PROC TPSPLINE decomposes the regressor contributions to the mean function into parametric components and into smooth functional components. Suppose that the regressor variables are collected into the vector and that this vector is partitioned as . The relationship between Y and is linear (parametric), and the relationship between Y and is nonparametric. PROC TPSPLINE fits models of the form

The function can be represented as a sequence of spline basis functions.

The parameters are estimated by a penalized least squares method. The penalty is applied to the usual least squares criterion to obtain a regression estimate that fits the data well and to prevent the fit from attempting to interpolate the data (fit the data too closely).