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On Using Principal Components before Separating a Mixture of Two Multivariate Normal Distributions

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  • Wei‐Chien Chang

Abstract

In applying principal components for reducing the dimension of the data before clustering, it has ordinarily been the practice to use components with the largest eigenvalues. We prove, by means of a mixture of two multivariate normal distributions, that this practice is not justified in general. A relationship between the distance of the two sub populations and any subset of principal components is derived, showing that the components with the larger eigenvalues do not necessarily contain more information (distance). This result is further demonstrated through hypothetical as well as real situations which use actual data. The effect of scaling the variables on the distribution of the information to different components is investigated. An application to a mixture of two normal distributions is illustrated by utilizing a set of generated data in which the information is concentrated in the components with the largest and the smallest eigenvalues.

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  • Wei‐Chien Chang, 1983. "On Using Principal Components before Separating a Mixture of Two Multivariate Normal Distributions," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 32(3), pages 267-275, November.
    • Handle: RePEc:bla:jorssc:v:32:y:1983:i:3:p:267-275
    DOI: 10.2307/2347949
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    4. Ahlquist, John S. & Breunig, Christian, 2009. "Country clustering in comparative political economy," MPIfG Discussion Paper 09/5, Max Planck Institute for the Study of Societies.
    5. McLachlan, G. J. & Peel, D. & Bean, R. W., 2003. "Modelling high-dimensional data by mixtures of factor analyzers," Computational Statistics & Data Analysis, Elsevier, vol. 41(3-4), pages 379-388, January.
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    8. Robert Kapłon, 2006. "A retrospective review of categorical data analysis – theory and marketing practice," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 16(1), pages 55-72.
    9. Andrews, Jeffrey L., 2018. "Addressing overfitting and underfitting in Gaussian model-based clustering," Computational Statistics & Data Analysis, Elsevier, vol. 127(C), pages 160-171.
    10. Bouveyron, Charles & Brunet, Camille, 2012. "Theoretical and practical considerations on the convergence properties of the Fisher-EM algorithm," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 29-41.
    11. Douglas Steinley & Lawrence Hubert, 2008. "Order-Constrained Solutions in K-Means Clustering: Even Better Than Being Globally Optimal," Psychometrika, Springer;The Psychometric Society, vol. 73(4), pages 647-664, December.
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    15. Zhiliang Ma & Adam Cardinal-Stakenas & Youngser Park & Michael Trosset & Carey Priebe, 2010. "Dimensionality Reduction on the Cartesian Product of Embeddings of Multiple Dissimilarity Matrices," Journal of Classification, Springer;The Classification Society, vol. 27(3), pages 307-321, November.
    16. Heungsun Hwang & Hec Montréal & William Dillon & Yoshio Takane, 2006. "An Extension of Multiple Correspondence Analysis for Identifying Heterogeneous Subgroups of Respondents," Psychometrika, Springer;The Psychometric Society, vol. 71(1), pages 161-171, March.
    17. Laura Anderlucci & Francesca Fortunato & Angela Montanari, 2022. "High-Dimensional Clustering via Random Projections," Journal of Classification, Springer;The Classification Society, vol. 39(1), pages 191-216, March.
    18. Cappozzo, Andrea & Greselin, Francesca & Murphy, Thomas Brendan, 2021. "Robust variable selection for model-based learning in presence of adulteration," Computational Statistics & Data Analysis, Elsevier, vol. 158(C).
    19. A. Penttinen & W. Krzanowski & J. Kettenring & F. Rohlf & William Day & B. Weir & John Kececioglu & N. Ohsumi & Peter Willett, 1993. "Book reviews," Journal of Classification, Springer;The Classification Society, vol. 10(1), pages 125-156, January.
    20. Lazhar Labiod & Mohamed Nadif, 2021. "Efficient regularized spectral data embedding," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 15(1), pages 99-119, March.
    21. Douglas Steinley, 2009. "F. Murtagh (2005). Correspondence analysis and data coding with Java and R. 230 pp., US$76.00. ISBN 1584885289," Psychometrika, Springer;The Psychometric Society, vol. 74(1), pages 181-183, March.
    22. Dirk Depril & Iven Mechelen & Tom Wilderjans, 2012. "Lowdimensional Additive Overlapping Clustering," Journal of Classification, Springer;The Classification Society, vol. 29(3), pages 297-320, October.
    23. Yoshikazu Terada, 2014. "Strong Consistency of Reduced K-means Clustering," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 913-931, December.
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    25. Alessandro Casa & Giovanna Menardi, 2022. "Nonparametric semi-supervised classification with application to signal detection in high energy physics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(3), pages 531-550, September.

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