Nonparametric Regression on a Graph

Kovac, Arne; Smith, Andrew D. A. C.

Abstract

The 'Signal plus Noise' model for nonparametric regression can be extended to the case of observations taken at the vertices of a graph. This model includes many familiar regression problems. This article discusses the use of the edges of a graph to measure roughness in penalized regression. Distance between estimate and observation is measured at every vertex in the L2 norm, and roughness is penalized on every edge in the L1 norm. Thus the ideas of total variation penalization can be extended to a graph. The resulting minimization problem presents special computational challenges, so we describe a new and fast algorithm and demonstrate its use with examples. The examples include image analysis, a simulation applicable to discrete spatial variation, and classification. In our examples, penalized regression improves upon kernel smoothing in terms of identifying local extreme values on planar graphs. In all examples we use fully automatic procedures for setting the smoothing parameters. Supplemental materials are available online. © 2011 American Statistical Association.

Citation

Kovac, A., & Smith, A. D. A. C. (2011). Nonparametric Regression on a Graph. Journal of Computational and Graphical Statistics, 20(2), 432-447. https://doi.org/10.1198/jcgs.2011.09203

• Journal Article Type: Article
• Acceptance Date: Oct 4, 2010
• Publication Date: Jun 1, 2011
• Deposit Date: Dec 2, 2015
• Publicly Available Date: Aug 18, 2016
• Journal: Journal of Computational and Graphical Statistics
• Print ISSN: 1061-8600
• Publisher: Taylor & Francis
• Peer Reviewed: Peer Reviewed
• Volume: 20
• Issue: 2
• Pages: 432-447
• DOI: https://doi.org/10.1198/jcgs.2011.09203
• Keywords: active set algorithm, image analysis, penalized regression, total variation
• Public URL: https://uwe-repository.worktribe.com/output/961729
• Publisher URL: http://dx.doi.org/10.1198/jcgs.2011.09203
• Additional Information: Additional Information : This is an Accepted Manuscript of an article published by Taylor & Francis in Journal of Computational and Graphical Statistics in April 2011, available online: http://www.tandfonline.com/10.1198/jcgs.2011.09203