Penalized triograms: total variation regularization for bivariate smoothing

• 成果类型: Article
• 署名作者: Koenker, R; Mizera, I
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign; University of Alberta
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/j.1467-9868.2004.00437.x
• 发表日期: 2004
• 页码: 145-163
• 关键词: polynomial splines; tensor-products

摘要

Hansen, Kooperberg and Sardy 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 Stone. 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.