Given a fixed set of smoothing parameters in which each controls the smoothness of each spline term, you can fit a generalized additive model by the penalized likelihood estimation. There are infinitely many sets of smoothing parameters. In order to search optimum models, some model evaluation criteria need to be defined to quantify the model goodness-of-fit. The GAMPL procedure uses the following model evaluation criteria:
Consider the optimization problem
The parameter estimate for can be represented as
And the smoothing matrix (also called the influence matrix or hat matrix) is thus represented as
With the defined smoothing matrix, you can form the model evaluation criteria as follows:
In the equations, (which corresponds to the GAMMA= suboption of the CRITERION= option) is the tuning parameter that is sometimes used to enforce smoother models.
The GAMPL procedure uses fitting algorithms that involve minimizing the model evaluation criterion with respect to unknown smoothing parameters .