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Uniform in Bandwidth Estimation of the Gradient Lines of a Density

  • David MasonEmail author
  • Bruno Pelletier
Conference paper
Part of the Progress in Probability book series (PRPR, volume 74)

Abstract

Let X1, …, Xn, n ≥ 1, be independent identically distributed (i.i.d.) \(\mathbb {R}^{d}\) valued random variables with a smooth density function f. We discuss how to use these Xs to estimate the gradient flow line of f connecting a point x0 to a local maxima point (mode) based on an empirical version of the gradient ascent algorithm using a kernel estimator based on a bandwidth h of the gradient ∇f of f. Such gradient flow lines have been proposed to cluster data. We shall establish a uniform in bandwidth h result for our estimator and describe its use in combination with plug in estimators for h.

Keywords

Gradient lines Density estimation Nonparametric clustering Uniform in bandwidth 

Notes

Acknowledgements

David Mason thanks the editors for inviting him to contribute to this proceedings. He had intended give a talk based on the results in this paper at the 2017 High Dimensional Probability VIII conference. Unfortunately he had to cancel his participation due heath reasons. Part of this paper was written while he was a guest of Australian National University. All of the authors acknowledge the associate editor and referee for their useful comments and suggestions.

References

  1. 1.
    E. Arias-Castro, D.M. Mason, B. Pelletier, On the estimation of the gradient lines of a density and the consistency of the mean-shift algorithm. J. Mach. Learn. Res. 17, Paper No. 43, 1487–1514 (2016)Google Scholar
  2. 2.
    E. Arias-Castro, D.M. Mason, B. Pelletier, Errata: on the estimation of the gradient lines of a density and the consistency of the mean-shift algorithm. J. Mach. Learn. Res. 17, Paper No. 206, 1487–1514 (2016)Google Scholar
  3. 3.
    R. Bhatia, Matrix Analysis. Graduate Texts in Mathematics, vol. 169 (Springer, New York, 1997)Google Scholar
  4. 4.
    P. Deheuvels, D.M. Mason, General asymptotic confidence bands based on kernel-type function estimators. Stat. Inference Stoch. Process. 7, 225–277 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    U. Einmahl, D.M. Mason, An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theor. Probab. 13, 1–37 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    U. Einmahl, D.M. Mason, Uniform in bandwidth consistency of kernel-type function estimators. Ann. Stat. 33, 1380–1403 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    K. Fukunaga, L.D. Hostetler, The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Trans. Inf. Theory 21, 32–40 (1975)MathSciNetCrossRefGoogle Scholar
  8. 8.
    E. Giné, A. Guillou, Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. Henri Poincaré Probab. Stat. 38, 907–921 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    J.A. Hartigan, Clustering Algorithms (Wiley, New York, 1975)zbMATHGoogle Scholar
  10. 10.
    M.W. Hirsch, S. Smale, R.L. Devaney, Differential Equations, Dynamical Systems & An Introduction to Chaos, 2nd edn. (Academic, Cambridge, 2004)zbMATHGoogle Scholar
  11. 11.
    D.M. Mason, Proving consistency of non-standard kernel estimators. Stoch. Inference Stoch. Process. 15, 151–176 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    D.M. Mason, J. Swanepoel, A general result on the uniform in bandwidth consistency of kernel-type function estimators. Test 20, 72–94 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D.M. Mason, J. Swanepoel, Errata: a general result on the uniform in bandwidth consistency of kernel-type function estimators. Test 24, 205–206 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Applied Economics and StatisticsUniversity of DelawareNewarkUSA
  2. 2.Département de Mathématiques, IRMAR – UMR CNRS 6625Université Rennes IIUpper BrittanyFrance

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