Self-consistent patterns for symmetric multivariate distributions

Journal of Classification

Volumn 15, Issue 1, 1998, Pages 57-79

  Tarpey, Thaddeus ^a  

^a WRIGHT STATE UNIVERSITY   (United States)

Author keywords

Elliptical distributions; K means clustering; Mean squared error; Principal components; Principal points; Spherical distributions; Voronoi regions

Indexed keywords

EID: 0041169479     PISSN: 01764268     EISSN: None     Source Type: Journal    
DOI: 10.1007/s003579900020     Document Type: Article

References (42)

1. BOFINGER, E. (1970), "Maximizing the Correlation of Grouped Observations," Journal of the American Statistical Association, 65, 1632-1638.
2. CAMBANIS, S., and GERR, N. (1983), "A Simple Class of Asymptotically Optimal Quantizers," IEEE Transactions on Information Theory, 29, 664-676.
3. CHAUDHURI, D., MURTHY, C., and CHAUDHURI, B. (1994), "Finding a Subset of Representative Points in a Data Set," IEEE Transactions on Systems, Man, and Cybernetics, 24, 1416-1424.
4. CONNOR, R. (1972), "Grouping for Testing Trends in Categorical Data," Journal of the American Statistical Association, 67, 601-604.
5. COX, D. R. (1957), "Note on Grouping," Journal of the American Statistical Association, 52, 543-547.
6. DALENIUS, T. (1950), "The Problem of Optimum Stratification," Skandinavisk Aktuarietidskrift, 33, 203-213.
7. DALENIUS, T., and GURNEY, M. (1951), "The Problem of Optimum Stratification II," Skandinavisk Aktuarietidskrift, 34, 203-213.
8. EUBANK, R. L. (1988), "Optimal Grouping, Spacing, Stratification, and Piecewise Constant Approximation," SIAM Review, 30, 3, 404-420.
9. 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," Department of Statistics Technical Report, Stanford University.
10. FANG, K., KOTZ, S., and NG, K. (1990), Symmetric Multivariate and Related Distributions, New York: Chapman and Hall.
11. FANG, K. T., and WANG, Y. (1993), Number Theoretic Methods in Statistics, New York: Chapman and Hall.
12. FANG, K. T., YUAN, K., and BENTLER, P. (1994), "Applications of Number-Theoretic Methods to Quantizers of Elliptically Contoured Distributions," in Multivariate Analysis and Its Applications, Eds., T. W. Anderson, K. T. Fang, and I. Olkin, Hayward California: IMS Lecture Notes - Monograph Series, 24, 237-251.
13. FLURY, B. (1990), "Principal Points," Biometrika, 77, 33-41.
14. FLURY, B. (1993), "Estimation of Principal Points," Applied Statistics, 42, 139-151.
15. FLURY, B., and RIEDWYL, H. (1988), Multivariate Statistics: a Practical Approach, London: Chapman and Hall.
16. FLURY, B., and TARPEY, T. (1993), "Representing a Large Collection of Curves: A Case for Principal Points," The American Statistician, 47, 304-306.
17. GERSHO, A. (1979), "Asymptotically Optimal Block Quantization," IEEE Transactions on Information Theory, 25, 373-380.
18. GRAY, R. M., and KARNIN, E. D. (1982), "Multiple Local Optima in Vector Quantizers," IEEE Trans actions in Information Theory, 28, 256-261.
19. HARTIGAN, J. A. (1975), Clustering Algorithms, New York: Wiley.
20. HASTIE, T., and STUETZLE, W. (1989), "Principal Curves," Journal of the American Statistical Association, 84, 502-516.
21. IBRAGIMOV, I. (1956), "On the Composition of Unimodal Distributions," Theory of Probability and its Applications, 1 255-260.
22. IYENGAR, S., and SOLOMON, H. (1983), "Selecting Representative Points in Normal Populations," Recent Advantages in Statistics: Papers in Honor of Herman Chernoff on his 60th Birthday, Academic Press, 579-591.
23. KARLIN, S. (1968), Total Positivity, Volume 1, California: Stanford University Press.
24. KIEFFER, J. (1982), "Exponential Rate of Convergence for Lloyd's Method I," IEEE Transactions in Information Theory, 28, 205-210.
25. LI, L., and FLURY, B. (1995), "Uniqueness of Principal Points for Univariate Distributions," Statistics and Probability Letters, 25, 323-327.
26. LINDE, Y., BUZO, A., and GRAY, R. (1980), "An Algorithm for Vector Quantizer Design," IEEE Transactions in Communication, COM-28, 84-95.
27. MUIRHEAD, R. J. (1982), Aspects of Multivariate Statistical Theory, New York: Wiley.
28. NEWMAN, D. (1982), "The Hexagon Theorem," IEEE Transactions in Information Theory, 28, 137-138.
29. POLLARD, D. (1981), "Strong Consistency of K-means Clustering," The Annals of Statistics, 9, 135-140.
30. PÖTZELBERGER, K., and FELSENSTEIN, K. (1994), ̈An Asymptotic Result on Principal Points for Univariate Distributions,̈ Optimization, 28, 397-406.
31. 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.
32. SU, Y. (1997), ̈On the Asymptotics of Quantizers in Two Dimensions,̈ Journal of Multivariate Analysis, 61, 67-85.
33. TARPEY, T. (1992), Principal Points, Unpublished Ph.D. thesis, Indiana University: Department of Mathematics.
34. TARPEY, T. (1994), "Two Principal Points for Symmetric, Strongly Unimodal Distributions," Statistics and Probability Letters, 20, No. 1, 253-257.
35. TARPEY, T. (1995), "Principal Points and Self-Consistent Points of Symmetric Multivariate Distributions," Journal of Multivariate Analysis, 53, 39-51.
36. TARPEY, T. (1997), "Estimating Principal Points of Univariate Distributions," The Journal of Applied Statistics, 24, 483-496.
37. TARPEY, T., LI, LUNING, and FLURY, B. (1995), "Principal Points and Self-Consistent Points of Elliptical Distributions," Annals of Statistics, 23, 103-112.
38. TARPEY, T., and FLURY, B. (1996), "Self-consistency: A Fundamental Concept in Statistics," Statistical Science, 11, 229-243.
39. THISTED, R. (1988), Elements of Statistical Computing, New York: Chapman and Hall.
40. TRUSKIN, A. (1982), "Sufficient Conditions for Uniqueness of a Locally Optimal Quantizer," IEEE Transactions in Information Theory, 28, 187-198.
41. ZOPPÈ, A. (1995), "Principal Points of Univariate Continuous Distributions," Statistics and Computing, 5, 127-132.
42. ZOPPÈ, A. (1997), "On Uniqueness and Symmetry of Self-Consistent Points of Univariate Continuous Distributions," Journal of Classification, 14, 147-158.