Generalized Isotonic Regression

• Published in: Journal of computational and graphical statistics Vol. 23; no. 1; pp. 192 - 210
• Main Authors: Luss, Ronny, Rosset, Saharon
• Format: Journal Article
• Language: English
• Published: Alexandria Taylor & Francis 01.03.2014; American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America; Taylor & Francis Ltd
• Subjects: Algorithms; Convex optimization; Datasets; Integers; Isotonic solutions; Isotonicity; Iterative solutions; Mathematical models; Nonparametric regression; Optimal solutions; Outliers; Regression analysis; Regression Methods; Regularization path; Simulation training; Software packages; Studies; Theorems
• ISSN: 1061-8600; 1537-2715
• DOI: 10.1080/10618600.2012.741550

We present a new computational and statistical approach for fitting isotonic models under convex differentiable loss functions through recursive partitioning. Models along the partitioning path are also isotonic and can be viewed as regularized solutions to the problem. Our approach generalizes and subsumes the well-known work of Barlow and Brunk on fitting isotonic regressions subject to specially structured loss functions, and expands the range of loss functions that can be used (e.g., adding Huber loss for robust regression). This is accomplished through an algorithmic adjustment to a recursive partitioning approach recently developed for solving large-scale ł 2 -loss isotonic regression problems. We prove that the new algorithm solves the generalized problem while maintaining the favorable computational and statistical properties of the l 2 algorithm. The results are demonstrated on both real and synthetic data in two settings: fitting count data using negative Poisson log-likelihood loss, and fitting robust isotonic regressions using Huber loss. Proofs of theorems and a MATLAB-based software package implementing our algorithm are available in the online supplementary materials.