Multivariate calibration with temperature interaction using two-dimensional penalized signal regression

The Penalized Signal Regression (PSR) approach to multivariate calibration (MVC) assumes a smooth vector of coefficients for weighting a spectrum to predict the unknown concentration of a chemical component. B-splines and roughness penalties, based on differences, are used to estimate the coefficients. In this paper, we extend PSR to incorporate a covariate like temperature. A smooth surface on the wavelength–temperature domain is estimated, using tensor products of B-splines and penalties along the two dimensions. A slice of this surface gives the vector of weights at an arbitrary temperature. We present the theory and apply multi-dimensional PSR to a published data set, showing good performance. We also introduce and apply a simplification based on a varying-coefficient model (VCM).