Standard error of prediction for multiway PLS: 1. Background and a simulation study

While a multitude of expressions has been proposed for calculating sample-specific standard errors of prediction when using partial least squares (PLS) regression for the calibration of first-order data, potential generalisations to multiway data are lacking to date. We have examined the adequacy of two approximate expressions when using unfold- or tri-PLS for the calibration of second-order data. The first expression is derived under the assumption that the errors in the predictor variables are homoscedastic, i.e., of constant variance. In contrast, the second expression is designed to also work in the heteroscedastic case. The adequacy of the approximations is tested using extensive Monte Carlo simulations while the practical utility is demonstrated in Part 2 of this series.