Standard error of prediction in parallel factor analysis of three-way data

A simple approach is described to calculate sample-specific standard errors for the concentrations predicted by a three-way parallel factor (PARAFAC) analysis model. It involves a first-order error propagation equation in which the correct sensitivity and leverage values are introduced. A comparison is made with a related unidimensional partial least-squares (PLS) model, specifically as regards the required leverage values. Monte Carlo simulation results obtained by adding random noise to both concentrations and instrumental signals for theoretical binary mixtures are in good agreement with the proposed approach. An experimental multicomponent example was studied by a similar Monte Carlo approach, and the obtained standard errors are also in agreement with the calculated values. Implications concerning the limit of detection are discussed.