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A recently developed fuzzy clustering technique is utilized to analyze the substructure of a well known set of 4-dimensional botanical data. A solution obtained without prior knowledge of labelled pattern structure is offered in support of our contention that the technique proposed affords a comparatively reliable criterion for a posteriori evaluation of cluster validity.

Pattern Recognition methods like the A-nearest neighbor technique and clustering have been successfully used to classify nonlinear systems to a predetermined set of well separated classes, demonstrating similarities in their input-output behavior. The cross-correlation function has been used as a feature vector which yielded a classification with a relatively small misclassification error. Further categorization of the nonlinearity within a class is not possible through the above approach. However, a membership factor in a fuzzy sense can be assigned to each nonlinear system within a certain class. This factor is dependent on the coefficients of the polynomial expansion of the nonlinearity involved and may be obtained through the cross-correlation vector. This way, every class of nonlinear systems may be defined as a fuzzy set. The investigation of the properties of such a set as well as its contribution to the further classification of nonlinear systems is discussed in this paper.

Many concepts of problem solving theory are better understood in an abstract algebraic framework which also applies in automata theory. Because many systems of practical interest fall outside the scope of linear theory, it is desirable to enlarge as much as possible the class of systems for which a complete structure theory is available. The fuzzy system approach is presented as a basis for the design of systems far superior in artificial intelligence to those we can conceive today. The concepts of controllability, observability and minimality are developed, and conditions for the realization of an input‐output map by such a system are given. Several problems, all directly or indirectly related to fuzzification, arise in considering this broader class of systems.

This paper develops the notion of fuzzy preference orderings for individuals and groups, on the basis of the theory of fuzzy (binary) relations as developed by L. A. Zadeh [4]. This concept is illustrated with well known methods of group decision-making and some of its formal properties are investigated. Finally an algorithm to map fuzzy orderings on non-fuzzy ones is proposed to guide the decision-maker in reaching decisions which reflect the fuzziness of the preferences over the available options.

A strictly binary approach to the treatment of switching circuits today is not always adequate to describe systems in the real world. This approach is partly due, to the relative simplicity of designing binary switching systems, and to the fact that basic switching modules in common use are two-positional. Consequently, every variable in Boolean logic is assumed to be two-valued. However, because of real-world constraints, the attributes of system variables are often ambiguously defined. In other words, quite often variables might have values other than falsehood and truth. Cases with such attributes arise, for example in artifical intelligence and related subjects. Ever since Zadeh introduced the idea of fuzzy set theory [11 by utilizing the concept of membership grade, a number of authors have been concerned with the analysis and applications of fuzzy models. Especially, the relation of fuzzy to switching systems have been discussed in [2]–[12] and by other researchers in relation to other topi...

The notion of fuzzy event is introduced in the theory of measurement in quantum mechanics by indicating in which sense measurements can be considered to yield fuzzy sets. The concept of probability measure on fuzzy events is defined, and its general properties are deduced from the operational meaning assigned to it. It is pointed out that such probabilities can be derived from the formalism of quantum mechanics. Any such probability on a given fuzzy set is related to the frequency of occurrence within that set of points in a random sample, where the sample points are themselves fuzzy sets obtained as outcomes of measurements of, in general, incompatible observables on replicas of the system in the same prepared state.

In this paper the fuzzy set as discussed by Zadeh is viewed as a multivalued logic with a continuum of truth values in the interval [0,1]. The concept of static hazard in combinational switching systems is related to fuzzy logic and various properties of this relation are established. The paper derives the necessary and sufficient conditions for a fuzzy function to adequately describe the steady-state and static hazard behavior of a combinational system, by extending the ternary method discussed by Yoeli and Rinon and using the resolution principle of mechanical theorem-proving.

This paper discusses the applicability of the ideas of fuzzy sets and grades of membership to problems encountered in the quantification of clinical (i.e., diagnostic and prognostic) judgment. The methods of constrained maximum likelihood are used to derive consensus estimates of grades of membership given a set of categorical data and an a priori set of specified pure types. A numerical example is given.

