Nonlinear PLS modeling

Abstract

The linear two block predictive PLS model (PPLS2) is often used to model the relation between two data matrices, X and Y. Applications include multivariate calibration, quantitative structure-activity relationships (QSAR), and process optimization. In each PPLS2 model dimension the matrices X and Y are decomposed as bilinear products plus residual matrices: X = tp′ + E Y = uq′ + F In addition, a linear model is assumed to relate the score vectors t and u (h denotes residuals): u = bt + h This allows Y to be modeled by t and q as: Y = tq′b + ƒ* In the present work the linear PPLS2 model is extended to the case when the inner model relating the block scores u and t is nonlinear (h is a vector of residuals): u = ƒ(t) + h An algorithm is outlined for the model where the inner relation is a quadratic polynomial: u = c0 + c1t + c2t2 + h This will be referred to as the QPLS2 model (standing for quadratic PLS with two blocks). Applications to cosmetics qualimetrics and a drug structure-activity relationship are used as illustrations.