Total least squares#Geometrical interpretation

In statistics, a best fit line for a finite amount scaled metric data pairs (xi, yi ) is calculated by the least squares method with the orthogonal regression. As in other regression models the sum of squared distances of ( xi, yi) is thereby minimized by the straight line. In contrast to the linear regression is not the distances in the x and y direction may be used, but the orthogonal distances.

The orthogonal regression is an important special case of the Deming regression. It was introduced in 1878 by Robert James Adcock in the statistics and made ​​known in a more general framework in 1943 by WE Deming for technical and economic applications.

Calculating method

A straight line is y = β0 β1x sought which (*, y * xi) minimizes the sum of squared distances of ( xi, yi) from the respective base points on the straight. Because yi * = β0 β1xi *, we can calculate these squared distances to ( yi - β0 - β1xi *) 2 ( xi - xi * ) 2, the sum of which is to be minimized:

For further calculation the following auxiliary values ​​are required:

Thus, the parameters for the solution of the minimization problem arise: