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.