Theory & Methods: Spatially‐adaptive Penalties for Spline Fitting

Abstract

The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are pth degree piecewise polynomials with p− 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the pth derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize the generalized cross validation (GCV) criterion. This locally‐adaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knot‐selection techniques for least squares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions, empirical Bayes confidence intervals using this prior achieve better pointwise coverage probabilities than confidence intervals based on a global‐penalty parameter. The method is developed first for univariate models and then extended to additive models.