Efficient Computation of Net Analyte Signal Vector in Inverse Multivariate Calibration Models

The net analyte signal vector has been defined by Lorber as the part of a mixture spectrum that is unique for the analyte of interest; i.e., it is orthogonal to the spectra of the interferences. It plays a key role in the development of multivariate analytical figures of merit. Applications have been reported that imply its utility for spectroscopic wavelength selection as well as calibration method comparison. Currently available methods for computing the net analyte signal vector in inverse multivariate calibration models are based on the evaluation of projection matrices. Due to the size of these matrices (p × p, with p the number of wavelengths) the computation may be highly memory- and time-consuming. This paper shows that the net analyte signal vector can be obtained in a highly efficient manner by a suitable scaling of the regression vector. Computing the scaling factor only requires the evaluation of an inner product (p multiplications and additions). The mathematical form of the newly derived expression is discussed, and the generalization to multiway calibration models is briefly outlined.