Skip to main content
SpringerLink
Log in

Recent trends and developments in computational multivariate analysis

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

Many traditional multivariate techniques such as ordination, clustering, classification and discriminant analysis are now routinely used in most fields of application. However, the past decade has seen considerable new developments, particularly in computational multivariate methodology. This article traces some of these developments and highlights those trends that may prove most fruitful for future practical implementation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Altman, N. S. (1992) An introduction to kernel and nearest-neighbor non-parametric regression. American Statistician 46, 175–85.

    Google Scholar 

  • Arnold, S. F. (1993) Gibbs sampling, In C. R. Rao (ed) Com-putational Statistics. Handbook of Statistics 9, pp. 599–625, North Holland, Amsterdam.

  • Babu, G. J. and Rao C. R. (1993) Bootstrap methodology. In C. R. Rao (ed.) Computational Statistics. Handbook of Statis-tics 9, pp. 627–59, North Holland, Amsterdam.

  • Bailey, T. C., Sapatinas, T., Powell, K. and Krzanowski, W. J. (1966) Use of wavelets in the detection of underwater sound signals. Computing Science and Statistics. Proceedings of the 28th Symposium on the Interface, Sydney, Australia.

  • Banfield, J. D. and Raftery, A. E. (1992) Ice floe identification in satellite images using mathematical morphology and clus-tering about principal curves. Journal of the American Sta-tistical Association 87, 7–15.

    Google Scholar 

  • Barnett, V. (1976) The ordering of multivariate data (with dis-cussion). Journal of the Royal Statistical Society, Series A 139, 318–55.

    Google Scholar 

  • Bartlett, M. S. (1951) An inverse matrix adjustment arising in discriminant analysis. Annals of Mathematical Statistics 22, 107–11.

    Google Scholar 

  • Becker, R. A., Chambers, J. and Wilks, A. (1988) The New S Language. Wadsworth, New York.

  • Bellman, R. E. (1961) Adaptive Control Processes. Princeton University Press, Princeton, New Jersey.

    Google Scholar 

  • Bhattacharyya, A. (1943) On a measure of divergence between two statistical populations defined by their probability dis-tributions. Bulletin of the Calcutta Mathematical Society 35, 99–109.

    Google Scholar 

  • Bollen, K. A. (1989) Structural Equations with Latent Variables. Wiley, New York.

    Google Scholar 

  • Boswell, M. T., Gore, S. D., Patil, G. P. and Taille, C. (1993) The art of computer generation of random variables. In C. R. Rao (ed.) Computational Statistics. Handbook of Statistics 9, pp. 661–721, North Holland, Amsterdam.

  • Breiman, L., Friedman, J. H., Olshen, R. A. and Stone, C. J. (1984) Classification and Regression Trees. Wadsworth, Bel-mont, California.

  • Campbell, N. A. (1985) Updating formulae for allocation of in-dividuals. Applied Statistics 34, 235–6.

    Google Scholar 

  • Casella, G. and George E. I. (1992) Explaining the Gibbs sampler. American Statistician 46, 167–74.

    Google Scholar 

  • Cox, D. R. and Wermuth, N. (1996) Multivariate Dependencies. Chapman and Hall, London.

    Google Scholar 

  • Cressie, N. A. (1991) Statistics for Spatial Data. Wiley, New York.

    Google Scholar 

  • Crowder, M. J. and Hand, D. J. (1990) Analysis of Repeated Measures. Chapman and Hall, London.

    Google Scholar 

  • Davison, A. C., Hinkley, D. V. and Schechtman E. (1986) Effi-cient bootstrap simulation. Biometrika 73, 555–66.

    Google Scholar 

  • Denham, M. C. and Brown, P. J. (1993) Calibration with many variables. Applied Statistics 42, 515–28.

    Google Scholar 

  • Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995) Wavelet shrinkage: asymptotia (with discussion). Journal of the Royal Statistical Society, Series B 57, 301–69.

