A geometric approach to maximum likelihood estimation of the functional principal components from sparse longitudinal data

Journal of Computational and Graphical Statistics

Volumn 18, Issue 4, 2009, Pages 995-1015

  Peng, Jie ^a   Paul, Debashis ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Covariance kernel; Cross validation; Newton Raphson algorithm; Stiefel manifold

Indexed keywords

EID: 72449195207     PISSN: 10618600     EISSN: None     Source Type: Journal    
DOI: 10.1198/jcgs.2009.08011     Document Type: Article

References (33)

1. Ash, R. B. (1972), Real Analysis and Probability, New York-London: Academic Press.
2. Besse, P., Cardot, H., and Ferraty, F. (1997), "Simultaneous Nonparametric Regression of Unbalanced Longitudinal Data," Computational Statistics and Data Analysis, 24, 255-270.
3. Boente, G., and Fraiman, R. (2000), "Kernel-Based Functional Principal Components Analysis," Statistics and Probability Letters, 48, 335-345.
4. Boyd, S., and Vandenberghe, L. (2004), Convex Optimization, Cambridge, U.K.: Cambridge University Press.
5. Burman, P. (1989), "A Comparative Study of Ordinary Cross-Validation, v-Fold Cross-Validation and the Repeated Learning-Testing Methods," Biometrika, 76, 503-514.
6. - (1990), "Estimation of Generalized Additive Models," Journal of Multivariate Analysis, 32, 230-255.
7. Cai, T., and Hall, P. (2006), "Prediction in Functional Linear Regression," Annals of Statistics, 34, 2159-2179.
8. Cardot, H. (2000), "Nonparametric Estimation of Smoothed Principal Components Analysis of Sampled Noisy Functions," Journal of Nonparametric Statistics, 12, 503-538.
9. - (2006), "Conditional Functional Principal Components Analysis," Scandinavian Journal of Statistics, 33, 317-335.
10. Cardot, H., Ferraty, F., and Sarda, P. (1999), "Functional Linear Model," Statistics and Probability Letters, 45, 11-22.
11. Chui, C. (1988), Multivariate Splines, Philadelphia: SIAM.
12. de Boor, C. (1978), A Practical Guide to Splines, New York: Springer-Verlag.
13. Diggle, P. J., Heagerty, P., Liang, K.-Y., and Zeger, S. L. (2002), Analysis of Longitudinal Data (2nd ed.), New York: Oxford University Press.
14. Dussault, J. P. (2000), "Convergence of Implementable Descent Algorithms for Unconstrained Problems," Journal of Optimization Theory and Applications, 104, 739-745.
15. Edelman, A., Arias, T. A., and Smith, S. T. (1998), "The Geometry of Algorithms With Orthogonality Constraints," SIAM Journal on Matrix Analysis and Applications, 20, 303-353.
16. Fan, J., and Zhang, J. T. (2000), "Two-Step Estimation of Functional Linear Models With Applications to Longitudinal Data," Journal of Royal Statistical Society, Ser. B, 62, 303-322.
17. Ferraty, F., and Vieu, P. (2006), Nonparametric Functional Data Analysis: Theory and Practice, New York: Springer.
18. Golub, G. H., Heath, M., andWahba, G. (1979), "Generalized Cross Validation as a Method for Choosing a Good Ridge Parameter," Technometrics, 21, 215-224.
19. Green, P. J., and Silverman, B. W. (1994), Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach, London: Chapman & Hall/CRC.
20. Hall, P., and Horowitz, J. L. (2007), "Methodology and Convergence Rates for Functional Linear Regression," Annals of Statistics, 35, 41-69.
21. Hall, P., Müller, H.-G., and Wang, J.-L. (2006), "Properties of Principal Component Methods for Functional and Longitudinal Data Analysis," Annals of Statistics, 34, 1493-1517.
22. James, G. M., Hastie, T. J., and Sugar, C. A. (2000), "Principal Component Models for Sparse Functional Data," Biometrika, 87, 587-602.
23. Kaslow, R. A., Ostrow, D. G., Detels, R., Phair, J. P., Polk, B. F., and Rinaldo, C. R. (1987), "The Multicenter AIDS Cohort Study: Rationale, Organization, and Selected Characteristics of the Participants," American Journal of Epidemiology, 126 (2), 310-318.
24. Marron, S. J.,Müller, H.-G., Rice, J.,Wang, J.-L.,Wang, N., andWang, Y. (2004), "Discussion of Nonparametric and Semiparametric Regression," Statistica Sinica, 14, 615-629.
25. Muirhead, R. J. (1982), Aspects of Multivariate Statistical Theory, New Jersey: Wiley.
26. Paul, D., and Peng, J. (2008), "Principal Components Analysis for Sparsely Observed Correlated Functional Data Using a Kernel Smoothing Approach," technical report (arXiv:0807.1106).
27. - (2009), "Consistency of Restricted Maximum Likelihood Estimators of Principal Components," Annals of Statistics, 37 (3), 1229-1271.
28. Ramsay, J., and Silverman, B. W. (2005), Functional Data Analysis (2nd ed.), New York: Springer.
29. Rice, J. A., andWu, C. O. (2001), "Nonparametric Mixed Effects Models for Unequally Sampled Noisy Curves," Biometrics, 57, 253-259.
30. Stone,M. (1977), "An Asymptotic Equivalence of Choice of Model by Cross Validation and Akaike's Information Criterion," Journal of Royal Statistical Society, Ser. B, 39, 44-47.
31. Wu, C., and Chiang, C. (2000), "Kernel Smoothing on Varying Coefficient ModelsWith Longitudinal Dependent Variables," Statistica Sinica, 10, 433-456.
32. Yao, F.,Müller, H.-G., andWang, J.-L. (2005), "Functional Data Analysis for Sparse Longitudinal Data," Journal of the American Statistical Association, 100, 577-590.
33. - (2006), "Functional Linear Regression for Longitudinal Data," Annals of Statistics, 33, 2873-2903.