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This paper gives an introduction to multivariate calibration from a chemometrics perspective and reviews the various proposals to generalize the well-established univariate methodology to the multivariate domain. Univariate calibration leads to relatively simple models with a sound statistical underpinning. The associated uncertainty estimation and figures of merit are thoroughly covered in several official documents. However, univariate model predictions for unknown samples are only reliable if the signal is sufficiently selective for the analyte of interest. By contrast, multivariate calibration methods may produce valid predictions also from highly unselective data. A case in point is quantification from near-infrared (NIR) spectra. With the ever-increasing sophistication of analytical instruments inevitably comes a suite of multivariate calibration methods, each with its own underlying assumptions and statistical properties. As a result, uncertainty estimation and figures of merit for multivariate calibration methods has become a subject of active research, especially in the field of chemometrics.

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In this paper the general theory of multiway multiblock component and covariates regression models is explained. Unlike in existing methods such as multiblock PLS and multiblock PCA, in the new proposed method a different number of components can be selected for each block. Furthermore, the method can be generalized to incorporate multiway blocks to which any multiway model can be applied. The method is a direct extension of principal covariates regression and therefore works in a simultaneous fashion in which a clearly defined objective criterion is minimized. It can be tuned to fulfil the requirements of the user. Algorithms to calculate the components will be presented. The method will be illustrated with two three‐block examples and compared to existing approaches. The first example is with two‐way data and the second example is with a three‐way array. It will be shown that predictions are as good as with the existing methods, but because for most blocks fewer components are required, diagnostic properties of the method are improved. Copyright © 2000 John Wiley & Sons, Ltd.