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.
KeywordsEnvironmental synthesis Fuzzy set theory Gradient analysis Montana Ordination method Vegetation analysis
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- Allen, T. F. H. & Starr, T. B., 1982. Hierarchy: Perspectives for ecological complexity. The University of Chicago Press, Chicago. 310 pp.Google Scholar
- Beals, E. W., 1960. Forest bird communities of the Apostle Islands of Wisconsin. Wilson Bull. 72: 156–181.Google Scholar
- Beals, E. W., 1973. Ordination: Mathematical elegance and ecological naivete. J. Ecol. 61: 23–35.Google Scholar
- Bray, J. R. & Curtis, J. T., 1957. An ordination of the upland forest communities of southern Wisconsin. Ecol. Monogr. 27: 325–349.Google Scholar
- Dale, M. B., 1977. Planning and adaptive numerical classification. Vegetatio 35: 131–136.Google Scholar
- Dubois, D. & Prade, H., 1980. Fuzzy sets and systems: Theory and application. Academic Press, New York. 393 pp.Google Scholar
- Feoli, E., 1977. A criterion for monothetic classification of phytosociological entities on the basis of species ordination. Vegetatio 33: 147–152.Google Scholar
- Gauch, H. G., 1982. Multivariate analysis in community ecology. Cambridge Univ. Press, New York. 298 pp.Google Scholar
- Goldin, A. & Nimlos, T. J., 1977. Vegetation patterns on limestone and acid parent materials in the Garnet Mountains of western Montana. Northwest Sci. 51: 149–160.Google Scholar
- Greig-Smith, P., 1980. The development of numerical classification and ordination. Vegetatio 42: 1–9.Google Scholar
- Greig-Smith, P., 1983. Quantitative plant ecology. Third edition. Univ. of Calif. Press, Berkeley. 359 pp.Google Scholar
- Hitchcock, C. L. & Cronquist, A., 1973. Flora of the Pacific Northwest: An illustrated manual. Univ. of Wash. Press, Seattle. 730 pp.Google Scholar
- Kandel, A., 1982. Fuzzy techniques in pattern recognition. John Wiley and Sons, New York. 356 pp.Google Scholar
- Kaufmann, A., 1975. Introduction to the theory of fuzzy subsets: Vol. 1—Fundamental theoretical elements. Academic Press, New York. 409 pp.Google Scholar
- Loucks, O. L., 1962 Ordinating forest communities by means of environmental scalars and phytosociological indices. Ecol. Monogr. 32: 137–166.Google Scholar
- Orlóci, L. H., 1980. Preface. In: E. van der Maarel (ed.). Classification and ordination. Advances in vegetation science. Vol. 2. Dr. W. Junk, The Hague.Google Scholar
- Pfister, R. D. & Arno, S. F., 1980. Classifying forest habitat types based on potential climax vegetation. Forest Sci. 26: 52–70.Google Scholar
- Pfister, R. D., Kovalchik, B. L., Arno, S. F. & Presby, R. C., 1977. Forest habitat types of Montana. USDA Forest Service Intermt. For. and Range Exp. Stn., Gen. Tech. Rep. INT-34; 174 pp.Google Scholar
- Roberts, D. W., 1986. An anticommutative difference operator for fuzzy sets and relations. Int. J. Fuzzy Sets and Systems. In press.Google Scholar
- Whittaker, R. H., 1967. Gradient analysis of vegetation. Biol. Rev. 42: 207–264.Google Scholar
- Zadeh, L. A., 1965. Fuzzy sets. Inf. Control 8: 338–353.Google Scholar