Principal points of a multivariate mixture distribution
A set of n-principal points of a distribution is defined as a set of n points that optimally represent the distribution in terms of mean squared distance. It provides an optimal n-point-approximation of the distribution. However, it is in general difficult to find a set of principal points of a multivariate distribution. Tarpey et al. [T. Tarpey, L. Li, B. Flury, Principal points and self-consistent points of elliptical distributions, Ann. Statist. 23 (1995) 103-112] established a theorem which states that any set of n-principal points of an elliptically symmetric distribution is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem, called a "principal subspace theorem", is a strong tool for the calculation of principal points. In practice, we often come across distributions consisting of several subgroups. Hence it is of interest to know whether the principal subspace theorem remains valid even under such complex distributions. In this paper, we define a multivariate location mixture model. A theorem is established that clarifies a linear subspace in which n-principal points exist.
If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
Volume (Year): 102 (2011)
Issue (Month): 2 (February)
|Contact details of provider:|| Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description|
|Order Information:|| Postal: http://www.elsevier.com/wps/find/supportfaq.cws_home/regional|
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Su, Yingcai, 1997. "On the Asymptotics of Quantizers in Two Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 67-85, April.
- Tarpey, Thaddeus, 1994. "Two principal points of symmetric, strongly unimodal distributions," Statistics & Probability Letters, Elsevier, vol. 20(4), pages 253-257, July.
- Li, Luning & Flury, Bernard, 1995. "Uniqueness of principal points for univariate distributions," Statistics & Probability Letters, Elsevier, vol. 25(4), pages 323-327, December.
- Thaddeus Tarpey, 2007. "A parametric k-means algorithm," Computational Statistics, Springer, vol. 22(1), pages 71-89, April.
- Bali, Juan Lucas & Boente, Graciela, 2009. "Principal points and elliptical distributions from the multivariate setting to the functional case," Statistics & Probability Letters, Elsevier, vol. 79(17), pages 1858-1865, September.
- Kano, Y., 1994. "Consistency Property of Elliptic Probability Density Functions," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 139-147, October.
- Tarpey, Thaddeus, 2007. "Linear Transformations and the k-Means Clustering Algorithm: Applications to Clustering Curves," The American Statistician, American Statistical Association, vol. 61, pages 34-40, February.
- Thaddeus Tarpey, 1997. "Estimating principal points of univariate distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 24(5), pages 499-512.
- Tarpey, T., 1995. "Principal Points and Self-Consistent Points of Symmetrical Multivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 53(1), pages 39-51, April.
- Yamamoto, Wataru & Shinozaki, Nobuo, 2000. "On uniqueness of two principal points for univariate location mixtures," Statistics & Probability Letters, Elsevier, vol. 46(1), pages 33-42, January.
When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:102:y:2011:i:2:p:213-224. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Shamier, Wendy)
If references are entirely missing, you can add them using this form.