# Polynomial Regression¶

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.