Abstract
There is often more structure in the way two random variables are associated than a single scalar dependence measure, such as correlation, can reflect. Local dependence functions such as that of Holland and Wang (1987) are, therefore, useful. However, it can be argued that estimated local dependence functions convey information that is too detailed to be easily interpretable. We seek to remedy this difficulty, and hence make local dependence a more readily interpretable practical tool, by introducing dependence maps. Via local permutation testing, dependence maps simplify the estimated local dependence structure between two variables by identifying regions of (significant) positive, (not significant) zero and (significant) negative local dependence. When viewed in conjunction with an estimate of the joint density, a comprehensive picture of the joint behaviour of the variables is provided. A little theory, many implementational details and several examples are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Basak I., Balch W.R., and Basak P. 1992. Skewness: Asymptotic critical values for a test related to Pearson's measure. Journal of Applied Statistics 19: 479–487.
Bowman A.W. and Azzalini A. 1997. Applied Smoothing Techniques for Data Analysis; the Kernel Approach with S-Plus Illustrations. Oxford University Press, Oxford.
Cook R.D. and Weisberg S. 1999. Applied Statistics Including Computing and Graphics. Wiley, New York.
Hawkins D.M. 1994. Fitting monotonic polynomials to data. Computational Statistics 9: 233–247.
Hjort N.L. and Jones M.C. 1996. Locally parametric nonparametric density estimation. Annals of Statistics 24: 1619–1647.
Holland P.W. and Wang Y.J. 1987. Dependence function for continuous bivariate densities. Communications in Statistics-Theory and Methods 16: 863–876.
Joe H. 1997. Multivariate Models and Dependence Concepts. Chapman and Hall, London.
Jones M.C. 1996. The local dependence function. Biometrika 83: 899–904.
Jones M.C. 1998. Constant local dependence. Journal of Multivariate Analysis 64: 148–155.
Jones M.C. 2000. Rough-and-ready assessment of the degree and importance of smoothing in functional estimation. Statistica Neerlandica 54: 37–46.
Jones M.C. 2002. Marginal replacement in multivariate densities, with application to skewing spherically symmetric distributions. Journal of Multivariate Analysis 81: 85–99.
Loader C.R. 1996. Local likelihood density estimation. Annals of Statistics 24: 1602–1618.
Manly B.F.J. 1991. Randomization and Monte Carlo Methods in Biology. Chapman and Hall, London.
MathWorks Inc. 1996. MATLAB; the Language of Technical Computing. MathWorks, Natick, MA.
Molenberghs G. and Lesaffre E. 1997. Non-linear integral equations to approximate bivariate densities with given marginals and dependence function. Statistica Sinica 7: 713–738.
Scott D.W. 1992. Multivariate Density Estimation; Theory, Practice, and Visualization. Wiley, New York.
Simonoff J.S. 1996. Smoothing Methods in Statistics. Springer, New York.
Wand M.P. and Jones M.C. 1995. Kernel Smoothing. Chapman and Hall, London.
Wang Y.J. 1993. Construction of continuous bivariate density functions. Statistica Sinica 3: 173–187.
Wang Y.J. 1997. Multivariate normal integrals and contingency tables with ordered categories. Psychometrika 62: 267–284.
Whittaker, J. 1990. Graphical Models in Applied Multivariate Statistics. Wiley, Chichester.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jones, M.C., Koch, I. Dependence maps: Local dependence in practice. Statistics and Computing 13, 241–255 (2003). https://doi.org/10.1023/A:1024270700807
Issue Date:
DOI: https://doi.org/10.1023/A:1024270700807