Advertisement

Introduction: Examples of Functional Data

  • Pedro A. Morettin
  • Aluísio Pinheiro
  • Brani Vidakovic
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Wavelet-based functional data analysis (FDA) is a modern approach to dealing with statistical inference when observations are curves or images. Making inference (estimation and testing) in the wavelet domain is beneficial in several respects such as: reduction of dimensionality, decorrelation, localization, and regularization. This chapter gives an overview of theory for wavelet-based functional analysis, reviews relevant references, and provides some examples that will be used in the next chapters.

Bibliography

  1. P.J. Brown, T. Fearn, M. Vannucci, Bayesian wavelet regression on curves with application to a spectroscopic calibration problem. J. Am. Stat. Assoc. 96, 398–408 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  2. B. Brumback, J.A. Rice, Smoothing spline models for the analysis of nested and crossed samples of curves. J. Am. Stat. Assoc. 93, 961–994 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. V. Cahouët, L. Martin, D. Amarantini, Static optimal estimation of joint accelerations for inverse dynamic problem solution. J. Biomech. 35, 1507–1513 (2002)CrossRefGoogle Scholar
  4. M.W. Dewhirst, R.D. Braun, J.L. Lanzen, Temporal changes in PO2 of R3230Ac tumors in Fischer-344 rats. Int. J. Radiat. Oncol. Biol. Phys. 42, 723–726 (1998)CrossRefGoogle Scholar
  5. J. Fan, Test of significance based on wavelet thresholding and Neyman’s truncation. J. Am. Stat. Assoc. 91, 674–688 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  6. F. Ferraty, Y. Romain, The Oxford Handbook of Functional Data Analysis (Oxford University Press, New York, 2011)zbMATHGoogle Scholar
  7. F. Ferraty, P. Vieu, Nonparametric Functional Data Analysis (Springer, New York, 2006)zbMATHGoogle Scholar
  8. J.B. German, M.A. Roberts, S.M. Watkins, Genetics and metabolomics as markers for the interaction of diet and health: lessons from lipids. J. Nutr. 133, 2078S–2083S (2003)Google Scholar
  9. J.B. German, D.E. Bauman, D.G. Burrin, M.L. Failla, H.C. Freake, J.C. King, S. Klein, J.A. Milner, G.H. Pelto, K.M. Rasmussen, S.H. Zeisel, Metabolomics in the opening decade of the 21st century: building the roads to individualized health. J. Nutr. 134, 2729–2732 (2004)Google Scholar
  10. L. Horváth, P. Kokoszka, Inference for Functional Data with Applications (Springer, New York, 2012)CrossRefzbMATHGoogle Scholar
  11. J.L. Lanzen, R.D. Braun, A.L. Ong, M.W. Dewhirst, Variability in blood flow and po2 in tumors in response to carbogen breathing. Int. J. Radiat. Oncol. Biol. Phys. 42, 855–859 (1998)CrossRefGoogle Scholar
  12. P. Müller, G. Rosner, L. Inoue, M.W. Dewhirst, A Bayesian model for detecting changes in nonlinear profiles. J. Am. Stat. Assoc. 96, 1215–1222 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  13. J.O. Ramsay, B.W. Silverman, Applied Functional Data Analysis (Springer, New York, 2002)zbMATHGoogle Scholar
  14. J.O. Ramsay, B.W. Silverman, Functional Data Analysis, 2nd edn. (Springer, New York, 2006)zbMATHGoogle Scholar
  15. J.O. Ramsay, G. Hooker, S. Graves, Functional Data Analysis with R and MATLAB (Springer, New York, 2009)CrossRefzbMATHGoogle Scholar
  16. J. Raz, B. Turetsky, Wavelet ANOVA and fMRI, in Wavelet Applications in Signal and Image Processing VII, Proceedings of the SPIE (SPIE, Maui, HI, 1999), pp. 561–570CrossRefGoogle Scholar
  17. J.R. Sato, M.M. Felix, E. Amaro Jr., D.Y. Takahashi, M.J. Brammer, P.A. Morettin, A method to produce evolving functional connectivity maps during the course of an fMRI experiment using wavelet-based time-varying granger causality. NeuroImage 31, 187–196 (2006)CrossRefGoogle Scholar
  18. J.R. Sato, P.A. Morettin, P.R. Arantes, E. Amaro Jr., Wavelet based time-varying vector autoregressive models. Comput. Stat. Data Anal. 51, 5847–5866 (2007b)Google Scholar
  19. J.R. Sato, D.Y. Takahashi, S.M. Arcuri, K. Sameshima, P.A. Morettin, L.A. Baccala, Frequency domain connectivity identification: an application of partial directed coherence in fMRI. Hum. Brain Mapp. 30, 452–461 (2009)CrossRefGoogle Scholar
  20. B. Vidakovic, Wavelet-based functional data analysis: theory, applications and ramifications, ed. by T. Kobayashi, in Proceedings of PSFVIP-3 (3rd Pacific Symposium on Flow Visualization and Image Processing), PSFVIP-3, Maui, HI, 2001Google Scholar
  21. J.T. Zhang, Analysis of Variance for Functional Data (Chapman & Hall, Boca Raton, FL, 2014)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Pedro A. Morettin
    • 1
  • Aluísio Pinheiro
    • 2
  • Brani Vidakovic
    • 3
  1. 1.Department of StatisticsUniversity of São PauloSão PauloBrazil
  2. 2.Department of StatisticsUniversity of CampinasCampinasBrazil
  3. 3.The Wallace H. Coulter Department of Biomedical EngineeringGeorgia Inst Tech & Emory Univ Sch MedAtlantaUSA

Personalised recommendations