This paper reports research related to mathematics, philosophy, computer science and linguistics. It gives a system of axioms for a relatively simple form of fuzzy set theory, and uses these axioms to consider the accuracy of representing concepts in various ways by fuzzy sets. By-products of this approach include a number of new operations and laws for fuzzy sets, parallel to those for ordinary sets, and a demonstration that all the basic operations are intrinsically determined. In addition, the paper explores both hierarchical and algorithmic extensions of fuzzy sets, and then applications to problems in natural language semantics and combinatorics. Finally, the paper returns to the problem of representing concepts, and discusses some implications for artificial intelligence.

The characteristic function of a fuzzy set denoted by a phrase “much larger than 5,” or “heavy,” is investigated experimentally. The S-shaped curve for the characteristic function is derived from some plausible assumptions involving a differential equation, and its parameters are estimated statistically. It was found that the precision of the fuzzy set–measured by the maximum slope of the characteristic function–denoted by “much greater than 5” is less than that denoted by “very much greater than 5.” When the subject is allowed to be fuzzy in his response about grade of membership, there is greater consistency in the response. Anchoring was also found to make an adjective denoting fuzzy set more precise.

It is possible that a better model for the behavior of a nerve cell may be provided by what might be called a fuzzy neuron, which is a generalization of the McCulloch-Pitts model. The concept of a fuzzy neuron employs some of the concepts and techniques of the theory of fuzzy sets which was introduced by Zadeh [2, 3] and applied to the theory of automaton by Wee and Fu [6], Tanaka et al. [7], Santo [8] and others. In effect, the introduction of fuzziness into the model of a neuron makes it better adapted to the study of the behavior of systems which are imprecisely defined by virtue of their high degree of complexity. Many of the biological systems, economic systems, urban systems and more generally, large-scale systems fall into this category. In the nearly three decades since its publication, the pioneering work of McCulloch and Pitts [1], has had a profound influence on the development of the theory of neural nets, in addition to stimulating much of the early work in automata theory and regula...

The problem of optimal stopping has long bee of intrest to various scientific disciplines. Most notably, the approacbes taken have been based on sequential decision theory and modeling techniqes. Aiming at optimization algorithms, we propose in this paper to combine the present approaches and study the cut-off rule problem, using asymptotic convergence behavior of sequences. A unifying framework for the design and analysis of stopping rules for algorithms generating a monotonically convergent sequence is presented while methodology for optimal design is discussed. The concavity structure should be observed to play an important role in our analysis.

An important problem which frequently arises is the problem of measuring the degree to which a collection of objects has the empirical properties of its elements. Examples of this are found in the question of whether a network of switches is on or off, the degree to which a group of factories pollutes the air, and even the probability of success defined on an empirical frequency distribution. While each of these questions may be addressed with an esoteric formalism, there exists no general methodology for addressing the general problem because the properties of a set's elements are not defined on the set itself. For example, the set of all tall buildings is neither tall nor a building. Rather, it is an abstraction and has only those properties defined on such abstractions, e.g., cardinality. The purpose of the present paper is to develop a broad new formalism for addressing the noted general problem in a very general way. To begin with, the property p defined on the object x is formalized as a function p(x) = [0,1] which may be ``Boolean'' (binary valued) or ``formal'' (many valued or fuzzy). A set P may likewise have such a property to the degreep(P). Such a set is referred to as a ``property set,'' and these are partitioned into two general categories: ``property systems'' and ``collectives.'' A property system is characterized by the fact that its elements are related through the operators AND or OR.

The notion of “entropy” of a fuzzy set, introduced in a previous paper in the case of generalized characteristic functions whose range is the interval [0, 1] of the real line, is extended to the case of maps whose range is a poset L (or, in particular, a lattice). Some of the reasons giving rise to the non-comparability of the truth values and then the necessity of considering poset structures as range of the maps are discussed. The interpretative problems of the given mathematical definitions regarding the connections with decision theory are briefly analyzed.

