Errors-in-Variables Estimation in Multivariate Calibration

Abstract

A set of q responses, y = (y ₁, y ₂, …, y _q,) ^T , is related to a set of p explanatory variables, x = (x ₁, x ₂, …, x _p ) ^T , through the classical linear regression model, y ^T = a ^T + x ^T B + e ^T . First, the unknown parameters a and B are estimated using a calibrafion set. The statistical problem that is considered here is that of estimating the vector x _o, that underlies a new observed vector of responses y _o using the parameter estimates obtained from the first procedure. These two procedures are commonly referred to as calibration and prediction (or inverse prediction) and sometimes jointly referred to as calibration. The prediction procedure can be viewed as parameter estimation in errors-in-variables regression. The maximum likelihood estimator (assuming normally distributed measurement errors) is proposed for the prediction procedure. Unlike the classical estimator used in the prediction procedure, the proposed estimator is consistent with respect to the number of response variables. The performances of the maximum likelihood estimator and the classical estimator are compared both analytically and via Monte Carlo simulations. An example is given from infrared spectroscopy.

KEY WORDS:

• Chemometrics
• Infrared spectroscopy
• Linear functional model
• Maximum likelihood estimation