Uncertainty quantification for partial least squares regression

Introduction
Despite the fact that partial least squares regression (PLSR) is a standard method for multivariate calibration, there is no consensus on how to quantify the uncertainty in predictions. The aim here is to present an overview of the difficulties involved in tackling this problem, to briefly review some of the approaches that have been suggested, and to make some recommendations.

Experimental
Where appropriate, the conclusions are supported by simulations carried out using Matlab.

Results and discussion
The simplest approach is to attach the same uncertainty to all predictions, based on either a root mean square error of cross-validation (RMSECV) or a root mean square error of prediction (RMSEP). At the next level of complexity a standard result for multiple linear regression (MLR) can be used to express the uncertainty as a function of the distance of the spectrum from the mean in the PLS score space. This result is exact for MLR but only approximate for PLSR, because it ignores the fact that the factor loadings are data dependent. Attempts to provide better approximations involve either analytical approximations or simulation methods such as the bootstrap. All of these are complex, some of them can break down, and it is arguable whether they are worth implementing in most contexts. We suggest that the use of the MLR formula gets the trade off between accuracy and complexity about right, though care needs to be taken when estimating the variance that scales the whole formula. A further issue that does not affect MLR but arises with PLSR is the fact that there is another spectral distance that could be considered relevant to the prediction uncertainty, the distance of the spectrum from the PLS factor space or the so-called X-residual. Attempts in the literature to incorporate this into the uncertainty calculation seem to us to lack solid foundations, and we doubt whether anything useful can be done, given that the training data provide very little information about the X-y relationship outside the PLS factor space. We suggest that the common practice of monitoring the X-residual and flagging as doubtful any predictions for which this exceeds some threshold is probably the best that can be done.