Flexible smoothing with B-splines and penalties

TLDR

B‑splines are attractive for nonparametric modelling, yet selecting optimal knot number and placement is challenging, and equidistant knots provide limited smoothness control. The study proposes using a large number of knots together with a difference penalty on adjacent B‑spline coefficients. The authors employ penalized B‑splines with a difference penalty on adjacent coefficients and discuss criteria for selecting the optimal penalty parameter. They demonstrate that the difference penalty corresponds to the classic spline penalty on the integral of the squared second derivative, and illustrate the approach with nonparametric logistic regression, density estimation, and scatterplot smoothing, providing computational details.

Abstract

B-splines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent B-splines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of B-splines, of their construction and of penalized likelihood is presented. We discuss properties of penalized B-splines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.