Drift correction in multivariate calibration models using on-line reference measurements.

On-line measurements from first-order instruments such as spectrometers may be compromised by instrumental, process and operational drifts that are not seen during off-line calibration. This can render the calibration model unsuitable for prediction of key components such as analyte concentrations. In this work, infrequently available on-line reference measurements of the analytes of interest are used for drift correction. The drift-correction methods that include drift in the calibration set are referred to as implicit correction methods (ICM), while explicit correction methods (ECM) model the drift based on the reference measurements and make the calibration model orthogonal or invariant to the space spanned by the drift. Under some working assumptions such as linearity between the concentrations and the spectra, necessary and sufficient conditions for correct prediction using ICM and ECM are proposed. These so-called space-inclusion conditions can be checked on-line by monitoring the Q-statistic. Hence, violation of these conditions implies the violation of one or more of the working assumptions, which can be used, e.g. to infer the need for new reference measurements. These conditions are also valid for rank-deficient calibration data, i.e. when the concentrations of the various species are linearly dependent. A constraint on the kernel used in ECM follows from the space-inclusion condition. This kernel does not estimate the drift itself but leads to an unbiased estimate of the drift space. In a noise-free environment, it is shown that ICM and ECM are equivalent. However, in the presence of noise, a Monte Carlo simulation shows that ECM performs slightly better than ICM. A paired t-test indicates that this difference is statistically significant. When applied to experimental fermentation data, ICM and ECM lead to a significant reduction in prediction error for the concentrations of five metabolites predicted from infrared spectra.