A nonlinear PCA based on manifold approximation

Computational Statistics

Volumn 15, Issue 2, 2000, Pages 145-167

  Girard, Stéphane ^a  

^a UNIV MONTPELLIER   (France)

Author keywords

Nonlinear auto associative models; Principal component analysis; Projection pursuit

Indexed keywords

EID: 0034414165     PISSN: 09434062     EISSN: None     Source Type: Journal    
DOI: 10.1007/s001800000025     Document Type: Article

References (29)

1. Besse, P. & Ferraty, F. (1995). A fixed effect curvilinear model. Computational Statistics, 10(4), 339-351.
2. Chalmond, B. & Girard, S. (1999). Nonlinear modeling of scattered multivariate data and its application to shape change. IEEE Pattern Analysis and Machine Intelligence, 21 (5), 422-431.
3. Chalmond, B. (2000). Eléments de modélisation pour l'analyse d'images. Springer-Verlag, Mathématiques et Applications 33.
4. Cheng, B. & Titterington, D. M. (1994). Neural networks: A review from a statistical perspective. Statistical Science, 9(1), 2-54.
5. Cottrell, M. (1994). Analyse de données et réseaux de neurones. Technical report SAMOS-Université Paris I.
6. Craven, P. & Wahba, G. (1978). Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation. Numerische Mathematik, 31, 377-403.
7. Diaconis, P. & ShahShahani, M. (1984). On nonlinear functions of lineaar combinations. SIAM Journal of Scientifical Statistical Computation, 5 (1), 175-191.
8. Duflo, M. (1996). Stochastic algorithms. Springer Verlag.
9. Durand, J. F. (1993). Generalized principal component analysis with respect to instrumental variables via univariate spline transformations. Computational Statistics and Data Analysis, 16, 423-440.
10. Eubank, R. L. (1990). Spline smoothing and non-parametric regression. Marcel Decker.
11. Friedman, J. H. & Tukey, J. W. (1974). A projection pursuit algorithm for exploratory data analysis. IEEE Transactions on computers, C23 (9), 881-890.
12. Friedman, J. H. & Stuetzle, W. (1981). Projection pursuit regression. Journal of the American Statistical Association, 76 (376), 817-823.
13. Friedman, J.H. (1987). Exploratory projection pursuit. Journal of the American Statistical Association, 82 (397), 249-266.
14. Girard, S. (1996). Construction et apprentissage statistique de modèles auto-associatifs non-linéaires. PhD thesis, Université de Cergy-Pontoise.
15. Girard, S., Chalmond, B. & Dinten, J. M. (1998). Position of principal component analysis among auto-associative composite models. Comptes-Rendus de l'Académie des Sciences, t. 326, Série I, 763-768.
16. Girosi, F., Jones, M. & Poggio, T. (1995). Regularization theory and neural networks architectures. Neural Computation, 219-269.
17. Hastie, T. & Stuetzle, W. (1989). Principal curves. Journal of the American Statistical Association, 84 (406), 502-516.
18. Huber, P. J. (1985). Projection pursuit. The Annals of Statistics, 13 (2), 435-525.
19. Huber, P. J. (1990). Algorithms for Projection Pursuit. Technical report of the Massachusetts Institute of Technology PJH-90-3.
20. Karhunen, J. & Joutsensalo, J. (1995). Generalizations of principal component analysis, optimization problems, and neural networks. Neural Networks, 8(4), 549-562, 1995.
21. Kneip, A. & Gasser, T. (1992). Statistical tools to analyze data representing a sample of curves. The Annals of Statistics, 20 (3), 1266-1305.
22. Mardia, K.V. & Kent, J.T. & Bibby, J.M. Multivariate Analysis, Academic Press.
23. Milnor, J. (1965). Topology from the differentiable point of view. The university press of Virginia, Charlottesville.
24. Ramsay, J. O. & Silverman, B. W. Functional data analysis, Springer-Verlag, 1997.
25. Ramsay, J. O. & Li, X. (1998). Curve Registration. Journal of the Royal Statistical Society B, 60 (2), 351-363.
26. Rao, C. R. (1973). Linear statistical inference and its application. John Wiley and Sons.
27. Rice, J. A. & Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society B, 53 (1), 233-243.
28. Stone, M. (1974). Cross-validatory choice and assessment of statistical prediction. Journal of the Royal Statistical Society B, 36, 111-147.
29. Tzovaras, D., Strintzis, M. (1998). Use of nonlinear principal component analysis and vector quantization for image coding. IEEE Transactions on Image Processing, 7 (8), 1218-1223.