Two sides of the same coin: kernel partial least-squares (KPLS) for linear and non-linear multivariate calibration. A tutorial

A tutorial is presented on the operation and properties of the non-linear multivariate regression model kernel partial least-squares (KPLS). After the discussion of a simple non-linear univariate problem, solved by regressing a dependent variable on the projection of an independent variable onto a set of Gaussian functions, the principles of KPLS are introduced for processing non-linear multivariate data. The following aspects are covered: (1) the estimation of the model sensitivity as a function of analyte concentration from error propagation concepts, (2) the proposal of a parameter measuring the degree of non-linearity, to avoid a black-and-white description of data sets as either linear or non-linear, (3) the use of the model parameters for variable selection. The application of KPLS to both simulated and experimental data sets is discussed, in the latter case involving near infrared spectra employed for the determination of quality parameters in foodstuff samples and fluorescence spectroscopic data for the study of systems of biological relevance. Computer codes written in the popular MATLAB and R environments are also provided.