============================
Principal Component Analysis
============================
:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/MultiDimensionalAlgorithm/index`

*Principal Component Analysis* algorithm is a well-known statistical procedure that can identify patterns in data and express the data using a small number of principal components.

For details refer to the online document `A tutorial on Principal Components Analysis <http://www.sccg.sk/~haladova/principal_components.pdf>`__. 

.. rubric:: Input Parameters

+--------------------+--------------------------------------------------------+--------------------------------------------------------+--------------------------------------------------------------+---------+
| Parameter          | Type                                                   | Constraint                                             | Description                                                  | Remarks |
+====================+========================================================+========================================================+==============================================================+=========+
| :math:`Y`          | :math:`Y \in \mathbb R^{N \times p}`                   | :math:`N \in \mathbb{N}, N \geq p`                     | Normalized input data of size :math:`N \times p`             |         |
+--------------------+--------------------------------------------------------+--------------------------------------------------------+--------------------------------------------------------------+---------+

.. rubric:: Output Parameters

+------------------------------------+------------------------------------------------------------------------+------------+-----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------+
| Parameter                          | Type                                                                   | Constraint | Description                                   | Remarks                                                                                                                                            |
+====================================+========================================================================+============+===============================================+====================================================================================================================================================+
| :math:`\text{COEFF}`               | :math:`\text{COEFF} \in \mathbb R^{p \times p}`                        |            | The computed principal component coefficients | The columns are summarized by following the decreasing order of component variance. Each column contains coefficients for one principal component. |
+------------------------------------+------------------------------------------------------------------------+------------+-----------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------+

.. rubric:: Tool Support

* :doc:`/Tools/MapleTool/index`

  For details refer to the online documentation of the function `g03aac(nag_mv_prin_comp) <http://www.maplesoft.com/support/help/Maple/view.aspx?path=NAG/g03aac>`__.

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

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

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/DataReduction/DimensionalityReduction/DataReductionWithPrincipalComponentAnalysis/index`

.. rubric:: References

- I.T.\  Jolliffe, Principal Component Analysis, 2nd edition, Springer, 2002.