Distance Methods in the Analysis of Spatial PatternAuthor(s): Toby LewisSource: Advances in Applied Probability, Vol. 10, Supplement: Proceedings of the Conferenceon Spatial Patterns and Processes (Mar., 1978), pp. 131-132Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1427015 .

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Suppl. Adv. Appl. Prob. 10, 131-132 (1978) Printed in N. Ireland

? Applied Probability Trust 1978

DISTANCE METHODS IN THE ANALYSIS OF SPATIAL PATIERN

TOBY LEWIS, CSIRO Division of Mathematics and Statistics, Canberra

Spatial patterns are constituted by the positions in the plane or on a closed surface such as a sphere (e.g. the earth's surface), of point objects (plants, trees, towns etc). In more complex situations (multivariate case) each object carries concomitant information; this may be a discrete classification (e.g. species of plant) or a continuous measurement (e.g. age of tree); it may be scalar (age of tree) or vector (age and height of tree).

Problems in the analysis of spatial pattern include, as Bartlett (1975) has said, (1) the detection of departures from randomness, (2) the investigation of appropriate stochastic models in the non-random case, (2a) estimation and testing of parameters of a chosen model, and (3) robust estimation (e.g. of density) against a variety of models. See also Ripley (1977).

With multivariate spatial patterns the problems proliferate further. As two examples we mention the investigation of interaction between two species from their joint spatial pattern, and the analysis of clustering of mobs of cattle in relation to the sizes of these mobs.

Distance methods. A variety of distance measures can be used. The distance may be measured from a randomly chosen point in the area under examina- tion, or a randomly chosen object (tree, town etc.). It may be measured to the nearest neighbour object, the next nearest neighbour, etc. In some methods, conditioned distances are used; e.g. Besag and Gleaves (1973) use the distance from a point A to the nearest neighbour object P, together with the distance from P to the nearest neighbour object Q on the further side from A.

In the talk, some advantages and disadvantages of distance methods were indicated, and some distance-based methods for analysing departures from randomness were discussed. These included the methods due to Clark and Evans (1954), Hopkins and Skellam (1954), Pielou (1959) with modification by Mountford (1961), Holgate (1965a) (1965b), Besag and Gleaves (1973), and Cox and Lewis (1976).

An account was also given of distance methods for the robust estimation of density, or equivalently of its reciprocal, the mean area per object. Various estimators, due to Persson (1971), Diggle (1975), S. M. Lewis (1975) and Cox

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132 T. LEWIS

(1976) were mentioned. Aspects of the robustness of these and other es- timators were pointed out. For further references and discussion see Holgate (1972), Warren (1972), and Diggle, Besag and Gleaves (1976).

Finally, some problems in the analysis of bivariate spatial patterns were presented. In particular, Pielou's (1961) concepts of association and segrega- tion between species were described, and a modification suggested for one of her test procedures for segregation; details will be published elsewhere.

References

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relationships in populations. Ecology 35, 445-453. Cox, T. F. (1976) The robust estimation of the density of a forest stand using a new

conditioned distance method. Biometrika 63, 493-499. Cox, T. F. AND LEWIS, T. (1976) A conditioned distance ratio method for analyzing spatial

patterns. Biometrika 63, 483-491. DIGGLE, P. J. (1975) Robust density estimation using distance methods. Biometrika 62, 39-48. DIGGLE, P. J., BESAG, J. AND GLEAVES, J. T. (1976) Statistical analysis of spatial point patterns

by means of distance methods. Biometrics 32, 659-667. HOLGATE, P. (1965a) Tests of randomness based on distance methods. Bibmetrika 52, 345-

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Biometrika 62, 519-521. MOUNTFORD, M. D. (1961) On E. C. Pielou's index of non-randomness. J. Ecol. 49, 271-275. PERSSON, 0. (1971) The robustness of estimating density by distance methods. In Statistical

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172-212. WARREN, W. G. (1972) Point processes in forestry. In Stochastic Point Processes, ed. P. Lewis,

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