Linear Regression

Global Algorithm - One-Dimensional Algorithm

Linear Regression algorithm aims to find parameters \(p_0\) and \(p_1\) for a line, \(y = p_0 + p_1 \cdot t\), that best fits \(N\) data points. The task is equivalent to solve systems of linear equations

\[\begin{split}Ap = \begin{bmatrix} 1&t_1 \\ 1&t_2 \\ \vdots&\vdots \\ 1&t_N \end{bmatrix} \begin{bmatrix} p_0 \\ p_1 \end{bmatrix} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_N \end{bmatrix} = Y.\end{split}\]

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

\[p = (A^T A)^{-1} A^T Y\]

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 \(N\)

Output Parameters

Parameter	Type	Constraint	Description	Remarks
\(p\)	\(p \in \mathbb R^2\)
\(\hat{Y}\)	\(\hat{Y} \in \mathbb R^N\)	\(N \in \mathbb{N}\)	Output data vector of length \(N\)

Tool Support

• Maple

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

• Matlab

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

Single Steps using the Algorithm

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

References

• C.R. Rao, H. Toutenburg, A. Fieger, C. Heumann, T. Nittner and S. Scheid, Linear Models: Least Squares and Alternatives, Springer Series in Statistics, pp. 23-33, 1999.
• R.C. Aster, B. Borchers, C.H. Thurber, Parameter Estimation and Inverse Problems, Academic Press, 2005.