Skip to main content
SpringerLink
Log in

Strict L∞ Isotonic Regression

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Given a function f and weights w on the vertices of a directed acyclic graph G, an isotonic regression of (f,w) is an order-preserving real-valued function that minimizes the weighted distance to f among all order-preserving functions. When the distance is given via the supremum norm there may be many isotonic regressions. One of special interest is the strict isotonic regression, which is the limit of p-norm isotonic regression as p approaches infinity. Algorithms for determining it are given. We also examine previous isotonic regression algorithms in terms of their behavior as mappings from weighted functions over G to isotonic functions over G, showing that the fastest algorithms are not monotonic mappings. In contrast, the strict isotonic regression is monotonic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ayer, M., Brunk, H.D., Ewing, G.M., Reid, W.T., Silverman, E.: An empirical distribution function for sampling with incomplete information. Ann. Math. Stat. 5, 641–647 (1955)

    Article  MathSciNet  Google Scholar 

  2. Barlow, R.E., Bartholomew, D.J., Bremner, J.M., Brunk, H.D.: Statistical Inference Under Order Restrictions: The Theory and Application of Isotonic Regression. Wiley, New York (1972)

    MATH  Google Scholar 

  3. Gebhardt, F.: An algorithm for monotone regression with one or more independent variables. Biometrika 57, 263–271 (1970)

    Article  MATH  Google Scholar 

  4. Hanson, D.L., Pledger, G., Wright, F.T.: On consistency in monotonic regression. Ann. Stat. 1, 401–421 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  5. Robertson, T., Wright, F.T., Dykstra, R.L.: Order Restricted Statistical Inference. Wiley, New York (1988)

    MATH  Google Scholar 

  6. Ubhaya, V.A.: Isotone optimization, I. J. Approx. Theory 12, 146–159 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ubhaya, V.A.: Isotone optimization, II. J. Approx. Theory 12, 315–331 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  8. Punera, K., Ghosh, J.: Enhanced hierarchical classification via isotonic smoothing. In: Proc. Int’l. Conf. World Wide Web, pp. 151–160 (2008)

    Google Scholar 

  9. Zheng, Z., Zha, H., Sun, G.: Query-level learning to rank using isotonic regression. In: Proc. 46th Allerton Conf., pp. 1108–1115 (2008)

    Chapter  Google Scholar 

  10. Angelov, S., Harb, B., Kannan, S., Wang, L.-S.: Weighted isotonic regression under the L 1 norm. Symp. Discrete Algorithms, 783–791 (2006)

  11. Barlow, R.E., Ubhaya, V.A.: Isotonic approximation. In: Rustagi, J.S. (ed.) Proc. Optimizing Methods in Stat., pp. 77–86. Academic Press, San Diego (1971)

    Google Scholar 

  12. Stout, Q.F.: Weighted L∞ isotonic regression, submitted (2011)

  13. Kaufman, Y., Tamir, A.: Locating service centers with precedence constraints. Discrete Appl. Math. 47, 251–261 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cuesta, J.A., Matrán, C.: Conditional bounds and best L ∞-approximations in probability spaces. J. Approx. Theory 56, 1–12 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. Darst, R.B., Sahab, S.: Approximations of continuous and quasi-continuous functions by monotone functions. J. Approx. Theory 38, 9–27 (1983)

    Article  MathSciNet  Google Scholar 

  16. Legg, D., Townsend, D.: Best monotone approximation in L ∞[0,1]. J. Approx. Theory 42, 30–35 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pólya, G.: Sur une algorithme toujours convergent pour obtenir les polynomes de meilleure approximation de Tchebycheff pour un fonction continue quelconque. Comptes Rendus 157, 840–843 (1913)

    MATH  Google Scholar 

  18. Ahuja, R.K., Olin, J.B.: A fast scaling algorithm for minimizing separable convex functions subject ot chain constraints. Oper. Res. 49, 784–789 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Stout, Q.F.: Unimodal regression via prefix isotonic regression. Comput. Stat. Data Anal. 53, 289–297 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Thompson, W.A. Jr.: The problem of negative estimates of variance components. Ann. Math. Stat. 33, 273–289 (1962)

    Article  MATH  Google Scholar 

  21. Pardalos, P.M., Xue, G.: Algorithms for a class of isotonic regression problems. Algorithmica 23, 211–222 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Megido, N.: Linear-time algorithms for linear programming in R 3 and related problems. SIAM J. Comput. 12, 759–776 (1983)

    Article  MathSciNet  Google Scholar 

  23. Stout, Q.F.: Isotonic regression for multiple independent variables, submitted (2011)

  24. Jackson, D.: Note on the median of a set of numbers. Bull. Am. Math. Soc. 27, 160–164 (1921)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science and Engineering, University of Michigan, Ann Arbor, MI, USA

    Quentin F. Stout

Authors
  1. Quentin F. Stout
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quentin F. Stout.

Additional information

Communicated by Panos M. Pardalos.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stout, Q.F. Strict L∞ Isotonic Regression. J Optim Theory Appl 152, 121–135 (2012). https://doi.org/10.1007/s10957-011-9865-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9865-8

Keywords

Search

Navigation