Robust estimation in canonical correlation analysis for multivariate functional data

Abstract

In this paper, the canonical correlation analysis for multivariate functional data is considered. The analysis is based on the basis functions representation of the data. The use of non-orthogonal bases is available in contrast to the approach given in the literature. The robust estimation methods of the covariance matrix are also studied in the multivariate functional canonical correlation analysis. Simulation studies and breakdown analysis suggest that the proposed methods may perform better than the classical estimator under non-normal models and in the presence of outlying observations.

Keywords

• Basis functions representation,Canonical correlation,Functional data analysis,Multivariate analysis,Robust covariance estimation

References

1. Aneiros, G., Bongiorno, E.G., Cao, R. and Vieu, P. (Eds) Functional statistics and related fields. Contributions to Statistics (Springer, 2017).
2. Bobelyn, E., Serban, A.S., Nicu, M., Lammertyn, J., Nicolai, B.M. and Saeys, W. Postharvest quality of apple predicted by NIR-spectroscopy: Study of the effect of biological variability on spectra and model performance. Postharvest Biol. Technol. 55, 133-143, 2010.
3. Boente, G., Barrera, M.S. and Tyler, D.E. A characterization of elliptical distributions and some optimality properties of principal components for functional data. J. Multivariate Anal. 131, 254-264, 2014.
4. Bongiorno, E.G., Salinelli, E., Goia, A. and Vieu, P. Contributions in infinite-dimensional statistics and related topics (Societa Editrice Esculapio, 2014).
5. Branco, J.A., Croux, C., Filzmoser, P. and Oliveira, M.R. Robust canonical correlations: A comparative study. Comput. Statist. 20, 203-229, 2005.
6. Chen, D., Hall, P. and Müller, H.G. Single and multiple index functional regression models with nonparametric link. Ann. Statist. 39, 1720-1747, 2011.
7. Collazos, J.A.A., Dias, R. and Zambom, A.Z. Consistent variable selection for functional regression models. J. Multivariate Anal. 146, 63-71, 2016.
8. Croux, C. and Haesbroeck, G. Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. J. Multivariate Anal. 71, 161-190, 1999.