In problems of system analysis, it is customary to treat imprecision by the use of probability theory. It is becoming increasingly clear, however, that in the case of many real world problems involving large scale systems such as economic systems, social systems, mass service systems, etc., the major source of imprecision should more properly be labeled ‘fuzziness’ rather than ‘randomness.’ By fuzziness, we mean the type of imprecision which is associated with the lack of sharp transition from membership to nonmembership, as in tall men, small numbers, likely events, etc. In this paper our main concern is with the application of the theory of fuzzy sets to decision problems involving fuzzy goals and strategies, etc., as defined by R. E. Bellman and L. A. Zadeh [1]. However, in our approach, the emphasis is on mathematical programming and the use of the concept of a level set to extend some of the classical results to problems involving fuzzy constraints and objective functions.

In a previous paper ^[^1^] the use of the concept of fuzzy sets in clustering was proposed. The convenience of fuzzy clustering over conventional representation was then stressed. Assigning each point a degree of belongingness to each cluster provides a way of characterizing bridges, strays, and undetermined points. This is especially useful when considering scattered data. The classificatory process may be considered as the breakdown of the probability density function of the original set into the weighted sum of the component fuzzy set densities. Such decomposition should be performed so that the components really represent clusters. This is done by optimization of some functional defined over all possible fuzzy classifications of the data set. Several functionals were suggested in ^[^1^]. The bulk of this paper is concerned with numerical techniques useful in the solution of such problems. The first two formulas treated do not provide an acceptable fuzzy classification but yield good starting points for the minimization of a third functional. This last method obtains very good dichotomies and is characterized by slower convergence than the previous processes. Using that functional, a modification is suggested to obtain partitions in more than two sets. Numerous computational experiments are presented.

A method of classifying patterns using fuzzy relations is described. To start with, we give a suitable value of the measure of subjective similarity to each pair of patterns that is taken from the population of patterns to be classified. Then a similitude between any two patterns is calculated by using the composition of a fuzzy relation. The similitude induces an equivalence relation. Consequently, we can classify the present population of the patterns into some classes by the equivalence relation. An experiment of the classification of portraits has been performed to test the method proposed here.

A learning model of fuzzy formal language is proposed and discussed. We continue training the learning machine by giving sets of sentences sequentially. As a result of parsing of the given teaching sentences, the learning machine reinforces fuzzy grades of membership of productions in an inherent fuzzy grammar of the machine. The convergence of the proposed model is considered, and it is shown that the grades of membership of desired productions are intensified by choosing an adequate teaching sequence of the sentence set. Furthermore, a concept of ``strongly equivalent,'' in which two grammars are not distinguished by any teaching sequence, is introduced.

Fuzzy sets are defined as mappings from sets into Boolean lattices. The basic set theory type results for Zadeh's fuzzy sets are shown to carry over. Some results on convex fuzzy sets, star-shaped fuzzy sets, and arcwise connected fuzzy sets are given. Fuzzy sets with “holes,” bounded fuzzy sets, and connected fuzzy sets are also discussed.

We established an application of fuzzy set logic for the development of a decision support in medicine. In this case, we apply a model on the prognosis of a patient suffering heart failure treated with beta-blockers. It shows how the basic rules are represented in a system depending on fuzzy logic based on expert experience. Keywords: Left ventricular diastolic diametre (LVDD); Dyspnea; Asthenia; Telediastolic; Ischemia; Servical vertibra three (S3); Atrio Venticular Block (AVB); Pace maker (PM); Heart failure probability (HFP). DOI: http://dx.doi.org/10.3329/bjsir.v46i1.8103 Bangladesh J. Sci. Ind. Res. 46(1), 41-46, 2011

This paper is a study on a pattern recognition model of discrete picture information by the visual perception.The discrete picture information means that informations comprehended from picture elements, in themselves, are discrete, and the picture elements signify a part of frame as the minimum area which a man will be able to catch in a moment.A man does recognize one after another the whole of frame selectting a suitable set of picture elements accoding to duantity of information obtained by itself.In this paper, it is discussed on a pattern recognition model as a Fuzzy Automaton where obscurity of informations derived from picture elements is caught by way a membership space of the Fuzzy Set. Furthermore, a computer simulation and a experiment of eyecamera have been done applying to a man-model.

An abstract model for the execution of fuzzy programs using finite-state machines is described, Different ways of executing fuzzy programs are investigated. It is shown that the performance function ? of a fuzzy machine M? can be effectively constructed so that the program searching cost of an elementary fuzzy program, a number of elementary fuzzy programs, or a regular fuzzy program is minimized. Some application examples are discussed.

