Diagonal designs for a multi-component calibration experiment.

Modern spectroscopic and sensor technologies combined with multivariate modelling are increasingly used for the quantitative analysis of complex mixtures. Their performance depends directly on the data design chosen for model training and validation. A well-balanced calibration experiment with the fewest samples possible presents additional challenges when several mixture components (factors) need to be calibrated on the same dataset and subsequently quantified from the same multivariate measurement. This practically important problem stays poorly addressed by the theory of experimental design. This theoretical work systematically formulates the requirements to an optimal calibration/validation dataset and introduces a new family of calibration designs, where the samples are placed along the diagonals of an experimental space that is a hypercube. Such placement is appropriate due to reasonable assumptions about the linear nature of analytical response. Suggested filling schemes allow economical diagonal designs with intrinsic validation to be built for multiple factors presented in as many levels as the number of samples. The most important practical cases of two and three factors are considered in detail, and generalization to higher dimensions is outlined. Diagonal designs of any complexity can be generated using a simple geometrical scheme or with a supplied script.