Using Fuzzy Intervals to Represent Measurement Error and Scientific Uncertainty in Endangered Species Classification zy Scott Ferson, H. Resit Akqakaya and Amy Dunham Applied Biomathematics 100 North Country Road Setauket, New York 11733 USA redlist@ramas.com Abstract Although fuzzy numbers (including fuzzy intervals) are often used to capture semantic ambiguity, they are also useful to represent and propagate measurement error. In this application, a classification scheme used by international authorities for assigning biological species into categories of relative endangerment is generalized to accept intervals and triangular or trapezoidal fuzzy numbers as inputs representing empirical estimates of unknown quantities. Non-traditional definitions for fuzzy magnitude comparisons and logical operations were required but, otherwise, standard fuzzy arithmetic was used. A defuzzification step, which explicitly reveals the analyst’s attitudes regarding evidence, can condense the result from the fuzzified classification scheme to a single category. But this step is not required and may be counterproductive. Introduction The classification of biological species into categories of endangerment (critical, endangered, vulnerable, etc.) is essential for planning effective conservation strategies. The IUCN (World Conservation Union), a partner of the United Nations Environment Programme, has promulgated a comprehensive system [l] for making these classifications based on available information about a species’ abundance, spatial distribution and life history. The IUCN classification system has been very widely used around the world over the last decade to classify thousands of species and to make sensitive management and regulatory decisions. Such decision rules are attractive because of their wide applicability, objectivity, and simplicity of use [2]. Although the IUCN scheme is but one of several that have been suggested by conservation biologists [3-6], it alone has enjoyed broad international acceptance, and it has become one of the most important decision tools in conservation biology. The IUCN classification scheme has two conceptual problems. The first (which will be addressed in a separate contribution by H. Regan et al. [7]) is that the categories are defined in terms of precise scalar thresholds whose specificity has little biological justification. The second, addressed in this paper, is that the classification presumes that analysts can make precise numerical estimates for several important variables, even for extremely rare and poorly studied species. This is unlikely to be possible, even for well studied species, and calls into question the reliability of the resulting classification. Measurement error and scientific uncertainty is ubiquitous throughout the empirical sciences, but they can be particularly large in the life sciences, especially ecology, because of the importance of stochasticity and complex, often nonlinear dynamics in natural ecosystems. For this reason, it seemed clear that the IUCN criteria should be generalized to account for measurement error. We wanted a robust approach that could be intelligible and useful to biologists without special training in uncertainty propagation techniques. We considered several possible approaches, including Monte Carlo uncertainty propagation [8], probability bounds analysis [9], interval analysis [lo] and fuzzy arithmetic zyxwv [ 113. The most reasonable approach appeared to be one based on fuzzy arithmetic. The IUCN rules for classification of threatened species consist of three sets. These sets define the classes Critically Endangered (CR), Endangered (EN), and Vulnerable (VU). The evaluation of a given taxon under each of these sets of rules results in a Boolean (TRUE or FALSE) outcome and the rule sets are nested in such a way that there are only four classifications possible: CR, EN, VU and a none-of-the-above zy 0-7803-521 1-41991s 10.00 zyxwvuts 0 1999 IEEE 690