are grades of membership probabilities abstract a small sample is given of the difficulties with present day fuzzy set theory with respect to the requirements of a scientific theory and a summary of some probabilistic interpretations of grades of membership to be found in the literature the tee model for grades of membership is then suggested instead of starting out from mathematical postulates such as the assumed max and min operations for or and and the tee model uses a semantic physio logical psycho logical starting point by investigating the possibilities which a person has for assigning linguistic labels and partial grades of membership in a meaningful way the assignment of partial grades of membership is explained by the assumption that humans have the ability to take errors of observation and intersubject differences into consideration a membership value uex is identified with the subject s estimate of the probability that the label e g tall would be assigned to an object of exact attribute value uex in a natural language situation this situation being a labeling or a yes no one in which at least one of three possible sources of uncertainty is present almost all the accepted formulas of fuzzy set theory as well as the shapes of the grade of membership curves are then derived instead of having to be postulated however the derived formulas for or and and reduce to the max and min operations only in a number of special cases in another special case they reduce to the summation and the zero operation respectively the generally valid formulas are presented

existence and approximation of robust solutions of variational inequality problems over polytopes we study nonlinear variational inequality problems over polytopes from a viewpoint of stability and propose a new solution concept extending an earlier concept proposed by yang z yang siam j control optim 34 1996 pp 491 506 on the unit simplex we will introduce the concept of the robust stationary point which is a refinement of the concept of the stationary point though a stationary point need not be robust it is shown that every continuous function on a polytope has a robust stationary point we develop a simplicial algorithm to compute a robust stationary point of a continuous function on a polytope the algorithm can be briefly stated as follows starting with any point in the relative interior of a polytope the algorithm generates a piecewise linear path which leads to an approximate robust stationary point of any a priori chosen accuracy within a finite number of steps moreover we also discuss several numerical examples and apply the new concept to noncooperative games and economic equilibrium problems

continuity and identity laws in the axiomatic basis of fuzzy set operations abstract t norms and t cornorms are considered as the suitable axiomatic basis for the aggregate functions used for the definition of fuzzy set intersection and union operations when the domain of the function is infinite as in most cases continuity has also to be included as an axiom this paper proves that the identity law axioms in t norm and t cornorm are implied by continuity and the other axioms a simple example is given to show that in the absence of continuity the identity law condition is necessary for a plausible axiomatic basis of fuzzy set operations we also proved that the only idempotent t norm t cornorm functions is min max

multi peak solutions for nonlinear choquard equation in the plane in this paper we consider the nonlinear choquard equation in the plane with general nonlinearity satisfying berestycki lions condition by means of global penalization method we obtain the existence of multi peak solutions which concentrate near local minimal points of potential in this paper we consider the nonlinear choquard equation in the plane with general nonlinearity satisfying berestycki lions condition by means of global penalization method we obtain the existence of multi peak solutions which concentrate near local minimal points of potential

the concept of grade of membership abstract the concept of grade of membership is bound up with the general problem of fuzzy reasoning in this paper an approach to the latter problem is outlined and the consequences for the notion of grade of membership are described particular care is taken from the beginning to base the theory on concepts which have an immediate practical meaning for this reason assertions rather than sentences are taken as fundamental an assertion being characterized in terms of utility and decision theory fuzzy sentences and degrees of truth and then fuzzy sets and grades of membership are defined in terms of assertions the definitions being illustrated with examples the problem of defining unary and binary operations connectives for assertions sentences and fuzzy sets is discussed finally some other aspects of the logic of assertions are outlined and the extent to which the concepts of fuzzy set and grade of membership can be incorporated into the theory is discussed

a simplicial algorithm for computing robust stationary points of a continuous function on the unit simplex a simplicial algorithm is proposed to compute a robust stationary point of a continuous function f from the n 1 dimensional unit simplex s n 1 into r n the concept of robust stationary point is a refinement of the concept of stationary point of f on s n 1 starting from an arbitrarily chosen interior point v in s n 1 the algorithm generates a piecewise linear path of points in s n 1 this path is followed by alternating linear programming pivot steps and replacement steps in a specific simplicial subdivision of the relative interior of s n 1 in this way an approximate robust stationary point of any a priori chosen accuracy is reached within a finite number of steps the algorithm leaves the starting point along one out of n rays when the path approaches the boundary of s n 1 the mesh size of the triangulation along the path automatically goes to zero this makes the algorithm different from all simplicial restart algorithms and homotopy algorithms known so far roughly speaking the algorithm is a blend of a restart and a homotopy algorithm and maintains the basic properties of both however the algorithm does not need an extra dimension as homotopy algorithms do some examples are discussed

an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i case of type 1 fuzzy sets abstract the grade of membership is assumed to be determined as the relative size of evidence grounds supporting a given element s membership compared to all the necessary reference grounds relevant to the judgement of membership grade and is interpreted as conditional probability conditioned by the given element by assigning the dependence relations of the evidence grounds to those of the fuzzy subsets the set operators are defined as probabilistic operators with dependence relations the properties of the operators are studied and it is found that with type 1 fuzzy subsets they satisfy all the basic properties satisfied by the crisp set operators illustrative examples are given describing the construction of membership functions and the selection of set operators in relation to the situations to be modelled the proposed operators a kind of parametrized operators with an index value of the dependence relations as a parameter cover the entire operators ranging from the so called max min operators to the bold union intersection operators an extension of the above operator range is also discussed

