Skip to main content
Springer Nature Link
Log in

Integrated data depth for smooth functions and its application in supervised classification

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

This paper concerns depth functions suitable for smooth functional data. We suggest a modification of the integrated data depth that takes into account the shape properties of the functions. This is achieved by including a derivative(s) into the definition of the suggested depth measures. We then further investigate the use of integrated data depths in supervised classification problems. The performances of classification rules based on different data depths are investigated, both in simulated and real data sets. As the proposed depth function provides a natural alternative to the depth function based on random projections, the difference in the performances of these two methods are discussed in more detail.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  • Berlinet A, Thomas-Agnan C (2004) Reproducing kernel Hilbert spaces in probability and statistics. Kluwer, Boston

    Book  MATH  Google Scholar 

  • Claeskens G, Hubert M, Slaets L, Vakili K (2014) Multivariate functional halfspace depth. J Am Stat Assoc 109(505):411–423

    Article  MathSciNet  Google Scholar 

  • Cuevas A, Febrero M, Fraiman R (2007) Robust estimation and classification for functional data via projection-based depth notions. Comput Stat 22(3):481–496

    Article  MathSciNet  MATH  Google Scholar 

  • Delaigle A, Hall P, Bathia N (2012) Componentwise classification and clustering of functional data. Biometrika 99(2):299–313

    Article  MathSciNet  MATH  Google Scholar 

  • Febrero-Bande M, Oviedo de la Fuente M (2012) Statistical computing in functional data analysis: the R package fda.usc. J Stat Softw 51(4):1–28

    Article  Google Scholar 

  • Ferraty F, Romain Y (eds) (2011) The Oxford handbook of functional data analysis. Oxford University Press, Oxford

  • Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer series in statistics. Springer, New York

    Google Scholar 

  • Fraiman R, Muniz G (2001) Trimmed means for functional data. Test 10(2):419–440

    Article  MathSciNet  MATH  Google Scholar 

  • Ghosh AK, Chaudhuri P (2005) On maximum depth and related classifiers. Scand J Stat 32:327–350

    Article  MathSciNet  MATH  Google Scholar 

  • James GM, Hastie TJ (2001) Functional linear discriminant analysis for irregularly sampled curves. J R Stat Soc Ser B Stat Methodol 63(3):533–550

    Article  MathSciNet  MATH  Google Scholar 

  • Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus, volume 113 of graduate texts in mathematics, second edn. Springer, New York

    Google Scholar 

  • Lange T, Mosler K, Mozharovskyi P (2014) Fast nonparametric classification based on data depth. Stat Pap 55(1):49–69

    Article  MathSciNet  MATH  Google Scholar 

  • Li J, Cuesta-Albertos JA, Liu RY (2012) DD-classifier: nonparametric classification procedure based on DD-plot. J Am Stat Assoc 107(498):737–753

    Article  MathSciNet  MATH  Google Scholar 

  • Liu RY (1990) On a notion of data depth based on random simplices. Ann Stat 18(1):405–414

    Article  MATH  Google Scholar 

  • Liu Y, Rayens W (2007) PLS and dimension reduction for classification. Comput Stat 22(2):189–208

    Article  MathSciNet  MATH  Google Scholar 

  • López-Pintado S, Romo J (2006) Depth-based classification for functional data. In: Data depth: robust multivariate analysis, computational geometry and applications, volume 72 of DIMACS Series Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, Providence, RI, pp 103–119

  • López-Pintado S, Romo J (2007) Depth-based inference for functional data. Comput Stat Data Anal 51(10):4957–4968

    Article  MATH  Google Scholar 

  • López-Pintado S, Romo J (2009) On the concept of depth for functional data. J Am Stat Assoc 104(486):718–734

    Article  Google Scholar 

  • Maechler M (2013) sfsmisc: utilities from seminar fuer Statistik ETH Zurich. R package version 1.0-24

  • Mosler K (2002) Multivariate dispersion, central regions and depth: the lift zonoid approach, volume 165 of lecture notes in statistics. Springer, Berlin

  • R Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna

  • Ramsay JO, Silverman BW (2002) Applied functional data analysis: methods and case studies. Springer series in statistics. Springer, New York

    Book  Google Scholar 

  • Ramsay JO, Wickham H, Graves S, Hooker G (2013) fda: functional data analysis. R package version 2.3.8

  • Tukey JW (1975) Mathematics and the picturing of data. In: Proceedings of the international congress of mathematicians (Vancouver, BC, 1974), vol 2. Canadian Mathematical Congress, Montreal, QC, pp 523–531

  • Zuo Y, Serfling R (2000) General notions of statistical depth function. Ann Stat 28(2):461–482

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors thank the Editor and three reviewers for their valuable comments which led to a considerable improvement of the paper. This research was supported by the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy). The work of the D. Hlubinka was supported by the Grant GACR P402/14-07234S. The I. Gijbels gratefully acknowledges support from the GOA/12/014—project of the Research Fund KU Leuven. The work of the fourth author was partially supported by the Czech Science Foundation Project No. P402/12/G097 “DYME–Dynamic Models in Economics”. Currently he is a Research Assistant of the Research Foundation—Flanders, and acknowledges support from this foundation.

Author information

Authors and Affiliations

  1. Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic

    Daniel Hlubinka, Marek Omelka & Stanislav Nagy

  2. Department of Mathematics, Leuven Statistics Research Center (LStat), KU Leuven, Leuven, Belgium

    Irène Gijbels & Stanislav Nagy

Authors
  1. Daniel Hlubinka
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Irène Gijbels
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Marek Omelka
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Stanislav Nagy
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Daniel Hlubinka.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hlubinka, D., Gijbels, I., Omelka, M. et al. Integrated data depth for smooth functions and its application in supervised classification. Comput Stat 30, 1011–1031 (2015). https://doi.org/10.1007/s00180-015-0566-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-015-0566-x

Keywords