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Principal points for an allometric extension model

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Abstract

A set of \(n\)-principal points of a \(p\)-dimensional distribution is an optimal \(n\)-point-approximation of the distribution in terms of a squared error loss. It is in general difficult to derive an explicit expression of principal points. Hence, we may have to search the whole space \(R^p\) for \(n\)-principal points. Many efforts have been devoted to establish results that specify a linear subspace in which principal points lie. However, the previous studies focused on elliptically symmetric distributions and location mixtures of spherically symmetric distributions, which may not be suitable to many practical situations. In this paper, we deal with a mixture of elliptically symmetric distributions that form an allometric extension model, which has been widely used in the context of principal component analysis. We give conditions under which principal points lie in the linear subspace spanned by the first several principal components.

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References

  • Bali JL, Boente G (2009) Principal points and elliptical distributions from the multivariate setting to the functional case. Stat Probab Lett 79:1858–1865

    Article  MATH  MathSciNet  Google Scholar 

  • Bartoletti S, Flury BD, Nel DG (1999) Allometric extension. Biometrics 55:1210–1214

    Article  MATH  Google Scholar 

  • Fang KT, Kotz S, Ng KW (1990) Symmetric multivariate and related distributions. Chapman and Hall, London

    Book  MATH  Google Scholar 

  • Flury B (1990) Principal points. Biometrika 77:33–41

    Article  MATH  MathSciNet  Google Scholar 

  • Flury B (1993) Estimation of principal points. Appl Stat 42:139–151

    Article  MATH  MathSciNet  Google Scholar 

  • Flury B (1997) A first course in multivariate statistics. Springer, Berln

    Book  MATH  Google Scholar 

  • Graf L, Luschgy H (2000) Foundations of quantization for probability distributions. Springer, Berlin

    Book  MATH  Google Scholar 

  • Gu XN, Mathew T (2001) Some characterizations of symmetric two-principal points. J Stat Plan Inference 98:29–37

    Article  MATH  MathSciNet  Google Scholar 

  • Hills M (1982) Allometry. In: Kotz S, Johnson NL (eds) Encyclopedia of statistical sciences. Wiley, New York, pp 48–54

    Google Scholar 

  • Joarder AH (2008) Some useful integrals and their applications in correlation analysis. Stat Pap 49:211–224

    Article  MATH  MathSciNet  Google Scholar 

  • Khan S (2009) Optimal tolerance regions for future regression vector and residual sum of squares of multiple regression model with multivariate spherically contoured errors. Stat Pap 50:511–525

    Article  MATH  Google Scholar 

  • Klingenberg CP, Froese R (1991) A multivariate comparison of allometric growth patterns. Syst Biol 40:410–419

    Article  Google Scholar 

  • Kurata H (2008) On principal points for location mixtures of multivariate spherically symmetric distributions. J Stat Plan Inference 138:3405–3418

    Article  MATH  MathSciNet  Google Scholar 

  • Kurata H (2010) A theorem on the covariance matrix of a generalized least squares estimator under an elliptically symmetric error. Stat Pap 51:389–395

    Article  MATH  MathSciNet  Google Scholar 

  • Kurata H, Hoshino T, Fujikoshi Y (2008) Allometric extension model for conditional distributions. J Multivar Anal 99:1985–1998

    Article  MATH  MathSciNet  Google Scholar 

  • Kurata H, Qiu D (2011) Linear subspace spanned by principal points of a mixture of spherically symmetric distributions. Commun Stat Theory Methods 40:2737–2750

    Article  MATH  MathSciNet  Google Scholar 

  • Liu S, Heyde CC, Wong WK (2011) Moment matrices in conditional heteroskedastic models under elliptical distributions with applications in AR-ARCH models. Stat Pap 52:621–632

    Article  MATH  MathSciNet  Google Scholar 

  • Matsuura S (2011) Optimal partitioning of probability distributions under general convex loss functions in selective assembly. Commun Stat Theory Methods 40:1545–1560

    Article  MATH  MathSciNet  Google Scholar 

  • Matsuura S, Kurata H (2010) A principal subspace theorem for 2-principal points of general location mixtures of spherically symmetric distributions. Stat Probab Lett 80:1863–1869

    Article  MATH  MathSciNet  Google Scholar 

  • Matsuura S, Kurata H (2011) Principal points of a multivariate mixture distribution. J Multivar Anal 102:213–224

    Article  MATH  MathSciNet  Google Scholar 

  • Matsuura S, Kurata H (2013) Definition and properties of \(m\)-dimensional \(n\)-principal points. Commun Stat Theory Methods 42:267–282

    Article  MATH  MathSciNet  Google Scholar 

  • Matsuura S, Shinozaki N (2007) Optimal binning strategies under squared error loss in selective assembly with measurement error. Commun Stat Theory Methods 36:2863–2876

    Article  MATH  MathSciNet  Google Scholar 

  • Matsuura S, Shinozaki N (2010) Optimal binning strategies under squared error loss in selective assembly with a tolerance constraint. Commun Stat Theory Methods 39:592–605

    Article  MATH  MathSciNet  Google Scholar 

  • Mease D, Nair VN (2006) Unique optimal partitions of distributions and connections to hazard rates and stochastic ordering. Stat Sin 16:1299–1312

    MATH  MathSciNet  Google Scholar 

  • Mease D, Nair VN, Sudjianto A (2004) Selective assembly in manufacturing: statistical issues and optimal binning strategies. Technometrics 46:165–175

    Article  MathSciNet  Google Scholar 

  • Micheas AC, Dey DK, Mardia KV (2006) Complex elliptical distributions with application to shape analysis. J Stat Plan Inference 136:2961–2982

