Sundberg, Rolf; Brown, Philip J. (1989, ASQC and the American Statistical Association) University of Stockholm, Sweden; University of Liverpool, United Kingdom
Multivariate calibration involves the use of an estimated (linear) relationship between a q-variate response vector Y and a p-dimensional explanatory vector X to estimate or predict future unknown X from further observed Y. In some applications - for example, when Y represents automatic intensity measurements at q different wavelengths on n chemical standard samples - the feasible sample size n may be restricted, whereas the dimension q can be chosen quite large. In this article, the singular cases appearing when n < p + q + 1 are investigated. It is shown that, if n>q, the traditional solutions to the estimation and prediction problems - that is, the generalized least squares estimator and the estimated best linear predictor - are both unique, whereas for smaller n, (q - n + 1)-dimensional hyperplanes of solutions are obtained, the same in both problems. The properties of the predictor are also empirically studied in an example with p = 1, q = 6, and varying n.
Statistics,Collinearity,Least squares,Regression,Statistical methods