Advertisement

Advances in Data Analysis and Classification

, Volume 13, Issue 4, pp 1083–1103 | Cite as

A Kendall correlation coefficient between functional data

  • Dalia ValenciaEmail author
  • Rosa E. Lillo
  • Juan Romo
Regular Article
  • 129 Downloads

Abstract

Measuring dependence is a very important tool to analyze pairs of functional data. The coefficients currently available to quantify association between two sets of curves show a non robust behavior under the presence of outliers. We propose a new robust numerical measure of association for bivariate functional data. We extend in this paper Kendall coefficient for finite dimensional observations to the functional setting. We also study its statistical properties. An extensive simulation study shows the good behavior of this new measure for different types of functional data. Moreover, we apply it to establish association for real data, including microarrays time series in genetics.

Keywords

Concordance Dependence Functional data Kendall’s tau 

Mathematics Subject Classification

62-07 62G35 62G09 

Notes

References

  1. Borovskikh Y (1996) U-statistics in Banach space. VSP BV, Oud-BeijerlandzbMATHGoogle Scholar
  2. Cardot H, Ferraty F, Sarda P (1999) Functional linear model. Stat Probab Lett 45:11–22MathSciNetCrossRefGoogle Scholar
  3. Cuevas A, Febrero M, Fraiman R (2004) An ANOVA test for functional data. Comput Stat Data Anal 47:111–122MathSciNetCrossRefGoogle Scholar
  4. Delicado P (2007) Functional k-sample problem when data are density functions. Comput Stat 22:391–410MathSciNetCrossRefGoogle Scholar
  5. Dubin JA, Müller HG (2005) Dynamical correlation for multivariate longitudinal data. J Am Stat Assoc 100:872–881MathSciNetCrossRefGoogle Scholar
  6. Efron B (2004) Large-scale simultaneous hypothesis testing: the choice of a null hypothesis. J Am Stat Assoc 99:96–104MathSciNetCrossRefGoogle Scholar
  7. Efron B (2005) Local false discovery rates. Technical report, Department of Statistics, Stanford UniversityGoogle Scholar
  8. Escabias M, Aguilera A, Valderrama M (2004) Principal components estimation of functional logistic regression: discussion of two different approaches. J Non Parametr Stat 16(3–4):365–384MathSciNetCrossRefGoogle Scholar
  9. Febrero M, Galeano P, González-Manteiga W (2008) Outlier detection in functional data by depth measures, with application to identify abnormal \(NO_x\) levels. Envirometrics 19:331–345CrossRefGoogle Scholar
  10. He G, Müller HG, Wang JL (2000) Extending correlation and regression from multivariate to functional data. In: Puri ML (ed) Asymptotics in statistics and probability. VSP, Leiden, pp 197–210CrossRefGoogle Scholar
  11. Kendall M (1938) A new measure of rank correlation. Biometrika Trust 30(1/2):81–93CrossRefGoogle Scholar
  12. Leurgans SE, Moyeed RA, Silverman BW (1993) Canonical correlation analysis when data are curves. J R Stat Soc B 55:725–740MathSciNetzbMATHGoogle Scholar
  13. López-Pintado S, Romo J (2007) Depth-based inference for functional data. Comput Stat Data Anal 51:4957–4968MathSciNetCrossRefGoogle Scholar
  14. López-Pintado S, Romo J (2009) On the concept of depth for functional data. J Am Stat Assoc 104:718–734MathSciNetCrossRefGoogle Scholar
  15. Opgen-Rhein R, Strimmer K (2006) Inferring gene dependency networks from genomic longitudinal data: a functional data approach. REVSTAT 4(1):53–65MathSciNetzbMATHGoogle Scholar
  16. Pezulli S, Silverman B (1993) Some properties of smoothed components analysis for functional data. Comput Stat 8:1–16MathSciNetGoogle Scholar
  17. Ramsay JO, Silverman BW (2005) Functional data analysis, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  18. Rangel C, Angus J, Ghahramani Z et al (2004) Modelling T-cell activation using gene expression profiling and state-space models. Bioinformatics 20:1361–1372CrossRefGoogle Scholar
  19. Scarsini M (1984) On measure of concordance. Stochastica 8(3):201–218MathSciNetzbMATHGoogle Scholar
  20. Schwabik S, Guoju Y (2005) Topics in Banach space integration. World Scientific Publishing, SingaporeCrossRefGoogle Scholar
  21. Taylor MD (2007) Multivariate measures of concordance. Ann Inst Stat Math 59:789–806MathSciNetCrossRefGoogle Scholar
  22. Taylor MD (2008) Some properties of multivariate measures of concordance. arXiv:0808.3105 [math.PR]
  23. Whittaker J (1990) Graphical models in applied multivariate statistics. Wiley, New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of StatisticsUniversidad Carlos III de MadridMadridSpain
  2. 2.Department of Statistics, UC3M-Santander Big Data InstituteUniversidad Carlos III de MadridMadridSpain

Personalised recommendations