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Polynomial Regression
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:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

*Polynomial Regression* algorithm is a generalization of the linear regression algorithm that aims to find parameters :math:`p_1, p_2, \ldots, p_n` for a polynomial model of degree :math:`n`\ , i.e. :math:`y = p_0 + p_1 \cdot t + \ldots + p_n \cdot t^n`\ , that best fits :math:`N` data points. The task is equivalent to solve the following systems of linear equations

.. math::

   Ap = \begin{bmatrix} 1&t_1&t_1^2&\cdots&t_1^n \\ 1&t_2&t_2^2&\cdots&t_2^n \\ \vdots&\vdots&\vdots&\vdots&\vdots \\ 1&t_N&t_N^2&\cdots&t_N^n \end{bmatrix} \, \begin{bmatrix} p_0 \\ p_1 \\ \vdots \\ p_n \end{bmatrix} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_N \end{bmatrix} = Y. 

The method of least squares is the most common method for finding the fitted parameters. If :math:`A` is of full column rank, the least squares solution is given by

.. math::

   p = (A^T A)^{-1} A^T Y.



.. rubric:: Input Parameters

+----------------------------+------------------------------------------------+----------------------------------------+------------------------------------------------+---------+
| Parameter                  | Type                                           | Constraint                             | Description                                    | Remarks |
+============================+================================================+========================================+================================================+=========+
| :math:`[t_i]`              | :math:`[t_i] \in \mathbb R^N`                  | :math:`N \in \mathbb{N}`               |                                                |         |
+----------------------------+------------------------------------------------+----------------------------------------+------------------------------------------------+---------+
| :math:`Y`                  | :math:`Y \in \mathbb R^N`                      | :math:`N \in \mathbb{N}`               | Input data vector of length :math:`N`          |         |
+----------------------------+------------------------------------------------+----------------------------------------+------------------------------------------------+---------+
| :math:`n`                  | :math:`n \in \mathbb N`                        |                                        |                                                |         |
+----------------------------+------------------------------------------------+----------------------------------------+------------------------------------------------+---------+

.. rubric:: Output Parameters

+----------------------------+----------------------------------------------------+----------------------------------------+-------------------------------------------------+---------+
| Parameter                  | Type                                               | Constraint                             | Description                                     | Remarks |
+============================+====================================================+========================================+=================================================+=========+
| :math:`p`                  | :math:`p \in \mathbb R^n`                          |                                        |                                                 |         |
+----------------------------+----------------------------------------------------+----------------------------------------+-------------------------------------------------+---------+
| :math:`\hat{Y}`            | :math:`\hat{Y} \in \mathbb R^N`                    | :math:`N \in \mathbb{N}`               | Output data vector of length :math:`N`          |         |
+----------------------------+----------------------------------------------------+----------------------------------------+-------------------------------------------------+---------+

.. rubric:: Tool Support

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

  For details refer to the online documentation of the function `'polyfit' <http://www.mathworks.de/help/techdoc/ref/polyfit.html>`__.

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/DataReduction/NumerosityReduction/DataReductionWithPolynomialRegression/index`

* :doc:`/DataPreprocessing/DataCleaning/HandlingImproperValues/ReconstructingImproperValues/ReconstructingImproperValuesWithPolynomialRegression/index`

.. rubric:: References

- R.C.\  Aster, B. Borchers, C.H. Thurber, Parameter Estimation and Inverse Problems, Academic Press, 2005.