Polynomial RegressionΒΆ

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
\([t_i]\) \([t_i] \in \mathbb R^N\) \(N \in \mathbb{N}\) None None
\(Y\) \(Y \in \mathbb R^N\) \(N \in \mathbb{N}\) Input data vector of length \(N\) None
\(n\) \(n \in \mathbb N\) None None None

Output Parameters

Parameter Type Constraint Description Remarks
\(p\) \(p \in \mathbb R^n\) None None None
\(\hat{Y}\) \(\hat{Y} \in \mathbb R^N\) \(N \in \mathbb{N}\) Output data vector of length \(N\) None

Single Steps using the Algorithm

References

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