Major axis regression

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.

In the least squares method of data modeling, the objective function, S,

is minimized, where r is the vector of residuals and W is a weighting matrix. In linear least squares the model contains equations which are linear in the parameters appearing in the parameter vector ${\displaystyle {\boldsymbol {\beta }}}$, so the residuals are given by

There are m observations in y and n parameters in β with m>n. X is a m×n matrix whose elements are either constants or functions of the independent variables, x. The weight matrix W is, ideally, the inverse of the variance-covariance matrix ${\displaystyle \mathbf {M} _{y}}$ of the observations y. The independent variables are assumed to be error-free. The parameter estimates are found by setting the gradient equations to zero, which results in the normal equations

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