The discussion of fuzzy sets indicates that set theory should be extended to make it more suitable for the development of algebraic structures with wider applications. Stochastic optimization, a synthesis of the three areas of continuum mathematics, is a rapidly growing field particularly in the context of control theory. Applications of discrete mathematics have found a rich outlet in graph theory, particularly in the social sciences. However, there remains a wide gap between the sophisticated activities that are required for the development of optimum policies and the more difficult and intractable problems of organizational structures and performance.

At present fuzzy logic control is receiving increasing emphasis in process control applications. The paper describes the application of fuzzy logic control in a power system that uses a 12- pulse bridge converter associated with Superconductive Magnetic Energy Storage (SMES) unit. The fuzzy control is used in both the frequency and voltage control loops, replacing the conventional control method. The control algorithms have been developed in detail and simulation results are presented. These results clearly indicate the superior performance of fuzzy control during the dynamic period of energy transfer between the power system and SMES unit. Keywords: Fuzzy logic controller; power system dynamic performance; SMES unit. DOI: http://dx.doi.org/10.3329/diujst.v6i2.9343 DIUJST 2011; 6(2): 33-41

A basic idea suggested in this paper is that a linguistic hedge such as very, more or less, much, essentially. slightly, etc. may be viewed as an operator which acts on the fuzzy set representing the meaning of its operand. For example, in the case of the composite term very tall man, the operator very acts on the fuzzy meaning of the term tall man. To represent a hedge as an operator, it is convenient to define several elementary operations on fuzzy sets from which more complicated operations may be built up by combination or composition. In this way, an approximate representation for a hedge can be expressed in terms of such operations as complementation, intersection, concentration, dilation, contrast intensification, fuzzification, accentuation, etc. Two categories of hedges are considered. In the case of hedges of Type I, e.g., very, much, more or less, slightly, etc., the hedge can be approximated by an operator acting on a single fuzzy set. In the case of hedges of Type II, e.g., technical...

In this paper the concept of probabilistic events is introduced and their closure properties under the operations on the set of all fuzzy events are studied. It is shown that the mean of probabilistic events is a probabilistic event and the set of all probabilistic events is closed under the transposition and the operation of the convex-combination. A sufficient condition for the union and the intersection of two probabilistic events to be probabilistic events in the sense of fuzzy sets is given. In the last two sections we showed some families of probabilistic events closed or approximately closed under the operation of the union and the intersection and the complementation.

The paper presents a new point of view about the “relevance” concept. The logic of information retrieval is analysed (two‐valued, many‐valued and infinite‐valued logic) and the truth‐value is interpreted as the relevance value. The logical system which seems to be most appropriate to describe the process of information retrieval is the continuous logic suggested by McNoughton. In this case the answer of the information retrieval system is interpreted as a fuzzy set.

The approach described in this paper represents a substantive departure from the conventional quantitative techniques of system analysis. It has three main distinguishing features: 1) use of so-called ``linguistic'' variables in place of or in addition to numerical variables; 2) characterization of simple relations between variables by fuzzy conditional statements; and 3) characterization of complex relations by fuzzy algorithms. A linguistic variable is defined as a variable whose values are sentences in a natural or artificial language. Thus, if tall, not tall, very tall, very very tall, etc. are values of height, then height is a linguistic variable. Fuzzy conditional statements are expressions of the form IF A THEN B, where A and B have fuzzy meaning, e.g., IF x is small THEN y is large, where small and large are viewed as labels of fuzzy sets. A fuzzy algorithm is an ordered sequence of instructions which may contain fuzzy assignment and conditional statements, e.g., x = very small, IF x is small THEN Y is large. The execution of such instructions is governed by the compositional rule of inference and the rule of the preponderant alternative. By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

The extension of standard concepts of set theory (like union, intersection, etc.) to the theory of fuzzy sets is not obvious. Several policies for answering questions of this type have been discussed, but it appears that the operations of max and min play a central role in the arithmetic of fuzzy sets. Letting μA(x) denote our willingness to accept x as a member in the fuzzy set A, Zadeh defines intersection A ∩ B and union A ∪ B of the two fuzzy sets A and B by A ∩ B = {(x; min {μA(x), μB(x)})} and A ∪ B = {(x; max {μA(x), μb(x)})}. The object of this paper is to show that these definitions are not only natural, but under quite reasonable assumptions the only ones possible.