    Google Scholar 

  • Efron, B. (1992) Jackknife-after-bootstrap standard errors and influence functions (with discussion). Journal of the Royal Statistical Society, Series B 54, 83–127.

    Google Scholar 

  • Everitt, B. S. (1993) Cluster Analysis (3rd ed.) Arnold, London.

    Google Scholar 

  • Fang, K-T. and Zhang, Y-T. (1990) Generalized Multivariate Analysis. Science Press, Beijing and Springer-Verlag, Berlin.

    Google Scholar 

  • Fix, E. and Hodges, J. L. (1951) Discriminatory analysis-Non-parametric Discrimination: Consistency Properties. Report No. 4, U. S. Air Force School of Aviation Medicine, Ran-dolph Field, Texas.

  • Flury, B. (1995) Developments in principal component analysis: a review. In W. J. Krzanowski (ed.) Descriptive Multivariate Analysis. pp. 14–33 Clarendon Press, Oxford.

    Google Scholar 

  • Friedman, J. H. (1991) Multivariate adaptive regression splines (with discussion). Annals of Statistics 19, 1–141.

    Google Scholar 

  • Friedman, J. H. and Stuetzle, W. (1981) Projection pursuit re-gression. Journal of the American Statistical Association 76, 817–23.

    Google Scholar 

  • Friedman, J. H., Stuetzle, W. and Schroeder, A. (1984) Projection pursuit density estimation. Journal of the American Statistical Association 79, 599–608.

    Google Scholar 

  • Gabriel, K. R. (1962) Antedependence analysis of a set of ordered variables. Annals of Mathematical Statistics 33, 201–12.

    Google Scholar 

  • Gabriel, K. R. (1971) The biplot-graphic display of matrices with application to principal component analysis. Biometrika 58, 453–67.

    Google Scholar 

  • Gelfand, A. E. and Smith, A. F. M. (1990) Sampling-based ap-proaches to calculating marginal densities. Journal of the American Statistical Association 85, 398–409.

    Google Scholar 

  • Geman, S. and Geman, D. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6, 721–41.

    Google Scholar 

  • Gifi, A. (1990) Nonlinear Multivariate Analysis. Wiley, New York.

    Google Scholar 

  • Gnanadesikan, R. (1977) Methods for Statistical Data Analysis of Multivariate Observations. Wiley, New York.

    Google Scholar 

  • Goodall, C. R. (1991) Procrustes methods in the statistical anal-ysis of shape (with discussion). Journal of the Royal Statis-tical Society, Series. B 53, 285–339.

    Google Scholar 

  • Gower, J. C. (1966) Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53, 325–38.

    Google Scholar 

  • Gower, J. C. and Hand, D. J. (1996) Biplots. Chapman and Hall, London.

    Google Scholar 

  • Gower, J. C. and Harding, S. A. (1988) Non-linear biplots. Biometrika 73, 445–55.

    Google Scholar 

  • Green, P. J. (1980) Peeling bivariate data. In V. Barnett (ed.) Interpreting Multivariate Data. pp. 3–19 John Wiley and Sons, Chichester.

    Google Scholar 

  • Green, P. J. and Silverman, B. W. (1993) Nonparametric Re-gression and Generalized Linear Models: A Roughness Penalty Approach. Chapman and Hall, London.

    Google Scholar 

  • Hä rdle, W. (1991) Smoothing Techniques. Springer-Verlag, Berlin.

    Google Scholar 

  • Hastie, T. and Stuetzle, W. (1989) Principal curves. Journal of the American Statistical Association 84, 502–16.

    Google Scholar 

  • Hastie, T. J. and Tibshirani, R. J. (1986) Generalized Additive Models. Statistical Science, 1, 297–318.

    Google Scholar 

  • Hastie, T. J. and Tibshirani, R. J. (1990) Generalized Additive Models. Chapman and Hall, London.

