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Rank tests for functional data based on the epigraph, the hypograph and associated graphical representations


Visualization techniques are very useful in data analysis. Their aim is to summarize information into a graph or a plot. In particular, visualization is especially interesting when one has functional data, where there is no total order between the data of a sample. Taking into account the information provided by the down–upward partial orderings based on the hypograph and the epigragh indexes, we propose new strategies to analyze graphically functional data. In particular, combining the two indexes we get an alternative way to measure centrality in a bunch of curves, so we get an alternative measure to the statistical depth. Besides, motivated by the visualization in the plane of the two measures for two functional data samples, we propose new methods for testing homogeneity between two groups of functions. The performance of the tests is evaluated through a simulation study and we have applied them to several real data sets.

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  1. Arribas-Gil A, Romo J (2014) Shape outlier detection and visualization for functional data: the outliergram. Biostatistics 15:603–619

  2. Baringhaus L, Franz C (2004) On a new multivariate two-sample test. J Multivariate Anal 88:190–206

  3. Bathke AC, Harrar SW, Ahmad MR (2009) Some contributions to the analysis of multivariate data. Biom J 51:285–303

  4. Cuesta-Albertos J, Nieto-Reyes A (2008) The random Tukey depth. Comput Stat Data Anal 52:4979–4988

  5. Cuevas A, Febrero M, Fraiman R (2006) On the use of bootstrap for estimating functions with functional data. Comput Stat Data Anal 51:1063–1074

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

  7. Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Springer, New York

  8. Flores R, Lillo RE, Romo J (2018) Homogeneity test for functional data. J Appl Stat 45:868–883

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

  10. Franco-Pereira AM, Lillo RE, Romo J (2011) Extremality for functional data. In: Ferraty F (ed) Recent advances in functional data analysis and related topics. Springer, New York

  11. Górecki T, Smaga L (2015) A comparison of tests for the one-way ANOVA problem for functional data. Comput Stat 30:987–1010

  12. Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer, New York

  13. Hsing T, Eubank R (2015) Theoretical foundations of functional data analysis, with an introduction to linear operators. Wiley series in probability and statistics. Wiley, Chichester

  14. Inselberg A (1985) The plane with parallel coordinates. Invited paper. Vis Comput 1:69–91

  15. Liu R, Singh K (1993) A quality index based on data depth and multivariate rank test. J Am Stat Assoc 88:257–260

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

  17. López-Pintado S, Romo J (2011) A half-region depth for functional data. Comput Stat Data Anal 55:1679–1695

  18. Martín-Barragán B, Lillo RE, Romo J (2018) Functional boxplots based on half-regions. J Appl Stat 43:1088–1103

  19. Nordhausen K, Oja H (2011) Multivariate \(L_1\) methods: the package MNM. J Stat Softw 43:1–28

  20. Oja H (2010) Multivariate nonparametric methods with R. Springer, New York

  21. Ramsay JO, Dalzell CJ (1991) Some tools for functional data analysis. J R Stat Soc B 53:539–572

  22. Ramsay JO, Silverman BW (2005a) Applied functional data analysis; methods and case studies. Springer, New York

  23. Ramsay JO, Silverman BW (2005b) Functional data analysis, 2nd edn. Springer, New York

  24. Sguera C, Galeano P, Lillo RE (2014) Spatial depth-based classification for functional data. TEST 23:725–750

  25. Sguera C, Galeano P, Lillo RE (2016) Functional outlier detection by a local depth with application to nox levels. Stoch Environ Res Risk Assess 30:1115–1130

  26. Sun Y, Genton MG (2011) Functional boxplots. J Comput Graph Stat 20:316–334

  27. Vardi Y, Zhang CH (2000) The multivariate \(L_1\)-median and associated data depth. Proc Natl Acad Sci USA 97:1423–1426

  28. Wang JL, Chiou JM, Müller HG (2016) Functional data analysis. Annu Rev Stat Appl 3:257–295

  29. Zhang J-T (2013) Analysis of variance for functional data. Chapman & Hall/CRC, New York

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This work has received financial support from the project MTM2017-89422-P of the Spanish Ministry of Economy and Competitiveness, the Xunta de Galicia (Centro singular de investigación de Galicia accreditation 2016–2019) and the European Union (European Regional Development Fund—ERDF). We also acknowledge support from the project ECO2015-66593-P of the Spanish Ministry of Science and Technology.

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Correspondence to Alba M. Franco Pereira.

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Pereira, A.M.F., Lillo, R.E. Rank tests for functional data based on the epigraph, the hypograph and associated graphical representations. Adv Data Anal Classif (2019).

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  • Data depth
  • Rank test
  • Epigraph
  • Hypograph
  • Functional data
  • Order statistics

Mathematics Subject Classification

  • 62G10
  • 62H30
  • 65S05