Residual spaces in latent variables model inversion and their impact in the design space for given quality characteristics

Abstract The paper contains a discussion about the null spaces associated to linear prediction models for the particular case of Partial Least Squares regression models. The discussion separately considers the two existing null spaces: the one in the input space related to the projection onto the latent space and the null space, coming from the projection space, corresponding to the mapping of the scores onto the predicted responses. The paper also explores the impact of such null spaces in the definition of the design space around some feasible solutions obtained by inverting the prediction model, via several cases with simulated and real data from the literature. The case-studies serve to illustrate the discussion and the need of considering points in the two null spaces, rather than just take into account the null space within the latent space. They also serve to show how to address the use of the resulting vectors in the design space to maintain the desired quality by modifying the tunable (maneuverable) process variables to compensate for variations due to some other feature variables not so easy to control.