    Google Scholar 

  • Hastings, W. K. (1970) Monte Carlo simulation methods using Markov Chains and their applications. Biometrika 57, 97–109.

    Google Scholar 

  • Hertz, J., Krogh, A. and Palmer, R. G. (1991) Introduction to the Theory of Neural Computation. Addison-Wesley, Reading, Massachusetts.

    Google Scholar 

  • Jones, M. C. and Rice J. A. (1992) Displaying the important features of a large collection of similar curves. American Statistician 46, 140–5.

    Google Scholar 

  • Jones, M. C. and Sibson, R. (1987) What is projection pursuit? (with discussion). Journal of the Royal Statistical Society, Series A 150, 1–36.

    Google Scholar 

  • Jö reskog, K. G. (1967) Some contributions to maximum likeli-hood factor analysis. Psychometrika 32, 443–82.

    Google Scholar 

  • Jö reskog, K. G. (1969) A general approach to confirmatory maximum likelihood factor analysis. Psychometrika 34, 183–220.

    Google Scholar 

  • Kenward, M. G. (1987) A method for comparing profiles of re-peated measurements. Applied Statistics 36, 296–308.

    Google Scholar 

  • Krzanowski, W. J. (1988) Principles of Multivariate Analysis; a User's Perspective. Clarendon Press, Oxford.

    Google Scholar 

  • Krzanowski, W. J. (1993) Antedependence modelling in discrim-inant analysis of high-dimensional spectroscopic data. In C. M. Cuadras and C. R. Rao (eds) Multivariate Analysis: Future Directions 2. pp. 87–95, Elsevier Science Publishers, Amsterdam.

    Google Scholar 

  • Krzanowski, W. J. (1995) Selection of variables, and assessment of their performance, in mixed-variable discriminant analy-sis. Computational Statistics and Data Analysis 19, 419–31.

    Google Scholar 

  • Krzanowski, W. J., Jonathan, P., McCarthy, W. V. and Thomas, M. R. (1995) Discriminant analysis with singular covariance matrices: methods and applications to spectroscopic data. Applied Statistics 44, 101–15.

    Google Scholar 

  • Lauritzen, S. L. (1996) Graphical Models. Clarendon Press, Ox-ford.

    Google Scholar 

  • Matusita, K. (1956) Decision rule, based on distance, for the classification problem. Annals of the Institute of Statistical Mathematics 8, 67–77

    Google Scholar 

  • McCullagh, P. and Nelder, J. A. (1990) Generalized Linear Models, 2nd ed. Chapman and Hall, London.

    Google Scholar 

  • McLachlan, G. J. (1992) Discriminant Analysis and Statistical Pattern Recognition. Wiley, New York.

    Google Scholar 

  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953) Equations of state calculations by fast computing machines. Journal of Chemical Physics 21, 1087–92.

    Google Scholar 

  • Meulman, J. J. (1986) A Distance Approach to Nonlinear Multi-variate Analysis. DSWO Press, Leiden, the Netherlands.

    Google Scholar 

  • Meulman, J. J. (1992) The integration of multidimensional scaling and multivariate analysis with optimal transformations. Psychometrika 54, 539–65.

    Google Scholar 

  • Peddada, S. D. (1993) Jackknife variance estimation and bias re-duction. In C. R. Rao (ed.) Computational Statistics. pp. 723–44, Handbook of Statistics 9, North Holland, Amsterdam.

  • Quenouille, M. (1956) Notes on bias in estimation. Biometrika 43, 353–60.

    Google Scholar 

  • Ramsay, J. O. and Dalzell, C. J. (1991) Some tools for functional data analysis (with discussion). Journal of the Royal Statis-tical Society, Series B 53, 539–72.

    Google Scholar 

  • Rice, J. A. and Silverman, B. W. (1991) Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society, Series B 53, 233–43.

    Google Scholar 

  • Ripley, B. D. (1993) Statistical aspects of neural networks. In O. E. Barndorff-Nielsen, D. R. Cox, J. L. Jensen and W. S. Kendall (eds) Chaos and Networks-Statistical and Probabilistic Aspects. pp. 40–123, Chapman and Hall, London.

