A closed-form expression for computing the sensitivity in second-order bilinear calibration

Abstract

A general expression is derived for estimating the sensitivity of second-order bilinear calibration models, particularly parallel factor analysis (PARAFAC) and bilinear least-squares (BLLS), whether the second-order advantage is required or not. In the latter case, the sensitivity is correctly estimated either if the advantage is achieved by processing the unknown sample together with the calibration set (PARAFAC), or by post-calibration residual bilinearization (BLLS). The expression includes, as special cases, the sensitivity expressions already discussed by Messick, Kalivas and Lang (MKL) and by Ho, Christian and Davidson (HCD). The former one is the maximum achievable sensitivity in a given calibration situation, where all components are present in the calibration set of samples. The latter approach gives the lowest possible sensitivity, corresponding to only calibrating the analyte of interest, leaving the remaining components as uncalibrated constituents of the unknown sample. In intermediate situations, that is more than one calibrated analyte and presence of unexpected components in the unknown sample, only the present approach is able to provide a satisfactory sensitivity parameter, in close agreement with previously described Monte Carlo numerical simulations.