NONLINEAR DATA-ANALYSIS WITH LATENT-VARIABLES

A new approach to nonlinear empirical modelling is proposed. Let X and Y be two data matrices, centered and optionally scaled. The aim is to find a functional relationship Y = F(X, parameters). In this approach the function F is determined by min(w,c,b,)[parallel-to u - u parallel-to/parallel-to u parallel-to]2-alpha-sigma(t)2(alpha-1) with constraints parallel-to w parallel-to = parallel-to c parallel-to = 1, where t = Xw, u = Yc, and b are the parameters of the nonlinear function between t and u, i.e. u = f(t, b). Here sigma(t)2 is the variance of t. The parameter a defines a continuum of models and gives a weight between maximizing the fit in the inner relationship and the variance of the latent variable score t. Y is now calculated as in linear partial least squares (PLS). This procedure is repeated up to a chosen number of dimensions with residual matrices X <-- X <-- tp(T) and Y <-- Y - uq(T). The vector u is the fitted value of u and q is constructed from u as in linear PLS. The function f will be called the link function and it can be any parametric function between the two latent variables. This will be referred to as nonlinear (NL) PLS and its performance is illustrated by four examples using quadratic and logistic link functions: one synthetic, one from product development, one from near infrared spectroscopy and one from molecular pharmacology.