Advertisement

TEST

, Volume 19, Issue 3, pp 537–557 | Cite as

A simple multiway ANOVA for functional data

  • J. A. Cuesta-Albertos
  • M. Febrero-BandeEmail author
Original Paper

Abstract

We propose a procedure to test complicated ANOVA designs for functional data. The procedure is effective, flexible, easy to compute and does not require a heavy computational effort. It is based on the analysis of randomly chosen one-dimensional projections. The paper contains some theoretical results as well as some simulations and the analysis of some real data sets. Functional data include multidimensional data, so the paper contains a comparison between the proposed procedure and some usual MANOVA tests.

Keywords

ANOVA Functional data Random projections Testing Two-way ANOVA 

Mathematics Subject Classification (2000)

62H15 62J10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramovich F, Angelini C (2005) Testing in mixed-effects FANOVA models. J Stat Plan Inference 136:4326–4348 CrossRefMathSciNetGoogle Scholar
  2. Abramovich F, Antoniadis A, Sapatinas T, Vidakovic B (2004) Optimal testing in a fixed-effects functional analysis of variance model. Int J Wavelets Multiresolut Inf Process 2:323–349 zbMATHCrossRefMathSciNetGoogle Scholar
  3. Angelini C, Vidakovic B (2002) Some novel methods in wavelet data analysis: Wavelet ANOVA, F-test shrinkage and Γ-minimax wavelet shrinkage. Istituto per le Applicazioni del Calcolo “Mauro Picone.” Rapporti Tecnici RT 256/02 Google Scholar
  4. Antoniadis A, Sapatinas T (2007) Estimation and inference in functional mixed-effects models. Comput Stat Data Anal 51:4793–4813 zbMATHCrossRefMathSciNetGoogle Scholar
  5. Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc Ser B 57:289–300 zbMATHMathSciNetGoogle Scholar
  6. Benjamini Y, Yekutieli D (2001) The control of the false discovery rate in multiple testing under dependency. Ann Stat 29:1165–1188 zbMATHCrossRefMathSciNetGoogle Scholar
  7. Brumback BA, Rice JA (1998) Smoothing spline models for the analysis of nested and crossed samples of curves. J Am Stat Assoc 93:961–994 zbMATHCrossRefMathSciNetGoogle Scholar
  8. Brunner E, Dette H, Munk A (1997) Box-type approximations in nonparametric factorial designs. J Am Stat Assoc 92:1494–1502 zbMATHCrossRefMathSciNetGoogle Scholar
  9. Cuesta-Albertos JA, Nieto-Reyes A (2007) An infinite dimensional depth: The random Tukey depth. Technical Report Google Scholar
  10. Cuesta-Albertos JA, Nieto-Reyes A (2008) The random Tukey depth. Comput Stat Data Anal 52:4979–4988 zbMATHCrossRefMathSciNetGoogle Scholar
  11. Cuesta-Albertos JA, del Barrio T, Fraiman R, Matrán C (2007a) The random projection method in goodness of fit for functional data. Comput Stat Data Anal 51:4814–4831 zbMATHCrossRefGoogle Scholar
  12. Cuesta-Albertos JA, Fraiman R, Galves A, García J, Svarc M (2007b) Identifying rhythmic classes of languages using their sonority: a Kolmogorov–Smirnov approach. J Appl Stat 34:749–761 CrossRefMathSciNetGoogle Scholar
  13. Cuesta-Albertos JA, Fraiman R, Ransford T (2007c) A sharp form of the Cramér–Wold theorem. J Theor Probab 20:201–209 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Cuesta-Albertos JA, Cuevas A, Fraiman R (2008) On projection-based tests for spherical and compositional data. Stat Comput 19:367–380 CrossRefGoogle Scholar
  15. Cuevas A, Fraiman R (2009) On depth measures and dual statistics. A methodology for dealing with general data. J Multivar Anal 100:753–766 zbMATHCrossRefMathSciNetGoogle Scholar
  16. Cuevas A, Febrero M, Fraiman R (2004) An ANOVA test for functional data. Comput Stat Data Anal 47:111–122 zbMATHCrossRefMathSciNetGoogle Scholar
  17. Fan J (1996) Tests of significance based on wavelet thresholding and Neyman’s truncation. J Am Stat Assoc 91:674–688 zbMATHCrossRefGoogle Scholar
  18. Fan J, Lin SK (1998) Test of significance when data are curves. J Am Stat Assoc 93:254–261 MathSciNetGoogle Scholar
  19. Faraway JJ (1997) Regression analysis for a functional response. J R Stat Soc B 64:887–898 Google Scholar
  20. Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer series in statistics. Springer, New York zbMATHGoogle Scholar
  21. Ferraty F, Vieu P, Viguier-Pla S (2007) Factor-based comparison of groups of curves. Comput Stat Data Anal 51:4903–4910 zbMATHCrossRefMathSciNetGoogle Scholar
  22. Fujikoshi Y, Himeno T, Wakaki H (2004) Asymptotic results of a high dimensional MANOVA and power comparisons when the dimension is large compared to the sample size. J Jpn Stat Soc 34:19–26 zbMATHMathSciNetGoogle Scholar
  23. Gu C (2002) Smoothing spline ANOVA models. Springer, New York zbMATHGoogle Scholar
  24. Guo W (2002) Inference in smoothing spline analysis of variance. J R Stat Soc B 64:887–898 zbMATHCrossRefGoogle Scholar
  25. Krzanowski WJ (1988) Principles of multivariate analysis. A user’s perspective. Oxford University Press, Oxford zbMATHGoogle Scholar
  26. R Development Core Team (2007) R: a language and environment for statistical computing, Vienna, Austria. ISBN 3-900051-07-0, http://www.R-project.org
  27. Ramsay JO, Silverman BW (2002) Applied functional data analysis. Methods and case studies. Springer, New York zbMATHCrossRefGoogle Scholar
  28. Ramsay JO, Silverman BW (2005) Functional data analysis, 2nd edn. Springer, New York Google Scholar
  29. Schott JR (2007) Some high-dimensional tests for a one-way ANOVA. J Multivar Anal 98:1825–1839 zbMATHCrossRefMathSciNetGoogle Scholar
  30. Shohat JA, Tamarkin JD (1963) The problem of moments. American Mathematical Society, Providence zbMATHGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2010

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Estadística y Computación, Facultad de CienciasUniversidad de CantabriaSantanderSpain
  2. 2.Departamento de Estadística e Inv. Operativa, Facultad de MatemáticasUniversidad Santiago de CompostelaSantiago de CompostelaSpain

Personalised recommendations