Polynomial Regression

Global Algorithm - One-Dimensional Algorithm

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

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

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

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

Input Parameters

Parameter	Type	Constraint	Description	Remarks
$[t_i]$	$[t_i] \in \mathbb R^N$	$N \in \mathbb{N}$
$Y$	$Y \in \mathbb R^N$	$N \in \mathbb{N}$	Input data vector of length $N$
$n$	$n \in \mathbb N$

Output Parameters

Parameter	Type	Constraint	Description	Remarks
$p$	$p \in \mathbb R^n$
$\hat{Y}$	$\hat{Y} \in \mathbb R^N$	$N \in \mathbb{N}$	Output data vector of length $N$

Tool Support

• Matlab

  For details refer to the online documentation of the function ‘polyfit’.

Single Steps using the Algorithm

• Data Reduction With Polynomial Regression
• Reconstructing Improper Values with Polynomial Regression

References

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