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Unimodal regression via prefix isotonic regression

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  • Stout, Quentin F.

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 L2 metric requires only [Theta](n) time, while for the L1 metric it requires time. For unweighted points our algorithm for the L[infinity] metric requires only [Theta](n) time. All of these times are optimal. Previous algorithms were for the L2 metric and required [Omega](n2) 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.

Suggested Citation

  • Stout, Quentin F., 2008. "Unimodal regression via prefix isotonic regression," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 289-297, December.
    • Handle: RePEc:eee:csdana:v:53:y:2008:i:2:p:289-297
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    References listed on IDEAS

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    1. Qian, Shixian, 1996. "An algorithm for tree-ordered isotonic median regression," Statistics & Probability Letters, Elsevier, vol. 27(3), pages 195-199, April.
    2. Ravindra K. Ahuja & James B. Orlin, 2001. "A Fast Scaling Algorithm for Minimizing Separable Convex Functions Subject to Chain Constraints," Operations Research, INFORMS, vol. 49(5), pages 784-789, October.
    3. Zhi Geng & Ning‐Zhong Shi, 1990. "Isotonic Regression for Umbrella Orderings," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 39(3), pages 397-402, November.
    4. Turner, T. R. & Wollan, P. C., 1997. "Locating a maximum using isotonic regression," Computational Statistics & Data Analysis, Elsevier, vol. 25(3), pages 305-320, August.
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    Cited by:

    1. Quentin F. Stout, 2012. "Strict L∞ Isotonic Regression," Journal of Optimization Theory and Applications, Springer, vol. 152(1), pages 121-135, January.
    2. Oliver Y. Feng & Yining Chen & Qiyang Han & Raymond J. Carroll & Richard J. Samworth, 2022. "Nonparametric, tuning‐free estimation of S‐shaped functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1324-1352, September.
    3. Feng, Oliver Y. & Chen, Yining & Han, Qiyang & Carroll, Raymond J & Samworth, Richard J., 2022. "Nonparametric, tuning-free estimation of S-shaped functions," LSE Research Online Documents on Economics 111889, London School of Economics and Political Science, LSE Library.
    4. Cui, Zhenyu & Lee, Chihoon & Zhu, Lingjiong & Zhu, Yunfan, 2021. "Non-convex isotonic regression via the Myersonian approach," Statistics & Probability Letters, Elsevier, vol. 179(C).

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