Penalized Triograms

Hansen, Kooperberg, and Sardy (1998) introduced a family of continuous, piecewise linear functions defined over adaptively selected triangulations of the plane as a general approach to statistical modeling of bivariate densities, regression and hazard functions. These triograms enjoy a natural affine equivariance that offers distinct advantages over competing tensor product methods that are more commonly used in statistical applications. Triograms employ basis functions consisting of linear “tent functions” defined with respect to a triangulation of a given planar domain. As in knot selection for univariate splines, Hansen, et al adopt the regression spline approach of Stone (1994). Vertices of the triangulation are introduced or removed sequentially in an effort to balance fidelity to the data and parsimony. In this paper we explore a smoothing spline variant of the triogram model based on a simple roughness penalty adapted to the piecewise linear structure of the triogram model. The proposed roughness penalty may be interpreted as a total variation penalty on the gradient of the fitted function. “Goniolatry, or the worship of angles, ...” Pynchon (1997)