Advertisement

Statistics and Computing

, Volume 2, Issue 3, pp 153–160 | Cite as

Measuring multimodality

  • G. P. Nason
  • Robin Sibson
Papers

Abstract

Existing sample statistics do little to address the question of multimodality, a question which is interesting in itself and which also arises in exploratory multivariate data analysis using projection pursuit. We propose a new index more strongly geared to the specific task of measuring multimodality than other sample statistics known to us, we show how to compute it, explore its properties, and consider its generalisation to the multivariate case. The behaviour of the index is illustrated by some simple numerical examples.

Keywords

Index of multimodality projection pursuit density estimation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Friedman, J. H. (1987) Exploratory projection pursuit.Journal of the American Statistical Association,82, 249–266.Google Scholar
  2. Friedman, J. H. and Tukey, J. W. (1974). A projection pursuit algorithm for exploratory data analysis.IEEE Transactions on Computing,23, 881–889.Google Scholar
  3. Good, I. J. and Gaskins, R. A. (1980) Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data (with discussion).Journal of the American Statistical Association,75, 42–73.Google Scholar
  4. Hartigan, J. A. (1987) Estimation of a convex density contour in two dimensions.Journal of the American Statistical Association,82, 267–270.Google Scholar
  5. Huber, P. J. (1985) Projection pursuit (with discussion).Annals of Statistics 13, 435–525.Google Scholar
  6. Jones, M. C. (1989) Discretized and interpolated kernel density extimates.Journal of the American Statistical Association,84, 733–741.Google Scholar
  7. Jones, M. C. and Lotwick, H. W. (1983) On the errors involved in computing the empirical characteristic function.Journal of Statistical Computation and Simulation,17, 133–149.Google Scholar
  8. Jones, M. C. and Lotwick, H. W. (1984) Remark ASR50: A remark on algorithm AS176, kernel density estimation using the fast Fourier transform.Applied Statistics,33, 120–122.Google Scholar
  9. Jones, M. C. and Sibson, R. (1987) What is projection pursuit? (with discussion).Journal of the Royal Statistical Society,150, 1–36.Google Scholar
  10. Kingman, J. F. C. and Taylor, S. J. (1966)Introduction to Measure and Probability. Cambridge University Press, Cambridge.Google Scholar
  11. Marron, J. S. and Wand, M. P. (1992) Exact men integrated squared error.Annals of Statistics,20, (in press).Google Scholar
  12. Monro, D. M. (1975) Algorithm AS83: Complex discrete fast Fourier transform.Applied Statistics,24, 153–160.Google Scholar
  13. Monro, D. M. (1976) Algorithm AS97: Real discrete fast Fourier transform.Applied Statistics,25, 166–172.Google Scholar
  14. Müller, D. W. and Sawitzki, G. (1987) Using excess mass estimates to investigate the modality of a distribution. Preprint 398, January 1987, Universität Heidelberg, Sonderforschungsbereich 123 Stochastiche Mathematische Modelle.Google Scholar
  15. Müller, D. W. and Sawitzki, G. (1991) Excess mass estimates and tests for multimodality.Journal of the American Statistical Association,86, 738–746.Google Scholar
  16. Silverman, B. W. (1981) Using kernel density estimates to investigate multimodality.Journal of the Royal Statistical Society, B,43, 97–99.Google Scholar
  17. Silverman, B. W. (1982) Algorithm AS176: Kernel density estimation using the fast Fourier transform.Applied Statistics,31, 93–97.Google Scholar

Copyright information

© Chapman & Hall 1992

Authors and Affiliations

  • G. P. Nason
    • 1
  • Robin Sibson
    • 1
  1. 1.School of Mathematical SciencesUniversity of BathBathUK

Personalised recommendations