Single and Double Penalty Spline Regression Models with Applications

 Abstract:

Penalized spline regression models are a popular statistical tool for curve fitting problems due to their flexibility and computational efficiency. In particular, penalized cubic spline functions have received a great deal of attention. Cubic splines have good numerical properties and have proven extremely useful in a variety of applications. Typically, splines are represented as linear combinations of basis functions. However, such representations can lack numerical stability or be difficult to manipulate analytically. We propose a different parametrization for cubic spline functions that is intuitive and simple to implement. Moreover, integral based penalty functionals have simple interpretable expressions in terms of the components of the parametrization. Also, the curvature of the function is not constrained to be continuous everywhere on its domain, which adds flexibility to the fitting process. We consider not only models where smoothness is imposed by means of a single penalty functional, but also a generalization where a combination of different measures of roughness is built in order to specify the adequate limit of shrinkage for the problem at hand. The proposed methodology is illustrated in two distinct regression settings.