j where the J knot locations and the model coefficients bk are the free parameters of the spline function. There are K + J + 1 coefficients where K is the order of the polynomial. The term bj+3(t — Zj)\ denotes a function with values 0 or bj+3(t — zj)a depending on the value of t:

The desirable number of knots and degrees of polynomial pieces can be estimated using cross-validation. An initial value for J can be N/7 or y/(N) for N > 100 where N is the number of data points. Quadratic splines can be used for data without inflection points, while cubic splines provide a general approximation for most continuous data. To prevent over-fitting data with higher-order polynomials, models of lower degree and higher number of knots should be considered for lower prediction errors and improved stability [660]. B splines that are discussed in Section 6.3.3 provide an attractive alternative to quadratic and cubic splines when the number of knots is large [121].

Other nonlinear PLS models that rely on nonlinear inner relations have been proposed [209, 581]. Nonlinear relations within X or Y can also be modeled. Simple cures would include use of known functional relationships for specific variables based on process knowledge such as exponential relationships of temperature in reactions. More sophisticated approaches are also available, including use of artificial neural networks to generate empirical nonlinear relations.

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