Advertisement

Metric Considerations in Clustering: Implications for Algorithms

  • Stanley L. Sclove
Part of the Theory and Decision Library book series (TDLB, volume 8)

Abstract

Given measurements on p variables for each of n individuals, aspects of the problem of clustering the individuals are considered. Special attention is given to models based upon mixtures of distributions, esp. multivariate normal distributions. The relationship between the orientation(s) of the clusters and the nature of the within-cluster covariance matrices is reviewed, as is the inadequacy of transformation to principal components based on the overall (total) covariance matrix of the whole (mixed) sample. The nature of certain iterative algorithms is discussed; variations which result from allowing different covariance matrices within clusters are studied.

Key words and phrases

Cluster analysis Mahalanobis distance mixture model isodata k-means 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akaike, H. (1983). ‘Statistical Inference and Measurement of Entropy.’ In G.E.P. Box, T. Leonard, and C.-F. Wu (eds.), Scientific Inference, Data Analysis, and Robustness, 165–189. New York: Academic Press.Google Scholar
  2. Akaike, H.(1985). ‘Prediction and Entropy.’ In A.C. Atkinson and S.E. Fienberg (eds.), A Celebration of Statistics: the ISI Centenary Volume, 1–24. New York: Springer-Verlag.Google Scholar
  3. Anderson, E. (1935). ‘The Irises of the Gaspe Peninsula,’ Bulletin of the American Iris Society 59, 2–5.Google Scholar
  4. Anderson, T.W.(1984). An Introduct ion to Multivariate Statistical Analysis, 2nd ed. New York: John Wiley and Sons.Google Scholar
  5. Ball, G.H., and Hall, D.J.(1967). ‘A Clustering Technique for Summarizing Multivariate Data,’ Behavioral Science 12, 153–155.CrossRefGoogle Scholar
  6. Bryant, P. and Williamson, J.A. (1978). ‘Asymptotic Behavior of Classification Maximum Likelihood Estimates,’ Biometrika 65, 273–281.zbMATHCrossRefGoogle Scholar
  7. Chernoff, H. (1972). ‘Metric Considerations in Cluster Analysis,’ Proc. 6th Berkeley Symposium on Mathematical Statistics and Probability II, 621–630. Berkeley: University of California Press.Google Scholar
  8. Dixon, W.J., and Massey, F.J. (1969). Introduction to Statistical Analysis, 3rd ed. New York: McGraw-Hill.Google Scholar
  9. Fisher, R.A. (1936). ‘The Use of Multiple Measurements in Taxonomic Problems,’ Annals of Eugenics 7, 179–188.CrossRefGoogle Scholar
  10. Johnson, R.A., and Wichern, D.W. (1982). Applied Multivariate Statistical Analysis. New York: Prentice Hall.zbMATHGoogle Scholar
  11. Kashyap, R.L. (1982). ‘Optimal Choice of AR and MA Parts in Autoregressive Moving Average Models, IEEE Transactions on Pattern Analys is and Machine Intelligence 4, 99–104.zbMATHCrossRefGoogle Scholar
  12. MacQueen, J. (1966). ‘Some Methods for Classification and Analysis of Multivariate Observations.’ In Proc. 5th Berkeley Symposium on Mathematical Statistics and Probability I, 281–297. Berkeley: University of California Press.Google Scholar
  13. McLachlan, G.J. (1982). ‘The Classification and Mixture Maximum Likelihood Approaches to Cluster Analysis.’ In P.R. Krishnaiah and L.N. Kanal (eds.), Handbook of Statistics 2 (Classification, Pattern Recognition and Reduction of Dimensionality), 199–208. New York: North Holland.Google Scholar
  14. Marriott, F.H.C. (1975). ‘Separating Mixtures of Normal Distributions,’ Biometrics 31, 767–769.zbMATHCrossRefGoogle Scholar
  15. Sclove, S.L.(1977). ‘Population Mixture Models and Clustering Algorithms,’ Communications in Statistics (A) 6, 417–434.MathSciNetCrossRefGoogle Scholar
  16. Solomon, H. (1977). ‘Data Dependent Clustering Techniques,’ In J. Van Ryzin (ed.), Classification and Clustering, 155–174. New York: Academic Press.Google Scholar
  17. Van Ryzin, J., ed.(1977). Classification and Clustering. New York: Academic Press.Google Scholar
  18. Wolfe, J.H. (1970). ‘Pattern Clustering by Multivariate Mixture Analysis,’ Multivariate Behavi oral Research 5, 329–350.CrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1987

Authors and Affiliations

  • Stanley L. Sclove
    • 1
  1. 1.Department of Information and Decision Sciences College of Business Administration m/c 294University of Illinois at ChicagoChicagoUSA

Personalised recommendations