Multivariate calibration of non-replicated measurements for the factored noise model

The accuracy of a multivariate calibration (MVC) model for relating concentrations of multicomponent mixtures to their spectral measurements depends on effective handling of errors in the measured data. For the case when error variances vary along only one mode (either along mixtures or along wavelengths), a method to estimate the error variances simultaneously along with the spectral subspace was developed by Narasimhan and Shah (Control Engineering Practice, 16, (2008), 146–155). This method was exploited by Bhatt et al. (Chemom. Intell. Lab. Syst., 85, (2007), 70–81) to develop an iterative principal component regression (IPCR) MVC model, which was shown to be more accurate than models developed using PCR. In this work, the IPCR method is extended to deal with measurement errors whose variances vary along both modes, by using a factored noise model. As a first step, an iterative procedure is developed to estimate the error variance factors along with the spectral subspace, which is subsequently used in developing the regression model. Using simulated and experimental data, it is shown that the quality of the MVC model developed using the proposed method is better than that obtained using PCR, and is as good as the model obtained using Maximum Likelihood PCR, which requires knowledge of the error variances. For dealing with large data sets, a sub-optimal approach is also proposed for estimating the large number of error variances.