Triogram Models

In this article we introduce the Triogram method for function estimation using piecewise linear, bivariate splines based on an adaptively constructed triangulation. We illustrate the technique for bivariate regression and log-density estimation and indicate how our approach can be applied directly to model bivariate functions in the broader context of an extended linear model. The entire estimation procedure is invariant under affine transformations and is a natural approach for modeling data when the domain of the predictor variables is a polygonal region in the plane. Although our examples deal exclusively with estimating bivariate functions, the use of Triograms for modeling two-factor interactions in analysis of variance decompositions of functions depending on more than two variables is straightforward.