Let be the adjusted weight matrix at convergence, and let be the roughness penalty matrix with selected smoothing parameters. The degrees of freedom matrix is defined as in Wood (2006):
Given the adjusted response , the parameter estimate is shown to be for the model without penalization, and the parameter estimate is with penalization. is thus the matrix that projects or maps the unpenalized parameter estimates to the penalized ones.
The model degrees of freedom is given as
And the degrees of freedom for error is given as
For the jth spline term, the degrees of freedom for the component is defined to be the trace of the submatrix of that corresponds to parameter estimates :
The degrees of freedom for the smoothing component test of the jth term is defined similarly as