# Linear Regression¶

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

Single Steps using the Algorithm

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.