Showing papers on "Membership function published in 1986"
TL;DR: It is shown that the proposed representation exists for certain families of the conjugate pairs of t-norms and t-conorms and resolves some of the difficulties associated with particular interpretations of conjunction, disjuntion, and implication in fuzzy set theories.
1,041 citations
TL;DR: In this paper, a modified pair-comparison procedure was used in two experiments to empirically establish and assess membership functions for several probability terms, and the results support the claim that the scaled values represented the vague meanings of the terms to the individual subjects in the present experimental context.
Abstract: Can the vague meanings of probability terms such as doubtful probable or likely be expressed as membership functions over the [0 1] probability interval? A function for a given term would assign a membership value of zero to probabilities not at all in the vague concept represented by the term a membership value of one to probabilities definitely in the concept and intermediate membership values to probabilities represented by the term to some degree. A modified pair-comparison procedure was used in two experiments to empirically establish and assess membership functions for several probability terms. Subjects performed two tasks in both experiments: They judged (a) to what degree one probability rather than another was better described by a given probability term and (b) to what degree one term rather than another better described a specified probability. Probabilities were displayed as relative areas on spinners. Task a data were analyzed from the perspective of conjoint-measurement theory and membership function values were obtained for each term according to various scaling models. The conjoint-measurement axioms were well satisfied and goodness-of-fit measures for the scaling procedures were high. Individual differences were large but stable. Furthermore the derived membership function values satisfactorily predicted the judgments independently obtained in task b. The results support the claim that the scaled values represented the vague meanings of the terms to the individual subjects in the present experimental context. Methodological implications are discussed as are substantive issues raised by the data regarding the vague meanings of probability terms. (authors)
419 citations
TL;DR: It is shown that the method is capable of generating membership functions in accordance with the possibility-probability consistency principle for fuzzy sets whose elements have a defining feature with a known probability density function in the universe of discourse.
333 citations
TL;DR: Fuzzy set theory is 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.
Abstract: 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.
258 citations
01 Jul 1986
TL;DR: The results show that the FCM clustering can be used in the single-level segmentation; and that cluster membership function values derived using this algorithm can be utilized effectively as indicators of region homogeneity.
Abstract: A low-level segmentation methodology based upon fuzzy clustering principles is developed. The approach utilizes region growing concepts and a pyramid data structure for the hierarchical analysis of aerial images. It is assumed that measurement vectors corresponding to perceptually homogeneous regions cluster together in the measurement space. The fuzzy c-means (FCM) clustering algorithm is used in the formulation. Utilization of the fuzzy partitioning allows one to derive a correspondence between the cluster membership function values and (the proportions of) the classes constituting a region. Thus cluster membership values can be used to split mixture regions into smaller regions at a higher resolution level. The feasibility of the methodology is evaluated using a three-channel Landsat image. The results show that the FCM clustering can be used in the single-level segmentation; and that cluster membership function values derived using this algorithm can be utilized effectively as indicators of region homogeneity.
232 citations
TL;DR: Some fundamental properties of fuzzy binary relations and certain conditions of reasonable orderings of fuzzy Utilities are introduced and a method for constructing a fuzzy preference relation on a given set of fuzzy utilities is proposed for the sake of rational decision making.
231 citations
01 Jan 1986
TL;DR: This paper presents an outline of a theory of usuality based on fuzzy logic and applications of fuzzy subsets theory and mathematical programming, and some particular applications.
Abstract: 1: Some theoretical Aspects.- 1.1 Mathematics and fuzziness.- 1.2 Radon-Nikodym Theorem for fuzzy set-valued measures.- 1.3 Construction of a probability distribution from a fuzzy information.- 1.4 Convolution of fuzzyness and probability.- 1.5 Fuzzy sets and subobjects.- 2: From theory to applications.- 2.1 Outline of a theory of usuality based on fuzzy logic.- 2.2 Fuzzy sets theory and mathematical programming.- 2.3 Decisions with usual values.- 2.4 Support logic programming.- 2.5 Hybrid data - various associations between fuzzy subsets and random variables.- 2.6 Fuzzy relation equations : methodology and applications.- 3: Various particular applications.- 3.1 Multi criteria decision making in crisp and fuzzy environments.- 3.2 Fuzzy subsets applications in O.R. and management.- 3.3 Character recognition by means of fuzzy set reasoning.- 3.4 Computerized electrocardiography and fuzzy sets.- 3.5 Medical applications with fuzzy sets.- 3.6 Fuzzy subsets in didactic processes.
