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Penalized triograms: total variation regularization for bivariate smoothing

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  • Roger Koenker
  • Ivan Mizera

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. Copyright 2004 Royal Statistical Society.

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  • Roger Koenker & Ivan Mizera, 2004. "Penalized triograms: total variation regularization for bivariate smoothing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 145-163.
  • Handle: RePEc:bla:jorssb:v:66:y:2004:i:1:p:145-163
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    Cited by:

    1. Fritsch, Markus & Haupt, Harry & Ng, Pin T., 2016. "Urban house price surfaces near a World Heritage Site: Modeling conditional price and spatial heterogeneity," Regional Science and Urban Economics, Elsevier, vol. 60(C), pages 260-275.
    2. Kaplan, David M., 2015. "Improved quantile inference via fixed-smoothing asymptotics and Edgeworth expansion," Journal of Econometrics, Elsevier, vol. 185(1), pages 20-32.
    3. Charlier, Isabelle & Paindaveine, Davy & Saracco, Jérôme, 2015. "Conditional quantile estimation based on optimal quantization: From theory to practice," Computational Statistics & Data Analysis, Elsevier, vol. 91(C), pages 20-39.
    4. David M. Kaplan, 2013. "IDEAL Inference on Conditional Quantiles via Interpolated Duals of Exact Analytic L-statistics," Working Papers 1316, Department of Economics, University of Missouri.
    5. Yue, Yu Ryan & Rue, Håvard, 2011. "Bayesian inference for additive mixed quantile regression models," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 84-96, January.
    6. repec:eee:regeco:v:68:y:2018:i:c:p:204-225 is not listed on IDEAS
    7. Robert J. Hill & Miriam Steurer & Sofie R. Waltl, 2017. "Owner Occupied Housing in the CPI and Its Impact On Monetary Policy During Housing Booms and Busts," Graz Economics Papers 2017-12, University of Graz, Department of Economics.
    8. McMillen, Daniel, 2015. "Conditionally parametric quantile regression for spatial data: An analysis of land values in early nineteenth century Chicago," Regional Science and Urban Economics, Elsevier, vol. 55(C), pages 28-38.
    9. Lan Zhou & Huijun Pan, 2014. "Smoothing noisy data for irregular regions using penalized bivariate splines on triangulations," Computational Statistics, Springer, vol. 29(1), pages 263-281, February.
    10. Matúš Maciak & Ivan Mizera, 2016. "Regularization techniques in joinpoint regression," Statistical Papers, Springer, vol. 57(4), pages 939-955, December.
    11. Sophie Dabo-Niang & Sidi Ould-Abdi & Ahmedoune Ould-Abdi & Aliou Diop, 2014. "Consistency of a nonparametric conditional mode estimator for random fields," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 23(1), pages 1-39, March.
    12. Mele, Angelo, 2013. "Poisson indices of segregation," Regional Science and Urban Economics, Elsevier, vol. 43(1), pages 65-85.
    13. Elisabeth Waldmann & Thomas Kneib & Yu Ryan Yu & Stefan Lang, 2012. "Bayesian semiparametric additive quantile regression," Working Papers 2012-06, Faculty of Economics and Statistics, University of Innsbruck.
    14. Maarten Jansen & Guy P. Nason & B. W. Silverman, 2009. "Multiscale methods for data on graphs and irregular multidimensional situations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 97-125.
    15. Sofie R. Waltl, 2015. "Variation across price segments and locations: A comprehensive quantile regression analysis of the Sydney housing market," Graz Economics Papers 2015-09, University of Graz, Department of Economics.
    16. Maria Marino & Alessio Farcomeni, 2015. "Linear quantile regression models for longitudinal experiments: an overview," METRON, Springer;Sapienza Università di Roma, vol. 73(2), pages 229-247, August.

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