Ordination on the basis of fuzzy set theory

Fuzzy set theory is an extension of classical set theory where elements of a set have grades of membership ranging from zero for non-membership to one for full membership. Exactly as for classical sets, there exist operators, relations, and mappings appropriate for these fuzzy sets. This paper presents the concepts of fuzzy sets, operations, relations, and mappings in an ecological context. Fuzzy set theory is then established as a theoretical basis for ordination, and is employed in a sequence of examples in an analysis of forest vegetation of western Montana, U.S.A. The example ordinations show how site characteristics can be analyzed for their effect on vegetation composition, and how different site factors can be synthesized into complex environmental factors using the calculus of fuzzy set theory. In contrast to current ordination methods, ordinations based on fuzzy set theory require the investigator to hypothesize an ecological relationship between vegetation and environment, or between different vegatation compositions, before constructing the ordination. The plotted ordination is then viewed as evidence to corroborate or discredit the hypothesis. I am grateful to Dr R. D. Pfister (formerly USDA Forest Service) for kind permission to publish data from a Forest Service study. I would like to gratefully acknowledge the helpful comments and criticisms of Drs. G. Cottam, J. D. Aber, T. F. H. Allen, E. W. Beals, I. C. Prentice, C. G. Lorimer, and two anonymous reviewers.