200 citations
TL;DR: The concept of risk evaluation, using linguistic representation of the likelihood of the occurrence of a hazardous event, exposure, and possible consequences of that event, and the approximate reasoning technique based on fuzzy logic is used to derive fuzzy values of risk.
130 citations
TL;DR: Diverse hard and fuzzy clustering methods are used to form homogeneous segments of customers and sets of competitive brands, and external and internal validity of this classifications are determined.
113 citations
TL;DR: For establishing optimality criteria of testing, the definition of probability of a fuzzy event is used in order to extend both Neyman-Pearson and Bayes theories, to the fuzzy framework, and the goodness of optimal procedures in both the fuzzy and the nonfuzzy situation is compared for each criterion.
80 citations
TL;DR: It is shown that if the notion of fuzzy sets is further fuzzified by making equality (as well as membership) fuzzy, the resultant categories are indeed toposes.
Abstract: The relation between the categories of Fuzzy Sets and that of Sheaves is explored and the precise connection between them is expli cated. In particular, it is shown that if the notion of fuzzy sets is further fuzzified by making equality (as well as membership) fuzzy, the resultant categories are indeed toposes.
01 Mar 1986
TL;DR: An algorithm of ordering retrieved documents according to the values of the fuzzy thesaurus is proposed and it is shown that one can obtain documents of maximum relevance in a fixed time interval.
Abstract: A fuzzy bibliographic information retrieval based on a fuzzy thesaurus or on a fuzzy pseudothesaurus is described. A fuzzy thesaurus consists of two fuzzy relations defined on a set of keywords for the bibliography. The fuzzy relations are generated based on a fuzzy set model, which describes association of a keyword to its concepts. If the set of concepts in the fuzzy set model is replaced by the set of documents, the fuzzy relations are called a pseudothesaurus, which is automatically generated by using occurrence frequencies of the keywords in the set of documents. The fuzzy retrieval uses two fuzzy relations in addition, that is, a fuzzy indexing and a fuzzy inverted file: the latter is the inverse relation of the former. They are, however, related to different algorithms for indexing and retrieval, respectively. An algorithm of ordering retrieved documents according to the values of the fuzzy thesaurus is proposed. This method of the ordering is optimal in the sense that one can obtain documents of maximum relevance in a fixed time interval. An example of the fuzzy retrieval is shown on a prototype database. This method shows one of the simplest way to realize fuzzy retrieval in practical database systems.
TL;DR: An experimental construction of fuzzy set membership leads to a realization of stochastic fuzziness and a type II fuzzy set representation and axioms of measurement can be validated with a probabilistic interpretation.
Abstract: Empirical measurement of membership functions of fuzzy sets are considered with the fundamental axioms of measurement theory. An experimental construction of fuzzy set membership leads to a realization of stochastic fuzziness and a type II fuzzy set representation. Axioms of measurement can be validated with a probabilistic interpretation.
TL;DR: An interactive fuzzy decision-making method for solving multiobjective nonlinear programming problems is presented by assuming that the decision maker (DM) has fuzzy goals for each of the objective functions.
TL;DR: In this article, a base theorique for construire une fonction de qualite de membre au moyen de comparaisons par paire is proposed.
01 Jan 1986
TL;DR: It will be shown how crisp multi criteria problems can be solved if either criteria or solution spaces or both are fuzzy.
Abstract: Decision making when more than one evaluation scheme exists has become a major concern of scientists and decision analysts during the last decade. In the numerous models and methods suggested in the literature in this area it is, however, assumed that the criteria as well as the solution psace can be crisply defined. It will be shown how these problems can be solved if either criteria or solution spaces or both are fuzzy. In addition approaches will be presented in which crisp multi criteria problems can be solved efficiently by applying fuzzy set theory.
TL;DR: The concept of fuzzy predictions is introduced, which reveals a fuzzy valuation of the predicted concentrations by a membership function rather than associating approximate confidence intervals with the sought concentrations.
TL;DR: This work proposes, as a generalization of the principle of transitivity, the transitivity inequality, which states that given the degrees to which A is contained in B and B in C, n(A E B) and T-c(B c C), what can or would the authors like to be able to say of the degree to which
Book•
01 Jan 1986
TL;DR: Benefits of fuzzy theory methods compared to common mathematical methods with respect to data handling for calibration of analytical methods, to classification of Chromatographie and spectroscopic patterns, to component identification and multicomponent analysis, and to designing fuzzy expert systems for selection of analytical procedures are demonstrated.
