Bilinear Regression and Second Order Calibration

Abstract: We consider calibration of second-order (or "hyphenated") instruments for chemical analysis. Many such instruments generate bilinear two-way (matrix) type data for each specimen. The bilinear regression model is to be estimated from a number of specimens of known composition. Once we have established the regression model from the calibration with specimens of known composition, we can use the model to predict the composition of a new specimen. For the estimation part we propose a new, simple estimator, which we call the SVD (singular value decomposition) estimator, and illustrate how it works on real and simulated data, as compared with other estimation methods, in particular bilinear least squares (BLLS). The SVD estimator is usually only slightly less efficient than BLLS, because it is based on reweighted least squares normal equations. In particular the statistical precision of the SVD estimator and that of BLLS are theoretically investigated, and the methods are compared and illustrated on real and simulated data. The advantages of our method over bilinear least squares are that it is faster and easier to compute, its standard errors are more easily and explicitly obtained, and it has a simpler correlation structure. We also develop a prediction theory based on any of these two estimators. There are numerous alternative estimation/prediction methods in the literature. We give a review of such estimation/prediction methods and carry out an extensive simulation study to compare their estimation and prediction precision. Included in the comparison are BLLS, SVD, GRAM (the generalized rank annihilation method), TLD (the trilinear decomposition method), PARAFAC (parallel factor analysis), PCR (principal components regression) and PLS (partial least squares regression). We conclude that BLLS does best both as estimator and predictor. In some circumstances PARAFAC and TLD do as well as BLLS, but with little information in the calibration set they work badly. Some of the methods do better if they are allowed to predict by a proportional multiple linear regression, whereas other methods do best when they predict with simple proportional regression.