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Auto-Associative Models, Nonlinear Principal Component Analysis, Manifolds and Projection Pursuit

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Principal Manifolds for Data Visualization and Dimension Reduction

Part of the book series: Lecture Notes in Computational Science and Enginee ((LNCSE,volume 58))

Auto-associative models have been introduced as a new tool for building nonlinear Principal component analysis (PCA) methods. Such models rely on successive approximations of a dataset by manifolds of increasing dimensions. In this chapter, we propose a precise theoretical comparison between PCA and autoassociative models. We also highlight the links between auto-associative models, projection pursuit algorithms, and some neural network approaches. Numerical results are presented on simulated and real datasets.

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References

  1. Bellman, R.: Dynamic programming. Princeton university press, (1957)

    Google Scholar 

  2. Bosq, D. and Lecoutre, J. P.: Théorie de l’estimation fonctionnelle. Economie et Statistiques avancées. Economica, Paris (1987)

    Google Scholar 

  3. Carroll, J. D. and Arabie P.: Multidimensionnal scaling. Annual Rev. of Psy-chology, 31, 607-649 (1980)

    Article  Google Scholar 

  4. Caussinus, H. and Ruiz-Gazen, A.: Metrics for finding typical structures by means of principal component analysis. In: Data science and its Applications, pages 177-192. Harcourt Brace, Japan (1995)

    Google Scholar 

  5. Chalmond, B.: Modeling and inverse problems in image analysis. In: Applied Mathematics Science series, volume 155. Springer, New-York (2002)

    Google Scholar 

  6. Chalmond, B. and Girard, S.: Nonlinear modeling of scattered multivariate data and its application to shape change. IEEE Pattern Analysis and Machine Intelligence, 21 (5), 422-432 (1999)

    Article  Google Scholar 

  7. Cheng, B. and Titterington, D. M.: Neural networks: A review from a statistical perspective. Statistical Science, 9 (1), 2-54 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Delicado, P.: Another look at principal curves and surfaces. Journal of Multivariate Analysis, 77, 84-116 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Demartines, P. and Hérault, J.: Curvilinear component analysis: A selforganizing neural network for nonlinear mapping of data sets. IEEE Trans. on Neural Networks, 8 (1), 148-154 (1997)

    Article  Google Scholar 

  10. Diamantaras, K. L. and Kung, S. Y.: Principal component neural networks. Wiley, New-York (1996)

    MATH  Google Scholar 

  11. Eubank, R. L.: Spline smoothing and non-parametric regression. Decker (1990)

    Google Scholar 

  12. Ferraty, F. and Vieu, P.: Nonparametric modelling for functional data. Springer (2005)

    Google Scholar 

  13. Friedman, J. H.: Exploratory projection pursuit. Journal of the American Statistical Association, 82(397), 249-266 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Friedman, J. H. and Stuetzle, W.: Projection pursuit regression. Journal of the American Statistical Association, 76 (376), 817-823 (1981)

    Article  MathSciNet  Google Scholar 

  15. Friedman, J. H. and Tukey, J. W.: A projection pursuit algorithm for exploratory data analysis. IEEE Trans. on Computers, C23 (9), 881-890 (1974)

    Article  Google Scholar 

  16. Girard, S.: A nonlinear PCA based on manifold approximation. Computational Statistics, 15 (2), 145-167 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Girard, S., Chalmond, B., and Dinten, J. -M.: Position of principal compo-nent analysis among auto-associative composite models. Comptes-Rendus de l’Académie des Sciences, Série I, 326, 763-768 (1998)

    Article  MathSciNet  Google Scholar 

  18. Girard, S. and Iovleff, S.: Auto-associative models and generalized principal component analysis. Journal of Multivariate Analysis, 93 (1), 21-39 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gorban, A. and Zinovyev, A.: Elastic principal graphs and manifolds and their practical applications. Computing, 75 (4), 359-379 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Green, P. J. and Silverman, B. W.: Non-parametric regression and generalized linear models. Chapman and Hall, London (1994)

    Google Scholar 

  21. Hall, P.: On polynomial-based projection indices for exploratory projection pursuit. The Annals of Statistics, 17 (2), 589-605 (1990)

    Article  Google Scholar 

  22. Härdle, W.: Applied nonparametric regression. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  23. Hartigan, J. A.: Clustering algorithms. Wiley, New-York (1995)

    Google Scholar 

  24. Hastie, T. and Stuetzle, W.: Principal curves. Journal of the American Statistical Association, 84 (406), 502-516 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hastie, T., Tibshirani, R., and Friedman, J.: The elements of statistical learning. In: Springer Series in Statistics. Springer (2001)

    Google Scholar 

  26. Huber, P. J.: Projection pursuit. The Annals of Statistics, 13 (2), 435-475 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  27. Jolliffe, I.: Principal Component Analysis. Springer-Verlag, New York (1986)

    Google Scholar 

  28. Jones, M. C. and Sibson R.: What is projection pursuit? Journal of the Royal Statistical Society A, 150, 1-36 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  29. Karhunen, J. and Joutsensalo, J.: Generalizations of principal component analy-sis, optimization problems and neural networks. Neural Networks, 8, 549-562 (1995)

    Article  Google Scholar 

  30. Kégl, B.:“Principal curves: learning, design, and applications”. PhD thesis, Concordia University, Canada (1999)

    Google Scholar 

  31. Kégl, B., Krzyzak, A., Linder, T., and Zeger, K.: A polygonal line algorithm for constructing principal curves. In: Proceedings of 12h NIPS, pages 501-507, Denver, Colorado (1998)

    Google Scholar 

  32. Klinke, S. and Grassmann, J. Projection pursuit regression. In: Wiley Series in Probability and Statistics, pages 471-496. Wiley (2000)

    Google Scholar 

  33. Kohonen, T.: Self-organization of topologically correct feature maps. Biological cybernetics, 43, 135-140 (1982)

    Article  MathSciNet  Google Scholar 

  34. Kohonen, T.: Self-organization and associative memory, 3rd edition. Springer Verlag, Berlin (1989)

    Google Scholar 

  35. Kruskal, J. B. and Wish, M.: Multidimensional scaling. Sage, Beverly Hills (1978)

    Google Scholar 

  36. Lebart L.: Contiguity analysis and classification. In: Opitz O. Gaul W. and Schader M. (eds. ) Data Analysis, pages 233-244. Springer, Berlin (2000)

    Google Scholar 

  37. LeBlanc, M. and Tibshirani, R.: Adaptive principal surfaces. Journal of the American Statistical Association, 89 (425), 53-64 (1994)

    Article  MATH  Google Scholar 

  38. Lee, J. A., Lendasse, A., and Verleysen, M.: Curvilinear distance analysis versus isomap. In: European Symposium on Artifical Neural Networks, pages 185-192. Bruges, Belgium (2002)

    Google Scholar 

  39. Milnor, J.: Topology from the differentiable point of view. University press of Virginia, Charlottesville (1965)

    Google Scholar 

  40. Pan, J-X., Fung, W-K., and Fang, K-T.: Multiple outlier detection in multivari-ate data using projection pursuit techniques. Journal of Statistical Planning and Inference, 83 (1), 153-167 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  41. Parlett, B. N.: The symmetric eigenvalue problem. In: Classics in Applied Mathematics, vol. 20. SIAM, Philadelphia (1997)

    Google Scholar 

  42. Roweis, S. T. and Saul, L. K.: Nonlinear dimensionality reduction by locally linear embedding. Science, 290, 2323-2326 (2000)

    Article  Google Scholar 

  43. Sammson, J. W.: A nonlinear mapping algorithm for data structure analysis. IEEE Transactions on Computer, 18 (5), 401-409 (1969)

    Article  Google Scholar 

  44. Shepard, R. N. and Carroll, J. D.: Parametric representation of nonlinear data structures. In: Krishnaiah, P. R. (ed. ) Int. Symp. on Multivariate Analysis, pages 561-592. Academic-Press (1965)

    Google Scholar 

  45. Tarpey, T. and Flury, B.: Self-consistency: A fundamental concept in statistics. Statistical Science, 11 (3), 229-243 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  46. Tenenbaum, J. B., de Silva, V., and Langford, J. C.: A global geometric frame-work for nonlinear dimensionality reduction. Science, 290, 2319-2323 (2000)

    Article  Google Scholar 

  47. Tibshirani, R.: Principal surfaces revisited. Statistics and Computing, 2, 183-190(1992)

    Article  Google Scholar 

  48. Vlachos, M., Domeniconi, C., Gunopulos, D., Kollios, G., and Koudas, N.: Non-linear dimensionality reduction techniques for classification and visualization. In: Proceedings of 8th SIGKDD, pages 23-26. Edmonton, Canada (2002)

    Google Scholar 

  49. Wang, Y., Klijn, J. G., Zhang, Y., Sieuwerts, et al.: Gene-expression profiles to predict distant metastasis of lymph-node-negative primary breast cancer. Lancet 365, 671-679 (2005)

    Google Scholar 

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Author information

Authors and Affiliations

  1. INRIA Rhône-Alpes, projet Mistis, Inovallée, av. de l'Europe, 655, 38334, Saint-Ismier cedex, France

    Stéphane Girard

  2. Laboratoire Paul Painlevé, 59655, Villeneuve d'Ascq Cedex, France

    Serge Iovleff

Authors
  1. Stéphane Girard
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  2. Serge Iovleff
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Editor information

Editors and Affiliations

  1. University of Leicester, University Road, LE1 7RH, Leicester, UK

    Alexander N. Gorban

  2. University of Paris-Sud - CNRS, Bâtiment 2000, 91898, Orsay, France

    Balázs Kégl

  3. University of Missouri - Rolla, 1870 Miner Circle, 65409, Rolla, MO, USA

    Donald C. Wunsch

  4. Institut Curie Service Bioinformatique, rue d'Ulm 26, 75248, Paris, France

    Andrei Y. Zinovyev

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Girard, S., Iovleff, S. (2008). Auto-Associative Models, Nonlinear Principal Component Analysis, Manifolds and Projection Pursuit. In: Gorban, A.N., Kégl, B., Wunsch, D.C., Zinovyev, A.Y. (eds) Principal Manifolds for Data Visualization and Dimension Reduction. Lecture Notes in Computational Science and Enginee, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73750-6_8

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