The study of fuzziness in combinational switching systems by means of a suitable fuzzy algebra is discussed. The insufficiency of the methods for simplification of fuzzy functions as presented by Lee and Chang [9] and by Siy and Chen [10] is discussed. A new technique for minimization of fuzzy functions is developed. Special properties of fuzzy functions are discussed and their relationships to two-valued logic are investigated.

This paper deals with a new search technique for optimum using a fuzzy automaton, which has been developed for the purpose of determining a desirable rod pattern for smoothing the core power distribution. Using the algorithm thus developed, the optimal input vector applied to a controlled system whose characteristics are unknown is determined automatically from a learning operation of the fuzzy automaton when an initial guess and a hypothetical range of optimal input vector are given. The characteristics of this algorithm are examined for an example of its application to a simple problem. The optimal control rod pattern for smoothing the core power distribution is also determined by this algorithm in a simulation study using the three-dimensional boiling water reactor simulator “FLARE”. It is concluded that optimum points are obtained with sufficient convergence and with only a small increase in the number of iterations and computation time accompanying an increase in the number of components of the input...

Bellman and Zadeh recently considered fuzzy multi-stage decision processes, the concept of fuzziness emanating from Zadeh's earlier work on fuzzy sets. Here we extend their results to processes inv...

The provision of appropriate quality rail services has an important role in terms of railway infrastructure: quality of infrastructure maintenance, regulation of railway traffic, line capacity, speed, safety, train station organization, the allowable lines load and other infrastructure parameters.The analysis of experiences in transforming the railway systems points to the conclusion that there is no unique solution in terms of choice for institutional rail infrastructure management modes, although more than nineteen years have passed from the beginning of the implementation of the Directive 91/440/EEC. Depending on the approach to the process of restructuring the national railway company, adopted regulations and caution in its implementation, the existence or absence of a clearly defined transport strategy, the willingness to liberalize the transport market, there are several different ways for institutional management of railway infrastructure.A hybrid model for selection of modes of institutional rail infrastructure management was developed based on the theory of artificial intelligence, theory of fuzzy sets and theory of multicriteria optimization.KEY WORDSmanagement, railway infrastructure, organizational structure, hybrid model

This paper deals with a new clustering technique using the concept of fuzzy set introduced by Zadeh. A cluster is defined as a subset characterized by a membership function which associates with each element a real number in the unit interval. A membership function is proposed and a method to select the cluster elements is derived using the separation theorem of the fuzzy sets. The clusters obtained by this way are controlled overlapping groupings.

A functional defined on the class of generalized characteristic functions (fuzzy sets), called “entropy≓, is introduced using no probabilistic concepts in order to obtain a global measure of the indefiniteness connected with the situations described by fuzzy sets. This “entropy≓ may be regarded as a measure of a quantity of information which is not necessarily related to random experiments. Some mathematical properties of this functional are analyzed and some considerations on its applicability to pattern analysis are made.

A probabilistic event is a fuzzy set of tapes accepted by a finite probabilistic automation. In this paper, a mapping induced by a PGSM-mapping is introduced and is shown to be one of the operations which preserve a probabilistic event. Using mappings induced by a GSM-mapping, some recursively unsolvable problems with respect to rational finite probabilistic automata are studied.

Given a finite, unlabelled set of real vectors X, one often presumes the existence of (c) subsets (clusters) in X, the members of which somehow bear more similarity to each other than to members of adjoining clusters. In this paper, we use membership function matrices associated with fuzzy c-partitions of X, together with their values in the Euclidean (matrix) norm, to formulate an a posteriori method for evaluating algorithmically suggested clusterings of X. Several numerical examples are offered in support of the proposed technique.

