Total Variation Regularization for Bivariate Smoothing

Roger Koenker and Ivan Mizera

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, {\it et al} adopt the regression spline approach:
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 roughness penalty
adapted to the piecewise linear structure of the triogram model.
We show that the proposed roughness penalty may be interpreted
as a total variation penalty
on the gradient of the fitted function.  The methods are illustrated
with two artificial examples and with an application to estimated quantile
surfaces of land value in the Chicago metropolitan area.