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Principal points of a multivariate mixture distribution

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  • Matsuura, Shun
  • Kurata, Hiroshi

Abstract

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.

Suggested Citation

  • Matsuura, Shun & Kurata, Hiroshi, 2011. "Principal points of a multivariate mixture distribution," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 213-224, February.
    • Handle: RePEc:eee:jmvana:v:102:y:2011:i:2:p:213-224
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    References listed on IDEAS

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    1. Thaddeus Tarpey, 1997. "Estimating principal points of univariate distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 24(5), pages 499-512.
    2. Tarpey, Thaddeus, 1994. "Two principal points of symmetric, strongly unimodal distributions," Statistics & Probability Letters, Elsevier, vol. 20(4), pages 253-257, July.
    3. 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.
    4. 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.
    5. Bernard D. Flury, 1993. "Estimation of Principal Points," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 42(1), pages 139-151, March.
    6. Thaddeus Tarpey, 2007. "A parametric k-means algorithm," Computational Statistics, Springer, vol. 22(1), pages 71-89, April.
    7. Tarpey T. & Petkova E. & Ogden R.T., 2003. "Profiling Placebo Responders by Self-Consistent Partitioning of Functional Data," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 850-858, January.
    8. Kano, Y., 1994. "Consistency Property of Elliptic Probability Density Functions," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 139-147, October.
    9. 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.
    10. 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.
    11. Su, Yingcai, 1997. "On the Asymptotics of Quantizers in Two Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 67-85, April.
    12. Li, Luning & Flury, Bernard, 1995. "Uniqueness of principal points for univariate distributions," Statistics & Probability Letters, Elsevier, vol. 25(4), pages 323-327, December.
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    Cited by:

    1. Tarpey, Thaddeus & Loperfido, Nicola, 2015. "Self-consistency and a generalized principal subspace theorem," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 27-37.
    2. Shun Matsuura, 2014. "Effectiveness of a random compound noise strategy for robust parameter design," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(9), pages 1903-1918, September.
    3. Shun Matsuura & Thaddeus Tarpey, 2020. "Optimal principal points estimators of multivariate distributions of location-scale and location-scale-rotation families," Statistical Papers, Springer, vol. 61(4), pages 1629-1643, August.
    4. Santanu Chakraborty & Mrinal Kanti Roychowdhury & Josef Sifuentes, 2021. "High Precision Numerical Computation of Principal Points for Univariate Distributions," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 558-584, November.
    5. Long-Hao Xu & Kai-Tai Fang & Ping He, 2022. "Properties and generation of representative points of the exponential distribution," Statistical Papers, Springer, vol. 63(1), pages 197-223, February.
    6. Loperfido, Nicola, 2014. "A note on the fourth cumulant of a finite mixture distribution," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 386-394.
    7. Shun Matsuura & Hiroshi Kurata, 2014. "Principal points for an allometric extension model," Statistical Papers, Springer, vol. 55(3), pages 853-870, August.

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