The point of departure in this paper is the definition of a language, L, as a fuzzy relation from a set of terms, T = x, to a universe of discourse, U = y. As a fuzzy relation, L is characterized by its membership function @m"L:T x U -> [0,1], which associates with each ordered pair (x,y) its grade of membership, @m"L(x,y), in L. Given a particular x in T, the membership function @m"L(x,y) defines a fuzzy set, M(x), in U whose membership function is given by @m"M"("x")(y) = @m"L(x,y). The fuzzy set M(x) is defined to be the meaning of the term x, with x playing the role of a name for M(x). If a term x in T is a concatenation of other terms in T, that is, x = x"1 ... x"n, x"i @e T, i = 1,...,n, then the meaning of x can be expressed in terms of the meanings of x"1,...,x"n through the use of a lambda-expression or by solving a system of equations in the membership functions of the x"i which are deduced from the syntax tree of x. The use of this approach is illustrated by examples.

Ever since the introduction of fuzzy sets by Zadeh [1] , the fuzzyness invaded almost all the branches of crisp mathematics. Deng [3] kaleva and Seikalla [2] and Kramosil and Michalek [5]Have introduced the concept of fuzzy metric space, George and Veeramani [4] modified the concept of fuzzy metric space introduced by kramosil and michalek [5]. In thia paper effort has been made to obtain some results on fixed points of expansion type mapping in Intuitionistic fuzzy metric space DOI: http://dx.doi.org/ 10.3126/kuset.v7i1.5420 KUSET 2011; 7(1): 38-47

This paper extends the concept of fuzzy set, introduced by L. A. Zadeh, to a time dependent setting. Time dependence enters into a number of areas including planning and pattern recognition. Results are given for topologies of fuzzy sets and applied to shadows. Time dependent fuzzy relations are defined and discussed.

An algorithm for classifying a data set into an initially unknown number of categories is presented. It is composed of procedure for selecting initial points, a mode estimation procedure, and a classification rule. An integer valued function is defined on the sample space and a gradient search technique is used for estimating its modes. A procedure for mode estimation in the case of an infinite data set is also proposed. Sufficient conditions for the convergence to the neighborhood of the modes have been stated. The algorithm was used for clustering multicategory artificially generated data sets and was compared with an optimal classification scheme.

This paper is mainly devoted to the description of experiments using improved techniques for clustering data in fuzzy sets. Previous methods provided good fuzzy dichotomies but failed for partitions in a larger number of sets. The modifications presented here resulted in good fuzzy classifications in any, previously established, number of clusters. The modifications are described both theoretically and computationally. The effect of variations in the function that maps the original distance structure in the data space and that prescribed for the classification space is examined. Numerous computational examples are presented.

An algorithm is presented which partitions a given sample from a multimodal fuzzy set into unimodal fuzzy sets. It is proven that if certain assumptions are satisfied, then the algorithm will derive the optimal partition in the sense of maximum separation.

Since comprehensive evaluation model was introduced by Wang in 1984, the comprehensive evaluation model has many successful applications in real life. But some examples show the inefficiency of the comprehensive evaluation model. In order to process these problems, in this paper, we introduce two concepts of standardized fuzzy set and normalized fuzzy set, propose a novel method to improve the comprehensive evaluation model and its algorithm. Finally, we use three numerical examples to illustrate our proposed model reasonable and point out that our proposed method can avoid some inefficiencies in applications.

The purpose of this paper is to investigate the issue systematically. We identify 9 generalized types of absolute continuity which possess desirable properties, such as reflexivity and transitivity. We also study the relationship between these distinct types of absolute continuity and determine which of them are possessed by the fuzzy measure (or the lower semicontinuous fuzzy measure) defined by the fuzzy integral. >

This paper surveys the developments in the field of pattern classification to describe the current state-of-the-art. The author divides the pattern classification problem into deterministic and statistical approaches although, in many instances, both converge to the same result. The approaches are further subdivided into: Adalines and Madalines, linear discriminant functions, mathematical programming, mode-seeking, nearest neighbor, Bayes and minimax, statistical criteria, and fuzzy sets subclasses. A major effort is devoted to show relationships among procedures. The conditions under which procedures are equivalent are discussed. Such relationships are summarized through a graph. The problems of supervision, adaptive property, sample size, and sequential analysis are discussed.

This correspondence points out the insufficiency of the method for minimization of fuzzy functions as presented in a recent short note.1 Two lemmas, which are the basis for a new minimization technique [2], are represented.