Abstract: The concept of fuzzy theory is described in order to provide the analyst with the means for dealing with vague statements, uncertain observations or the fuzziness of human perception and interpretation, in general. In a theoretical part, basic notions of fuzzy theory are given, such as types of membership functions, operations with fuzzy sets, definitions of fuzzy numbers, points, functions, and relations, and the use of linguistic variables. The difference between fuzziness and probability is outlined. The applications section demonstrates advantages of fuzzy theory methods compared to common mathematical methods with respect to data handling for calibration of analytical methods, to classification of Chromatographie and spectroscopic patterns, to component identification and multicomponent analysis, and to designing fuzzy expert systems for selection of analytical procedures.
TL;DR: Zadeh's suggested measure of fuzzy cardinality, the sigma-count, is adopted and shown to generalize classical counting measure, which allows many combinatorial structures and counting techniques to be fuzzified, and hence used in knowledge representation and pattern recognition models.
Abstract: The notion of fuzzy set cardinality is examined. Zadeh's suggested measure of fuzzy cardinality, the sigma-count, is adopted and shown to generalize classical counting measure. This allows many combinatorial structures and counting techniques to be fuzzified, and hence used in knowledge representation and pattern recognition models. A fuzzy set review is found in the Appendix.
01 Sep 1986
TL;DR: An adaptive learning scheme, based on a fuzzy approximation to the gradient descent method for training a pattern classifier using unlabeled samples, is described and an inductive entropy measure is defined in terms of induced possibility distribution to measure the extent of learning.
Abstract: An adaptive learning scheme, based on a fuzzy approximation to the gradient descent method for training a pattern classifier using unlabeled samples, is described. The objective function defined for the fuzzy ISODATA clustering procedure is used as the loss function for computing the gradient. Learning is based on simultaneous fuzzy decisionmaking and estimation. It uses conditional fuzzy measures on unlabeled samples. An exponential membership function is assumed for each class, and the parameters constituting these membership functions are estimated, using the gradient, in a recursive fashion. The induced possibility of occurrence of each class is useful for estimation and is computed using 1) the membership of the new sample in that class and 2) the previously computed average possibility of occurrence of the same class. An inductive entropy measure is defined in terms of induced possibility distribution to measure the extent of learning. The method is illustrated with relevant examples.
TL;DR: Fuzzy multistage decision making (control) models are presented and newer approaches which additionally allow for a more general aggregation of control stage scores (outcomes), e.g., “…. at most of the control stages…” instead of “… at all …” conventionally used.
TL;DR: Concepts of semi-Ti (i = 0, 1, 2) spaces and semi-Ri spaces and some fuzzy topological properties are investigated under the above mentioned axioms.
01 Jan 1986
TL;DR: A treatment of linguistic entities in medicine, allowing the assignment of graded diagnoses or types to patients, using possibility distributions and fuzzy intervals to give a meaning to fuzzy propositions issued from the knowledge base.
Abstract: This tutorial paper presents a treatment of linguistic entities in medicine, allowing the assignment of graded diagnoses or types to patients. A knowledge base can consist of a tableau with linguistic entries or of a set of propositions within rules and, in both cases, relationships between attributes like plasma lipids, serum proteins, and diagnoses or types, are not expressed by numbers but by labels of fuzzy sets. Possibility distributions and fuzzy intervals are simply introduced to give a meaning to the fuzzy propositions issued from the knowledge base. Finally, it is shown how possibility measures can be used to assign different types to patients after examination of their condition.
TL;DR: A method for constructing a membership function for the fuzzy sets that expert systems deal with and the systematic methodology presented will facilitate effective use of expert systems is put forth.
TL;DR: Fuzzy σ-algebras of events are introduced in an analogous way to the classical theory of probability by means of the above relation and specific properties of the fuzzy probability measures presented here are given.
TL;DR: This paper provides a methodology for using information regarding the probability of some variable being equal to a fuzzy subset to obtain the expected value of the variable.
TL;DR: One proves that every L -subset with L totally ordered is a nonstandard fuzzy set, and observes that the same operations and relations defined for fuzzy sets are definable for non standard fuzzy sets.