9. Davies, P.L. Asymptotic behavior of S-estimators of multivariate location parameters and dispersion matrices. Ann. Statist. 15, 1269-1292, 1987.
10. Davies, P.L. An efficient Fréchet-differentiable high breakdown multivariate location and dispersion estimator. J. Multivariate Anal. 40, 311-327, 1992.
11. Delaigle, A. and Hall, P. Achieving near perfect classification for functional data. J. R. Stat. Soc. Ser. B Stat. Methodol. 74, 267-286, 2012.
12. Febrero-Bande, M., Galeano, P. and González-Manteiga, W. Outlier detection in functional data by depth measures, with application to identify nbnormal NOx levels. Environmetrics 19, 331-345, 2008.
13. Ferraty, F. and Vieu, P. Nonparametric functional data analysis: Theory and practice (Springer, New York, 2006).
14. Fremdt, S., Horváth, L., Kokoszka, P. and Steinebach, J.G. Functional data analysis with increasing number of projections. J. Multivariate Anal. 124, 313-332, 2014.
15. Górecki, T., Krzyśko, M., Waszak, Ł. and Wołyński, W. Selected statistical methods of data analysis for multivariate functional data. Statist. Papers 59, 153-182, 2018.
16. Górecki, T. and Smaga, Ł. A comparison of tests for the one-way ANOVA problem for functional data. Comput. Stat. 30, 987-1010, 2015.
17. Górecki, T. and Smaga, Ł. Multivariate analysis of variance for functional data. J. Appl. Stat. 44, 2172-2189, 2017.
18. Hilgert, N., Mas, A. and Verzelen, N. Minimax adaptive tests for the functional linear model. Ann. Statist. 41, 838-869, 2013.
19. Horváth, L. and Kokoszka, P. Inference for functional data with applications (Springer, New York, 2012).
20. Huber, P.J. Robust estimation of a location parameter. Ann. Math. Statist. 35, 73-101, 1964.
21. James, G.M. and Hastie, T.J. Functional linear discriminant analysis for irregularly sampled curves. J. R. Stat. Soc. Ser. B Stat. Methodol. 63, 533-550, 2001.
22. Kent, J.T. and Tyler, D.E. Constrained M-estimation for multivariate location and scatter. Ann. Statist. 24, 1346-1370, 1996.
23. Kokoszka, P., Oja, H., Park, B. and Sangalli, L. Special issue on functional data analysis. Econometrics and Statistics 1, 99-100, 2017.
24. Lopuhaä, H.P. On the relation between S-estimators and M-estimators of multivariate location and covariance. Ann. Statist. 17, 1662-1683, 1989.
25. Lopuhaä, H.P. Multivarite τ-estimators for location and scatter. Can. J. Statist. 19, 307-321, 1991.
26. Long, W., Li, N., Wang, H. and Cheng, S. Impact of US financial crisis on different countries: Based on the method of functional analysis of variance. Procedia Computer Science 9, 1292-1298, 2012.
27. Maronna, R.A. Robust M-estimators of multivariate location and scatter. Ann. Statist. 1, 51-67, 1976.
28. Martínez-Camblor, P. and Corral, N. Repeated measures analysis for functional data. Comput. Statist. Data Anal. 55, 3244-3256, 2011.
29. Matsui, H. and Konishi, K. Variable selection for functional regression models via the $L_1$ regularization. Comput. Statist. Data Anal. 55, 3304-3310, 2011.
30. Ogden, R.T., Miller, C.E., Takezawa, K. and Ninomiya, S. Functional regression in crop lodging assessment with digital images. J. Agric. Biol. Environ. Stat. 7, 389-402, 2002.
31. Ramsay, J.O. and Silverman, B.W. Functional data analysis, 2nd edition. (Springer, New York, 2005).
32. Ramsay, J.O., Wickham, H., Graves, S. and Hooker, G. fda - Functional data analysis. R package version 2.4.7, 2017. http://CRAN.R-project.org/package=fda
33. R Core Team R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2017. https://www.R-project.org/
34. Reimherr, M. Functional regression with repeated eigenvalues. Statist. Probab. Lett. 107, 62-70, 2015.
35. Rousseeuw, P.J. Multivariate estimation with high breakdown point. in: W. Grossmann, G. Pflug, I. Vincze, W. Wertz (Eds), Mathematical Statistics and Applications, volume B (Reidel Publishing, Dordrecht, 1985), 283-297.
36. Rousseeuw, P.J. and Van Driessen, K. A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212-223, 1999.
37. Rousseeuw, P.J. and Yohai, V.J. Robust regression by means of S-estimators. in: Robust and Nonlinear Time Series Analysis. Lecture Notes in Statistics 26 (Springer, New York, 1984), 256-272.
38. Ruppert, D. Computing S-estimators for regression and multivariate location/dispersion. J. Comput. Graph. Statist. 1, 253-270, 1992.
39. Salibian-Barrera, M. and Yohai, V. A fast algorithm for S-regression estimates. J. Comput. Graph. Statist. 15, 414-427, 2006.
40. Shmueli, G. To explain or to predict? Statist. Sci. 25, 289-310, 2010.
41. Smaga, Ł. Repeated measures analysis for functional data using Box-type approximation - with applications. REVSTAT, 2017. (To appear)
42. Todorov, V. and Filzmoser, P. An object-oriented framework for robust multivariate analysis. Journal of Statistical Software 32, 1-47, 2009.
43. Tyler, D.E. Finite sample breakdown points of projection based multivariate location and scatter statistics. Ann. Statist. 22, 1024-1044, 1994.
44. Yamamoto, M. and Terada, Y. Functional factorial K-means analysis. Comput. Statist. Data Anal. 79, 133-148, 2014.
45. Zhang, J.T. Analysis of variance for functional data (Chapman & Hall, London, 2013).
46. Zuo, Y. Robust location and scatter estimators in multivariate analysis. in: J. Fan and H.L. Koul (Eds) Frontiers of Statistics (in honor of Professor P.J. Bickel’s 65th Birthday), Imperial College, 467-490, 2006.