AN.ROGERS@SCIENCE.UTAH.EDU (Alan Rogers) (05/19/89)
This is a reply to Una Smith's question about methods for measuring the extent to which spatial patterns are clumped, or over-dispersed. I have always thought that methods that give a single number measuring clumping or over-dispersion are potentially misleading since a spatial pattern may be clumped at one scale of distance, and over-dispersed at another. For example, individuals may avoid being closer than 3 feet from each other, and also avoid being farther than 300 feet, generating overdispersion at small scales and clumping at larger scales. The only method I know that measures this sort of thing is two dimensional spectral analysis. If the data are randomly distributed, the spectrum will be flat---equal to the mean everywhere. Clumping elevates the spectrum, while over-dispersion depresses it, and the spectrum will tell you how these effects are distributed across scales of distance. Some relevant references are: Bartlett, M. S. (1963) The spectral analysis of point processes. J. R. Stat. Soc. B, 25:264-96. Bartlett, M. S. (1964) The spectral analysis of two-dimensional point processes. Biometrika, 51:299-311. Cox, D. R. and P. A. W. Lewis (1972) Multivariate point processes. Proc of the 6th Berkeley Symposium on Mathematical Statistics & Probability, Vol 3, pp. 401-- Rogers, A. R. (1982) Data collection and information loss in the study of spatial pattern. World Archaeology, 14 (2): 249-258. Unfortunately, most of these are pretty heavy going. There may well be a better introduction to this subject published somewhere---I don't try to keep up with this field. I have a C program that does 2D fourier transforms of point patterns by brute force (not FFT), if you are interested. I doubt that it is much slower than the FFT because, with point processes, the data matrix is very sparse, and its zeroes speed the brute force algorithm more than the FFT. -------