Latent variable model inversion for intervals. Application to tolerance intervals in class-modelling situations, and specification limits in process control

Abstract

The paper deals with the inversion of intervals when a PLS (Partial Least Squares) model is used. However, instead of discretizing the interval, it is proved that the region resulting from the inversion of a PLS model is a convex set bounded by two parallel hyperplanes, each corresponding to the direct inversion of each endpoint of the given interval.

When the domain of the input variables is a convex set, any feasible solution with predictions within the interval set in the response can be obtained as a convex combination of a point on each of the two hyperplanes. In this way, the new solutions preserve the internal structure of the input variables.

This methodology can be of interest in several domains where the response under study is defined in terms of an interval of admissible values, such as specifications for a product in an industrial process, or tolerance intervals for computing compliant class-models.

The inversion of the corresponding fitted model defines a region in the input space (predictor variables) whose predictions fall within the specified interval. Then, estimating and exploring this region will increase the information about the problem under study.