Iteratively reweighted generalized rank annihilation method: 1. Improved handling of prediction bias

The generalized rank annihilation method (GRAM) is a method for curve resolution and calibration that uses two bilinear matrices simultaneously, i.e., one for the unknown and one for the calibration sample. A GRAM calculation amounts to solving an eigenvalue problem for which the eigenvalues are related to the predicted analyte concentrations. Previous studies have shown that random measurement errors bring about a bias in the eigenvalues, which directly translates into prediction bias. In this paper, accurate formulas are derived that enable removing most of this bias. Two bias correction methods are investigated. While the first method directly subtracts bias from the eigenvalues obtained by the original GRAM, the second method first applies a weight to the data matrices to reduce bias. These weights are specific for the analyte of interest and must be determined iteratively from the data. Consequently, the proposed modification is called iteratively reweighted GRAM (IRGRAM). The results of Monte Carlo simulations show that both methods are effective in the sense that the standard error in the bias-corrected prediction compares favourably with the root mean squared error (RMSE) that accompanies the original quantity. However, IRGRAM is found to perform best because the increase of variance caused by subtracting bias is minimised. In the original formulation of GRAM only a single calibration sample is exploited. The error analysis is extended to cope with multiple calibration samples.