Unimodal regression via prefix isotonic regression

Authors:
Quentin F. Stout
Affiliations:
University of Michigan, 2260 Hayward, Ann Arbor, MI 48109-2121, United States
Venue:
Computational Statistics & Data Analysis
Year:
2008

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Abstract

This paper gives algorithms for determining real-valued univariate unimodal regressions, that is, for determining the optimal regression which is increasing and then decreasing. Such regressions arise in a wide variety of applications. They are shape-constrained nonparametric regressions, closely related to isotonic regression. For unimodal regression on n weighted points our algorithm for the L"2 metric requires only @Q(n) time, while for the L"1 metric it requires @Q(nlogn) time. For unweighted points our algorithm for the L"~ metric requires only @Q(n) time. All of these times are optimal. Previous algorithms were for the L"2 metric and required @W(n^2) time. All previous algorithms used multiple calls to isotonic regression, and our major contribution is to organize these into a prefix isotonic regression, determining the regression on all initial segments. The prefix approach reduces the total time required by utilizing the solution for one initial segment to solve the next.