Self-Consistency Algorithms

Abstract The k-means algorithm and the principal curve algorithm are special cases of a self-consistency algorithm. A general self-consistency algorithm is described and results are provided describing the behavior of the algorithm for theoretical distributions, in particular elliptical distributions. The results are used to contrast the behavior of the algorithms when applied to a theoretical model and when applied to finite datasets from the model. The algorithm is also used to determine principal loops for the bivariate normal distribution.

[1]  Thaddeus Tarpey.  Self-Consistent Patterns for Symmetric Multivariate Distributions , 1998 .

[2]  Bernard D. Flury,et al.  Principal Points and Self-Consistent Points of Elliptical Distributions , 1995 .

[3]  Bernard D. Flury,et al.  Self-Consistency: A Fundamental Concept in Statistics , 1996 .

[4]  T. Tarpey.  Self-Consistency and Principal Component Analysis , 1999 .

[5]  B. Efron.  The two sample problem with censored data , 1967 .

[6]  Bernhard N Flury.  Multivariate Statistics: A Practical Approach , 1988 .

[7]  Werner Stuetzle,et al.  Geometric Properties of Principal Curves in the Plane , 1996 .

[8]  John A. Hartigan,et al.  Clustering Algorithms , 1975 .

[9]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[10]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[11]  W. Stuetzle,et al.  Extremal properties of principal curves in the plane , 1996 .

[12]  H. Riedwyl,et al.  Multivariate Statistics: A Practical Approach , 1988 .

[13]  Thaddeus Tarpey,et al.  Estimating Principal Points of Univariate Distributions , 1997 .

[14]  Ernesto Salinelli.  Nonlinear principal components I. Absolutely continuous random variables with positive bounded densities , 1998 .

[15]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[16]  Yingcai Su.  On the Asymptotics of Quantizers in Two Dimensions , 1997 .

[17]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[18]  V. Yohai,et al.  Nonlinear principal components , 1985 .

[19]  Bernard D. Flury,et al.  Estimation of Principal Points , 1993 .