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Computational Statistics

, Volume 28, Issue 4, pp 1775–1811 | Cite as

No effect tests in regression on functional variable and some applications to spectrometric studies

  • Laurent DelsolEmail author
Original Paper

Abstract

Recent advances in structural tests for regression on functional variable are used to construct test of no effect. Various bootstrap procedures are considered and compared in a simulation study. These tests are finally applied on real world datasets dealing with spectrometric studies using the information collected during this simulation study. The results obtained for the Tecator dataset are relevant and corroborated by former studies. The study of a smaller dataset concerning corn samples shows the efficiency of our method on small size samples. Getting information on which derivatives (or which parts) of the spectrometric curves have a significant effect allows to get a better understanding of the way spectrometric curves influence the quantity to predict. In addition, a better knowledge of the structure of the underlying regression model may be useful to construct a relevant predictor.

Keywords

No effect test Regression Functional variable Bootstrap Spectrometric curves 

References

  1. Alsberg BK (1993) Representation of spectra by continuous functions. J Chemom 7:177–193Google Scholar
  2. Aneiros-Perez G, Vieu P (2006) Semi-functional partial linear regression. Stat Probab Lett 76(11): 1102–1110MathSciNetzbMATHCrossRefGoogle Scholar
  3. Borggaard C, Thodberg HH (1992) Optimal minimal neural interpretation of spectra. Anal Chem 64(5): 545–551Google Scholar
  4. Bosq D (2000) Linear processes in function spaces: theory and applications Lecture Notes in Statistics 149. Springer, New YorkCrossRefGoogle Scholar
  5. Burba F, Ferraty F, Vieu P (2009) k-Nearest neighbor method in functional nonparametric regression. J Nonparametric Stat 21:453–469MathSciNetzbMATHCrossRefGoogle Scholar
  6. Cao R (1991) Rate of convergencefor the wild bootstrap in nonparametric regression. Ann Stat 19: 2226–2231zbMATHCrossRefGoogle Scholar
  7. Cardot H, Ferraty F, Sarda P (1999) Functional Linear Model. Stat Probab Lett 45(1):11–22Google Scholar
  8. Cardot H, Ferraty F, Mas A, Sarda P (2003) Testing hypothesys in the functional linear model. Scand J Stat 30:241–255MathSciNetzbMATHCrossRefGoogle Scholar
  9. Cardot H, Goia A, Sarda P (2004) Testing for no effect in functional linear regression models, some computational approaches. Commun Stat Simul Comput 33(1):179–199MathSciNetzbMATHCrossRefGoogle Scholar
  10. Chen SX, Van Keilegom I (2009) A goodness-of-fit test for parametric and semiparametric models in multiresponse regression. Bernoulli 15:955–976MathSciNetzbMATHCrossRefGoogle Scholar
  11. Crambes C, Kneip A, Sarda P (2009) Smoothing splines estimators for functional linear regression. Ann Stat 37:35–72MathSciNetzbMATHCrossRefGoogle Scholar
  12. Cuevas A, Fraiman R (2004) On the bootstrap methodology for functional data. In: Antoch J (ed) (English summary) COMPSTAT 2004—proceedings in computational statistics. Physica, Heidelberg, pp 127–135Google Scholar
  13. Cuevas A, Febrero M, Fraiman R (2006) On the use of the bootstrap for estimating functions with functional data. Comput Stat Data Anal 51(2):1063–1074Google Scholar
  14. Dabo-Niang S, Ferraty F, Vieu P (2006) Mode estimation for functional random variable and its application for curves classification. Far East J Theor Stat 18(1):93–119MathSciNetzbMATHGoogle Scholar
  15. Davidian M, Lin X, Wang J-L (2004) Introduction [Emerging issues in longitudinal and functional data analysis]. Stat Sinica 14(3):613–614Google Scholar
  16. Delsol L, Ferraty F, Vieu P (2011) Structural test in regression on functional variables. J Multivar Anal 102(3):422–447MathSciNetzbMATHCrossRefGoogle Scholar
  17. Efron B (1979) Bootstrap methods: another look at the Jackknife. Ann Stat 7(1):1–26MathSciNetzbMATHCrossRefGoogle Scholar
  18. Fernandez de Castro B, Guillas S (2005) Functional samples and bootstrap for predicting sulfur dioxide levels. Technometrics 47(2):212–222MathSciNetCrossRefGoogle Scholar
  19. Ferraty F (2010) Editorial to the special issue statistical methods and problems in infinite-dimensional spaces. J Multivar Anal 101(2):305–306MathSciNetCrossRefGoogle Scholar
  20. Ferraty F, Vieu P (2000) Dimension fractale et estimation de la régression dans des espaces vectoriels semi-normés. Compte Rendus de l’Académie des Sciences Paris 330:403–406MathSciNetGoogle Scholar
  21. Ferraty F, Vieu P (2002) The functional nonparametric model and application to spectrometric data. Comput Stat 17(4):545–564MathSciNetzbMATHCrossRefGoogle Scholar
  22. Ferraty F, Vieu P (2006) Nonparametric modelling for functional data. Springer, New YorkGoogle Scholar
  23. Ferré L, Villa N (2006) Multi-Layer perceptron with functional inputs: an inverse regression approach. Scand J Stat 33(4):807–823zbMATHCrossRefGoogle Scholar
  24. Ferraty F, Vieu P (2009) Additive prediction and boosting for functional data. Comput Stat Data Anal 53(4):1400–1413MathSciNetzbMATHCrossRefGoogle Scholar
