Total least squares

The bivariate (Deming regression) case of total least squares. The red lines show the error in both x and y. This is different from the traditional least squares method which measures error parallel to the y axis. The case shown, with deviations measured perpendicularly, arises when errors in x and y have equal variances.
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Background

In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models.

The total least squares approximation of the data is generically equivalent to the best, in the Frobenius norm, low-rank approximation of the data matrix.[1]

  1. ^ I. Markovsky and S. Van Huffel, Overview of total least squares methods. Signal Processing, vol. 87, pp. 2283–2302, 2007. preprint