## Nonlinear PLS Models

Linear PLS decomposes two variable blocks X and Y as X = TPt and Y = UQt such that X is modeled effectively by TPt and T predicts U well by a linear model (inner relation) U = TB where B is a diagonal matrix. To model nonlinear relationships between X and Y, their projections should be nonlinearly related to each other . One possibility is to use a polynomial function such as ua = c0a + ciata + c2at2a + ha (4.182)

where a represents the model dimension, CoOJ Cia, and c2a are constants, and ha is a vector of residuals. This quadratic function can be generalized to other nonlinear functions of tQ:

where /(•) may be a polynomial, exponential, or logarithmic function.

Another framework for expressing a nonlinear relationship between X and Y can be based on splines  or smoothing functions . Splines are piecewise polynomials joined at knots (denoted by Zj) with continuity constraints on the function and all its derivatives except the highest. Splines have good approximation power, high flexibility and smooth appearance as a result of continuity constraints. For example, if cubic splines are used for representing the inner relation: