Self-consistency: A fundamental concept in statistics

Statistical Science

Volumn 11, Issue 3, 1996, Pages 229-243

  Tarpey, Thaddeus ^a   Flury, Bernard ^b  

^a WRIGHT STATE UNIVERSITY   (United States)

^b INDIANA UNIVERSITY   (United States)

Author keywords

Elliptical distribution; EM algorithm; K means algorithm; Mean squared error; Principal components; Principal curves; Principal modes of variation; Principal points; Principal variables; Regression; Self organizing maps; Spherical distribution; Voronoi region

Indexed keywords

EID: 0001645194     PISSN: 08834237     EISSN: None     Source Type: Journal    
DOI: 10.1214/ss/1032280215     Document Type: Article

References (45)

1. ANDERSON, T. W. (1963). Asymptotic theory for principal component analysis. Ann. Math. Statist. 34 122-148.
2. Bandeen-Roche, K. (1994). Resolution of additive mixtures into source components and contributions: a compositional approach. J. Amer. Statist. Assoc. 89 1450-1458.
3. BANFIELD, J. and RAFTERY, A. (1992). Ice floe identification in satellite images using mathematical morphology and clustering about principal curves. J. Amer. Statist. Assoc. 87 7-16.
4. BICKEL, P. J. and DOKSUM, K. A. (1977). Mathematical Statistics. Holden-Day, San Francisco.
5. CASELLA, G. and BERGER, R. L. (1990). Statistical Inference. Duxbury Press, Belmont, CA.
6. Castro, P. E., Lawton, W. H. and Sylvestre, E. A. (1986). Principal modes of variation for processes with continuous sample curves. Technometrics 28 329-337.
7. Cox, D. R. (1957). Note on grouping. J. Amer. Statist. Assoc. 52 543-547.
8. COX, D. R. and OAKES, D. (1984). Analysis of Survival Data. Chapman and Hall, New York.
9. CUESTA, J. A. and Matran, C. (1988). The strong law of large numbers for k-means and best possible nets of Banach valued random variables. Probab. Theory Related Fields 78 523-534.
10. DALENIUS, T. (1950). The problem of optimum stratification. Skandinavisk Aktuarietidskrift 33 203-213.
11. Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1-38.
12. Efron, B. (1967). The two sample problem with censored data. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 4 831-853. Univ. California Press, Berkeley.
13. Eubank, R. L. (1988). Optimal grouping, spacing, stratification, and piecewise constant approximation. SIAM Rev. 30 404-420.
14. Fang, K. and He, S. (1982). The problem of selecting a given number of representative points in a normal population and a generalized Mill’s ratio. Technical report, Dept. Statistics, Stanford Univ.
15. Fang, K., Kotz, s. and Ng, K. (1990). Symmetric Multivariate and Related Distributions. Chapman and Hall, New York.
16. Flury, B. (1990). Principal points. Biometrika 77 33-41.
17. Flury, B. (1993). Estimation of principal points. J. Roy. Statist. Soc. Ser. C 42 139-151.
18. Flury, B. and Tarpey, T. (1993). Representing a large collection of curves: a case for principal points. Amer. Statist. 47 304-306.
19. FRIEDMAN, A. (1982). Foundations of Modern Analysis. Dover, New York.
20. HARTIGAN, J. A. (1975). Clustering Algorithms. Wiley, New York.
21. Hastie, T. and Stuetzle, W. (1989). Principal curves. J. Amer. Statist. Assoc. 84 502-516.
22. IEEE (1982). IEEE Trans. Inform. Theory 28. (Special issue on quantization.)
23. IYENGAR, S. and Solomon, H. (1983). Selecting representative points in normal populations. In Recent Advances in Statistics: Papers in Honor of Herman Chernoff on his 60th Birthday (M. H. Rizvi, J. Rustagi and D. Siegmund, eds.) 579-591. Academic Press, New York.
24. JOLICOEUR, P. (1968). Interval estimation of the slope of the major axis of a bivariate normal distribution in the case of a small sample. Biometrics 24 679-682.
25. JOLICOEUR, P. and MOSIMANN, J. E. (1960). Size and shape variation in the painted turtle; a principal component analysis. Growth 24 339-354.
26. KOHONEN, T. (1995). Self-Organizing Maps. Springer, Berlin.
27. Kshirsagar, A. M. (1961). The goodness of fit of a single (nonisotropic) hypothetical principal component. Biometrika 48 397-407.
28. LAIRD, N. (1988). Self-Consistency. In Encyclopedia of Statistical Sciences, 8 347-351. Wiley, New York.
29. LEBLANC, M. and Tibshirani, R. (1994). Adaptive principal surfaces. J. Amer. Statist. Assoc. 89 53-64.
30. LITTLE, J. and Rubin, D. B. (1987). Statistical Analysis with Missing Data. Wiley, New York.
31. Lloyd, S. (1982). Least squares quantization in PCM. IEEE Trans. Inform. Theory 28 129-149.
32. MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 3 281-297. Univ. California Press, Berkeley.
33. MALLOWS, C. L. (1961). Latent vectors of random symmetric matrices. Biometrika 48 133-149.
34. McCabe, G. P. (1984). Principal variables. Technometrics 26 137-144.
35. PÁÍRNA, K. (1990). On the existence and weak convergence of k-centres in Banach spaces. Acta et Commentationes Universi-tatis Tartuensis 893 17-28.
36. PEARSON, K. (1901). On lines and planes of closest fit to systems of points in space. Philosophical Magazine 2 559-572.
37. POLLARD, D. (1981). Strong consistency of K-means clustering Ann. Statist. 9 135-140.
38. ROWE, S. (1996). An algorithm for computing principal points with respect to a loss function in the unidimensional case Statistics and Computing 6 187-190.
39. SCHOTT, J. R. (1991). A test for a specific principal component oi a correlation matrix. J. Amer. Statist. Assoc. 86 747-751.
40. TARPEY, T. (1995). Principal points and self-consistent points oi symmetric multivariate distributions. J. Multivariate Anal 53 39-51.
41. TARPEY, T. (1996). Self-consistent patterns for symmetric, multivariate distributions. Unpublished manuscript.
42. TARPEY, T., Li, L. and FLURY, B. (1995). Principal points and self consistent points of elliptical distributions. Ann. Statist. 23 103-112.
43. Tibshirani, R. (1992). Principal curves revisited. Statistics ano. Computing 2 183-190.
44. TYLER, D. (1983). A class of asymptotic tests for principal component vectors. Ann. Statist. 11 1243-1250.
45. Zoppé, A. (1995). Principal points of univariate continuous distributions. Statistics and Computing 5 127-132.