Sample-Specific Prediction Error Measures in Spectroscopy

In applied spectroscopy, the purpose of multivariate calibration is almost exclusively to relate analyte concentrations and spectroscopic measurements. The multivariate calibration model provides estimates of analyte concentrations based on the spectroscopic measurements. Predictive performance is often evaluated based on a mean squared error. While this average measure can be used in model selection, it is not satisfactory for evaluating the uncertainty of individual predictions. For a calibration, the uncertainties are sample specific. This is especially true for multivariate calibration, where interfering compounds may be present. Consider in-line spectroscopic measurements during a chemical reaction, production, etc. Here, reference values are not necessarily available. Hence, one should know the uncertainty of a given prediction in order to use that prediction for telling the state of the chemical reaction, adjusting the process, etc. In this paper, we discuss the influence of variance and bias on sample-specific prediction errors in multivariate calibration. We compare theoretical formulae with results obtained on experimental data. The results point towards the fact that bias contribution cannot necessarily be neglected when assessing sample-specific prediction ability in practice.