\chi^2`             | :math:`\chi^2 \in \mathbb R`                   |            |             |         |
+----------------------------+------------------------------------------------+------------+-------------+---------+

.. rubric:: Tool Support

* :doc:`/Tools/MatlabTool/index`

  For details refer to the online documentation of the function `'chi2gof' <http://www.mathworks.de/help/toolbox/stats/chi2gof.html>`__.

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/DataDiscretization/DataDiscretizationWithChiSquaredTest/index`

* :doc:`/DataPreprocessing/DataCleaning/OutlierDetection/OutlierDetectionWithChiSquaredTest/index`

* :doc:`/DataPreprocessing/DataIntegration/RedundancyDetection/RedundancyDetectionWithChiSquaredTest/index`

.. rubric:: References

- P.E.\  Greenwood, M.S. Nikulin, A guide to chi-squared testing, Wiley, New York, 1996.