Demingregressiota

Demingregressiota is a generalized errors-in-variables regression concept inspired by the classical Deming regression. It is used to describe the relationship between a response vector Y and a predictor vector X when both sides are measured with error, and when the predictor dimension may exceed one. The term is encountered in theoretical discussions of high-dimensional calibration and measurement-error models, as a natural extension of the univariate Deming approach to multivariate settings.

In this framework, the true latent variables X and Y are related by a linear operator Y = B X + c, while the observed data are X̂ = X + E_x and Ȳ = Y + E_y, with E_x and E_y carrying a known or estimable error-covariance structure. The estimation objective is to determine B and c by minimizing a weighted sum of squared discrepancies that accounts for the measurement error in both X and Y. The weighting is derived from the error covariances, and the resulting optimization often reduces to a generalized eigenvalue problem or a trace-minimization criterion. When the predictor and response are both univariate (p = q = 1), the formulation reduces to the classical Deming regression.

Practical implementations require an identifiable error-covariance model and may involve eigen-decomposition or iterative algorithms. Assumptions about homoscedasticity or heteroscedasticity of the errors influence the method chosen. Applications span physical sciences, astronomy, metrology, and other fields where accurate calibration demands accounting for errors in both variables. The term demingregressiota is not universally standardized; it is often described in the literature as a multivariate Deming regression or as an errors-in-variables regression with a specified error structure.