  25. Ferraty F, Romain Y (2011) The Oxford handbook of functional data analysis. Oxford University Press, OxfordGoogle Scholar
  26. Ferraty F, Laksaci A, Vieu P (2006) Estimating some characteristics of the conditional distribution in nonparametric functional models. Stat Inference Stoch Process 9(1):47–76MathSciNetzbMATHCrossRefGoogle Scholar
  27. Ferraty F, Mas A, Vieu P (2007) Advances on nonparametric regression for fonctionnal data. ANZ J Stat 49:267–286MathSciNetzbMATHGoogle Scholar
  28. Ferraty F, Vieu P, Viguier-Pla S (2007) Factor-based comparison of groups of curves. Comput Stat Data Anal 51(10):4903–4910MathSciNetzbMATHCrossRefGoogle Scholar
  29. Ferraty F, Van Keilegom I, Vieu P (2010) On the validity of the bootstrap in nonparametric functionl regression. Scand J Stat 37:286–306MathSciNetzbMATHCrossRefGoogle Scholar
  30. Ferré L, Yao A-F (2005) Smoothed functional inverse regression. Stat Sinica 15(3):665–683zbMATHGoogle Scholar
  31. Gadiaga D, Ignaccolo R (2005) Test of no-effect hypothesis by nonparametric regression. Afr Stat 1(1): 67–76MathSciNetzbMATHGoogle Scholar
  32. González-Manteiga W, Vieu P (2007) Editorial of the special issue statistics for functional data. Comput Stat Data Anal 51(10):4788–4792Google Scholar
  33. Gao J, Gijbels I (2008) Bandwidth Selection in Nonparametric Kernel Testing, Journal of the American Statistical Association. Am Stat Assoc 103(484):1584–1594Google Scholar
  34. Gonzalez-Manteiga W, Quintela-del-Río A, Vieu P (2002) A note on variable selection in nonparametric regression with dependent data. Stat Probab Lett 57(3):259–268Google Scholar
  35. González-Manteiga W, Martinez Miranda MD, Perez Gonzalez A (2004) The choice of smoothing parameter in nonparametric regression through wild bootstrap comp. Stat Data Anal 47:487–515zbMATHCrossRefGoogle Scholar
  36. Hall P (1990) Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems. J Multivar Anal 32:177–203zbMATHCrossRefGoogle Scholar
  37. Hall P (1992) On bootstrap confidence intervals in nonparametric regression. Ann Stat 20:695–711zbMATHCrossRefGoogle Scholar
  38. Hall P, Hart J (1990) Bootstrap test for differene between means in nonparametric regression. J Am Stat Assoc 85:1039–1049Google Scholar
  39. Härdle W, Marron JS (1990) Semiparametric comparison of regression curves. Ann Stat 18(1):63–89zbMATHCrossRefGoogle Scholar
  40. Härdle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21(4):1926–1947zbMATHCrossRefGoogle Scholar
  41. Hernandez N, Biscay RJ, Talavera I (2008) Support vector regression methods for functional data. Lecture Notes Comput Sci 4756:564–573CrossRefGoogle Scholar
  42. James G, Silverman BW (2005) Functional adaptative model estimation. J Am Stat Assoc 100:565–576MathSciNetzbMATHCrossRefGoogle Scholar
  43. Laloë T (2007) A \(k\)-nearest neighbor approach for functional regression. Stat Probab Lett 78(10):1189–1193CrossRefGoogle Scholar
  44. Lavergne P, Patilea V (2007) Breaking the curse of dimensionality in nonparametric testing. J Econom 143(1):103–122Google Scholar
  45. Leardi R (2003) Nature-inspired methods in chemometrics: genetic algorithms and artificial neural networks. Elsevier, AmsterdamGoogle Scholar
  46. Leurgans SE, Moyeed RA, Silverman BW (1993) Canonical correlation analysis when the data are curves. J R Stat Soc Ser B 55(3):725–740MathSciNetzbMATHGoogle Scholar
  47. Mammen E (1993) Bootstrap and wild bootstrap for high-dimensional linear models. Ann Stat 21(1): 255–285MathSciNetzbMATHCrossRefGoogle Scholar
  48. Mas A, Pumo B (2007) Functional linear regression with derivatives (submitted)Google Scholar
  49. Müller H-G, Stadtmüller U (2005) Generalized functional linear models. Ann Stat 33(2):774–805zbMATHCrossRefGoogle Scholar
  50. Ramsay J, Dalzell C (1991) Some tools for functional data analysis. J R Stat Soc B 53:539–572MathSciNetzbMATHGoogle Scholar
  51. Ramsay J, Silverman B (1997) Functional data analysis. Springer, New YorkzbMATHCrossRefGoogle Scholar
  52. Ramsay J, Silverman B (2002) Applied functional data analysis: methods and case studies. Spinger, New YorkCrossRefGoogle Scholar
  53. Ramsay J, Silverman B (2005) Functional data analysis, 2nd edn. Spinger, New YorkGoogle Scholar
  54. Rossi F, Delannay N, Conan-Guez B, Verleysen M (2005) Representation of dunctional data in neural networks. Neurocomputing 64:183–210CrossRefGoogle Scholar
  55. Stute W (1997) Nonparametric model checks for regression. (English summary) Ann Stat 25(2):613–641Google Scholar
  56. Stute W, Gonzalez Manteiga W, Presedo Quindimil M (1998) Bootstrap approximations in model checks for regression. J Am Stat Assoc 93(441):141–149MathSciNetzbMATHCrossRefGoogle Scholar
  57. Valderrama M (2007) An overview to modelling functional data. Comput Stat Data Anal 22(3):331–334MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut de Statistique, U.C.L.Louvain-la-NeuveBelgium
  2. 2.MAPMO, Université d’OrléansOrléans Cedex 2France

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