A unified framework of constrained regression

Generalized additive models (GAMs) play an important role in modeling and understanding complex relationships in modern applied statistics. They allow for flexible, data-driven estimation of covariate effects. Yet researchers often have a priori knowledge of certain effects, which might be monotonic or periodic (cyclic) or should fulfill boundary conditions. We propose a unified framework to incorporate these constraints for both univariate and bivariate effect estimates and for varying coefficients. As the framework is based on component-wise boosting methods, variables can be selected intrinsically, and effects can be estimated for a wide range of different distributional assumptions. Bootstrap confidence intervals for the effect estimates are derived to assess the models. We present three case studies from environmental sciences to illustrate the proposed seamless modeling framework. All discussed constrained effect estimates are implemented in the comprehensive R package mboost for model-based boosting. DOI: https://doi.org/10.1007/s11222-014-9520-y Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-103599 Accepted Version Originally published at: Hofner, Benjamin; Kneib, Thomas; Hothorn, Torsten (2016). A unified framework of constrained regression. Statistics and Computing, 26(1-2):1-14. DOI: https://doi.org/10.1007/s11222-014-9520-y Noname manuscript No. (will be inserted by the editor) A Unified Framework of Constrained Regression Benjamin Hofner · Thomas Kneib · Torsten Hothorn Received: date / Accepted: date Abstract Generalized additive models (GAMs) play an important role in modeling and understanding com-Generalized additive models (GAMs) play an important role in modeling and understanding complex relationships in modern applied statistics. They allow for flexible, data-driven estimation of covariate effects. Yet researchers often have a priori knowledge of certain effects, which might be monotonic or periodic (cyclic) or should fulfill boundary conditions. We propose a unified framework to incorporate these constraints for both univariate and bivariate effect estimates and for varying coefficients. As the framework is based on (functional gradient descent) boosting methods, variables can be selected intrinsically, and effects can be estimated for a wide range of different distributional assumptions. We present three case studies from environmental sciences. The first on air pollution illustrates the use of monotonic and periodic effects in the context of an additive Poisson model. The second case study highlights the use of bivariate cyclic splines to model activity profiles of roe deer. The third case study demonstrates how to estimate the complete conditional distribution function of deer–vehicle collisions with the help of monotonicity constraints, and a cyclic constraint is considered for the seasonal variation of collision numbers. All discussed constrained effect estimates are imB. Hofner Institut für Medizininformatik, Biometrie und Epidemiologie, Friedrich-Alexander-Universität Erlangen-Nürnberg, Waldstraße 6, D-91054 Erlangen, Germany. E-mail: [email protected] T. Kneib Lehrstuhl für Statistik, Georg-August-Universität Göttingen, Platz der Göttinger Sieben 5, D-37073 Göttingen, Germany. T. Hothorn Institut für Sozialund Präventivmedizin, Abteilung Biostatistik, Universität Zürich, Hirschengraben 84, CH-8001 Zürich, Switzerland. plemented in the comprehensive R package mboost for model-based boosting.