Penalized triograms: Total variation regularization for bivariate smoothing

Roger Koenker, Ivan Mizera

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)145-163
Number of pages19
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Volume66
Issue number1
DOIs
StatePublished - 2004

Keywords

  • Penalty methods
  • Regularization
  • Spline smoothing
  • Total variation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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