Chemometrics and Intelligent Laboratory Systems ( IF 2.895 ) Pub Date : 2019-09-17 , DOI: 10.1016/j.chemolab.2019.103848 Daniel Palací-López, Pierantonio Facco, Massimiliano Barolo, Alberto Ferrer
Latent Variable Regression Model (LVRM) inversion can be an efficient tool to find the so-called Design Space (DS), i.e. the different combinations of inputs (e.g. process conditions, raw materials properties …) that lead to the desired outputs (e.g. product quality, benefits …). This is especially critical when first-principles models cannot be resorted to, running experimental designs is unfeasible and only data from daily production (i.e. historical data) are available. Since data-driven methods are not free of uncertainty, different approaches have been proposed in the literature to delimit a subspace that is expected to contain the DS of a product. However, some of these methods are computationally costly or depend on the existence of at least one combination of inputs that provides, according to the model, the desired values for all output variables simultaneously. Furthermore, no approach to date offers an analytical expression for the confidence region limits for this subspace. In this paper a new way to find the DS is proposed, so the above limitations are overcome. To this end, the analytical definition of the estimation of the DS, and its confidence region limits, as well as a way to transfer restrictions on the original space to the latent space are suggested. An extension of these methods to quality attributes defined as linear combinations of outputs is also provided. The proposed methodology is illustrated using three simulated case studies.