    Google Scholar 

  • Ripley, B. D. (1994) Neutral networks and related methods for classification (with discussion) Journal of the Royal Statistical Society, Series B B 409–56.

    Google Scholar 

  • SAS Institute, Inc, (1990) SAS User's Guide. Cary, North Caro-lina.

    Google Scholar 

  • Scott, D. W. (1992) Multivariate Density Estimation. Wiley, New York.

    Google Scholar 

  • Silverman, B. W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.

    Google Scholar 

  • Smith, A. F. M. (1996) Bayesian curves and CARTs. Address given at the Sydney International Statistical Congress, Sydney, Australia.

    Google Scholar 

  • Smith, A. F. M. and Roberts G. O. (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). Journal of the Royal Sta-tistical Society, Series B 55, 3–23.

    Google Scholar 

  • Smith, D. J., Bailey, T. C. and Munford, A. G. (1993) Robust classification of high-dimensional data using artificial neural networks. Statistics and Computing 3, 71–81.

    Google Scholar 

  • Solla, S.A., Levin, E. and Fleisher, M. (1988) Accelerated learning in layered neural networks. Complex Systems 2, 625–39.

    Google Scholar 

  • Stone, M. (1974) Cross-validatory choice and assessment of sta-tistical predictions (with discussion). Journal of the Royal Statistical Society, Series B 36, 111–48.

    Google Scholar 

  • Sundberg, R. and Brown, P. J. (1989) Multivariate calibration with more variables than observations. Technometrics 31, 365–71.

    Google Scholar 

  • Tanner, M. A. and Wong, W. (1987) The calculation of posterior distributions by data augmentation (with discussion). Jour-nal of the American Statistical Association 82, 528–50.

    Google Scholar 

  • Titterington, D. M. (1979) Estimation of correlation coefficients by ellipsoidal trimming. Applied Statistics 27, 227–34.

    Google Scholar 

  • van Ooyen, A. and Nienhuis, B. (1992) Improving the conver-gence of the back-propagation algorithm. Neural Networks 5, 465–71.

    Google Scholar 

  • Valchonikolis, I. G. and Vasdekis, V. G. S. (1994) On a class of change-point models in covariance structures for growth curves and repeated measurements. Communications in Sta-tistics, A-Theory and Methods 23, 1087–1102.

    Google Scholar 

  • Webb, A. R. (1996) An approach to non-linear principal com-ponents analysis using radially symmetric kernel basis func-tions. Statistics and Computing 6, 159–68.

    Google Scholar 

  • Wegman, E. J. and Carr, D. B. (1993) Statistical graphics and visualization. In C. R. Rao (ed.) Computational Statistics. Handbook of Statistics 9 pp. 857–958, North Holland, Am-sterdam.

  • Welch, B. L. (1939) Note on discriminant functions. Biometrika 31, 218–20.

    Google Scholar 

  • Whittaker, J. (1990) Graphical Models in Applied Multivariate Statistics. John Wiley and Son, Chichester.

    Google Scholar 

  • Young, F. W., Faldowski, R. A. and McFarlane M. M. (1993) Multivariate statistical visualization. In C. R. Rao (ed.) Computational Statistics. Handbook of Statistics 9 pp. 959–98, North Holland, Amsterdam.

  • Young, G. A. (1994) Bootstrap: more than a stab in the dark? (with discussion) Statistical Science 9, 382–415.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Statistics and Operational Research, University of Exeter, Laver Building, North Park Road, Exeter, EX4 4QE, UK

    W. J. Krzanowski

Authors
  1. W. J. Krzanowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krzanowski, W.J. Recent trends and developments in computational multivariate analysis. Statistics and Computing 7, 87–99 (1997). https://doi.org/10.1023/A:1018565514339

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018565514339

Search

Navigation