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Penalized triograms: total variation regularization for bivariate smoothing. (English) Zbl 1064.62038

Summary: M. Hansen, C. Kooperberg and S. Sardy [J. Am. Stat. Assoc. 93, No. 441, 101–119 (1998; Zbl 0902.62045)] introduced a family of continuous, piecewise linear functions defined over adaptively selected triangulations of the plane as a general approach to statistical modelling of bivariate densities and 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 and colleagues adopted the regression spline approach of C. Stone [Ann. Stat. 22, 118–184 (1994; Zbl 0827.62038)]. Vertices of the triangulation are introduced or removed sequentially in an effort to balance fidelity to the data and parsimony.
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 roughness penalty proposed may be interpreted as a total variation penalty on the gradient of the fitted function. The methods are illustrated with real and artificial examples, including an application to estimated quantile surfaces of land value in the Chicago metropolitan area.

MSC:

62G07 Density estimation
62G99 Nonparametric inference
65C60 Computational problems in statistics (MSC2010)
65C05 Monte Carlo methods

Software:

SparseM; ftnonpar
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