Polynomial multivariate least-squares regression for modeling nonlinear data applied to in-depth characterization of chromatographic resolution

Abstract

Well-established, linear multivariate calibration methods such as multivariate least-squares regression (MLR), principal component regression (PCR), or partial least squares (PLS) have two limitations: (i) measured data must be linearly related to the response variables and (ii) predictor variables xn = 1, …, N cannot be coupled to each other. For evaluation of nonlinear data, however, these restrictions need to be overcome and thus polynomial multivariate least-squares regression (PMLR or “response surfaces”) has been introduced here. PMLR is based on multivariate least squares but incorporates all combinations of predictor variables up to a user-selected polynomial order (e.g., including u or v = 0). Because of the inclusion of such coupled terms and their powers, PMLR models are better adapted to model nonlinear data and can help to enhance the prediction step's accuracy and precision. PMLR has been based on MLR because it facilitates—unlike PCR or PLS—a physical and chemical interpretation of the predictors. Hence, the origins and the relevance of nonlinear and/or coupled predictors can be investigated. The details of the PMLR algorithm and its implementation are presented along with a method for model optimization utilizing gradients of response surfaces. Newly developed PMLR models up to quintic order have been applied to predict a chromatograph's peak resolution as a function of six-instrument parameters. It has been demonstrated that PMLR is better capable than MLR and PCR to describe these nonlinear and coupled instrument parameters. In addition, the novel software tool has been utilized for model optimization to determine instrument parameters, which result in the best chromatographic resolution. Copyright © 2011 John Wiley & Sons, Ltd.