    Article  MATH  MathSciNet  Google Scholar 

  • Pötzelberger K, Felsenstein K (1994) An asymptotic result on principal points for univariate distributions. Optimization 28:397–406

    Article  MATH  MathSciNet  Google Scholar 

  • Qiu D (2010) A comparative study of the \(K\)-means algorithm and the normal mixture model for clustering: bivariate homoscedastic case. J Stat Plan Inference 140:1701–1711

    Article  MATH  Google Scholar 

  • Qiu D, Tamhane AC (2007) A comparative study of the \(K\)-means algorithm and the EM Algorithm for clustering under the normal mixture model: univariate case. J Stat Plan Inference 137:3722–3740

    Article  MATH  MathSciNet  Google Scholar 

  • Rowe S (1996) An algorithm for computing principal points with respect to a loss function in the unidimensional case. Stat Comput 6:187–190

    Article  Google Scholar 

  • Ruwet C, Haesbroeck G (2011) Impact of contamination on training and test error rates in statistical clustering. Commun Stat Simul Comput 40:394–411

    Article  MATH  MathSciNet  Google Scholar 

  • Ruwet C, Haesbroeck G (2013) Classification performance resulting from a 2-means. J Stat Plan Inference 143:408–418

    Article  MATH  MathSciNet  Google Scholar 

  • Shimizu N, Mizuta M (2007) Functional clustering and functional principal points. In: Apolloni B, Howlett RJ, Jain LC (eds) Knowledge-based intelligent information and engineering systems. Lect Notes Comput Sci 4693:501–508

  • Shimizu N, Mizuta M (2008) Functional principal points and functional cluster analysis. In: Jain LC, Sato-Ilic M, Virvou M, Tsihrintzis GA, Balas VE, Abeynayake C (eds) Computational intelligence paradigms, innovative applications. Stud Comput Intell 137:149–165

  • Stampfer E, Stadlober E (2002) Methods for estimating principal points. Commun Stat Simul Comput 31:261–277

    Article  MATH  MathSciNet  Google Scholar 

  • Su Y (1997) On the asymptotics of quantizers in two dimensions. J Multivar Anal 61:67–85

    Article  MATH  Google Scholar 

  • Tarpey T (1995) Principal points and self-consistent points of symmetric multivariate distributions. J Multivar Anal 53:39–51

    Article  MATH  MathSciNet  Google Scholar 

  • Tarpey T (2000) Parallel principal axes. J Multivar Anal 75:295–313

    Article  MATH  MathSciNet  Google Scholar 

  • Tarpey T, Ivey CT (2006) Allometric extension for multivariate regression. J Data Sci 4:479–495

    Google Scholar 

  • Tarpey T, Kinateder K (2003) Clustering functional data. J Classif 20:93–114

    Article  MATH  MathSciNet  Google Scholar 

  • Tarpey T, Li L, Flury B (1995) Principal points and self-consistent points of elliptical distributions. Ann Stat 23:103–112

    Article  MATH  MathSciNet  Google Scholar 

  • Tarpey T, Petkova E (2010) Principal point classification: applications to differentiating drug and placebo responses in longitudinal studies. J Stat Plan Inference 140:539–550

    Article  MATH  MathSciNet  Google Scholar 

  • Tarpey T, Petkova E, Lu Y, Govindarajulu U (2010) Optimal partitioning for linear mixed effects models: applications to identifying placebo responders. J Am Stat Assoc 105:968–977

    Article  MathSciNet  Google Scholar 

  • Tarpey T, Petkova E, Ogden RT (2003) Profiling placebo responders by self-consistent partitions of functional data. J Am Stat Assoc 98:850–858

    Article  MathSciNet  Google Scholar 

  • Trushkin A (1982) Sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions. IEEE Trans Inf Theory 28:187–198

    Article  MATH  MathSciNet  Google Scholar 

  • Yamamoto W, Shinozaki N (2000a) On uniqueness of two principal points for univariate location mixtures. Stat Probab 46:33–42

    Article  MATH  MathSciNet  Google Scholar 

  • Yamamoto W, Shinozaki N (2000b) Two principal points for multivariate location mixtures of spherically symmetric distributions. J Jpn Stat Soc 30:53–63

    Article  MATH  MathSciNet  Google Scholar 

  • Yamashita H, Suzuki H (2012) An approximation method of binary principal points based on \(k\)-means approach. Proceedings of the 10th Asian network for quality congress, pp 491–495.

  • Zoppe A (1995) Principal points of univariate continuous distributions. Stat Comput 5:127–132

    Article  Google Scholar 

  • Zoppe A (1997) On uniqueness and symmetry of self-consistent points of univariate continuous distributions. J Classif 14:147–158

    Article  MATH  MathSciNet  Google Scholar 

  • Zou C, Tsung F (2011) A multivariate sign EWMA control chart. Technometrics 53:84–97

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous reviewers for their constructive comments that have resulted in significant improvements in this paper. Matsuura’s portion of this work was supported by JSPS KAKENHI Grant Number 23700341. Kurata’s portion of this work was supported by JSPS KAKENHI Grant Number 20243016, 21500272.

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Authors and Affiliations

  1. Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa, 223-8522, Japan

    Shun Matsuura

  2. Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8902, Japan

    Hiroshi Kurata

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  1. Shun Matsuura
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  2. Hiroshi Kurata
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Correspondence to Shun Matsuura.

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Matsuura, S., Kurata, H. Principal points for an allometric extension model. Stat Papers 55, 853–870 (2014). https://doi.org/10.1007/s00362-013-0532-z

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  • DOI: https://doi.org/10.1007/s00362-013-0532-z

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