fuzzy linear regression and its applications to forecasting in uncertain environment abstract linear regression has been used for many years in developing mathematical models for application in marketing management and sales forecasting in this paper two different sales forecasting techniques are discussed the first technique involves non fuzzy abstract methods of linear regression and econometrics a study of the international market sales of cameras done in 1968 by john scott armstrong utilized these non fuzzy forecasting techniques the second sales forecasting technique uses fuzzy linear regression introduced by h tanaka s uejima and k asai in 1980 in this paper a study of the computer and peripheral equipment sales in the united states is discussed using fuzzy linear regression moreover fuzzy linear regression is applied to forecasting in an uncertain environment finally some possible improvements and suggestions for further study are mentioned

a measure theoretic axiomatization of fuzzy sets abstract using measurement theory this paper examines three empirical structures that underlie the representation of fuzzy sets the fuzzy membership structure the fuzzy component structure and the fuzzy system structure these qualitative structures justify the use of the standard min max system to represent fuzzy sets the results of this study facilitate the development of a sound measurement theoretic axiomatization of fuzzy systems

membership function as an evaluation abstract after giving an overview of the different kinds of mathematical forms of the membership functions we extract the different demands and determine the rational class of the membership functions from the generalisation of our result we can determine it giving four parameters the internal a b the sharpness and the decision level we show the correspondence of this membership function to the zimmermann zysno model and using their empirical data we also test our mathematical model we also show the connections between the operators of the evaluation and membership functions which gives the generalisation of this concept in a more general form as a special case we can get a linear form

fuzzy sets and vague environments abstract in this paper we propose a natural approach to handle imprecise numbers as they arise for example from measurements fuzzy sets turn out to be a canonical representation for such imprecise numbers that are induced by taking different tolerance or error bounds into account fuzzy sets are induced by scaling factors that describe the magnitude of the imprecision on the other the scaling factors can be derived from given fuzzy sets so that we have a correspondence between scaling factors and fuzzy sets when these concepts are applied to control problems the max min rule is rediscovered as an interpolation technique viewing fuzzy control as an interpolation technique in vague environments enables us to validate various concepts for the design and tuning of fuzzy controllers and also suggests new methods based on clear semantics

fuzzy p measure on the real line abstract the fuzzy measurable space on the real line say is defined by means of fuzzy relation less or equal 5 the main properties of are given these results are used for investigation of fuzzy p measure on the principal emphasis is laid on the connections between a fuzzy p measure and its cumulative distribution function a theorem extending a fuzzy p measure to is shown

reconstruction possibilities for fuzzy sets towards a new level of understanding abstract using the methodological approach of rational reconstruction some possible explications of the notion of fuzzy set are critically discussed with respect to their intuitive meaning plausibility and consistency together with a historical survey we try to provide a better understanding of the development of various concepts within traditional fuzzy set theory based on the approaches of 39 and 18 we show that zadeh s original intuition of a fuzzy set and its formalization as generalized membership function can be reinterpreted in a plausible and consistent way and thus are supported not only on a formal level by mathematical results but can also be justified in principle on a semantical level hence both the operational as well as the traditional valuational viewpoint have their own rights in their corresponding areas of modeling reality and it is shown how the latter can be given a sound foundation based upon the former

designing fuzzy sets with the use of the parametric principle of justifiable granularity this study is concerned with a design of membership functions of fuzzy sets the membership functions are formed in such a way that they are experimentally justifiable and exhibit a sound semantics these two requirements are articulated through the principle of justifiable granularity the parametric version of the principle is discussed in detail we show linkages with type 2 fuzzy sets which are constructed on a basis of type 1 fuzzy sets several experimental studies are reported which illustrate a behavior of the introduced method

fuzzy hypothesis testing for a population proportion based on set valued information abstract in this paper we propose fuzzy hypothesis testing for a proportion with crisp data as the exact generalized one tailed hypergeometric test with fuzzy non complementary hypotheses and incorporate the conventional hypergeometric test with crisp hypotheses into its framework in particular we formulate hypotheses as ontic or epistemic fuzzy sets since we assume that presumptions regarding the true population proportion can be expressed in the conjunctive or disjunctive reading further we suggest modeling of membership functions in terms of cost or frequency due to considerations of user s priorities or incomplete knowledge in relation to hypotheses formulation considering a hypothesis as union of its crisp and fuzzy areas we illustrate via real life examples that in contrast to classical test theory fuzzy hypothesis testing provides an additional partial and gradual consideration of the indifference zone for both complementary and non complementary hypotheses the test concept is introduced for both test methods a test of significance and an alternative test and leads to a crisp test decision considering further cost aspects like the sample size the generalized error criteria are derived and interpreted in relation to both hypotheses additionally a corresponding sensitivity analysis is presented for one of the most promising test types

possibility versus probability the main objective of this note is to try to indicate that a possibility distribution at least in some applications can be interpreted as a family of probabilities possibly subjective an advantage of the latter interpretation is that subsequent derivations can be made by well proven probabilistic arguments rather than relying on an intuition concerning the properties of possibility distributions which is not well developed multistate reliability theory is given as an example of an area of application where the probabilistic approach seems far superior

multi peak solution for nonlinear magnetic choquard type equation in this paper we study a class of nonlinear magnetic choquard type equation involving a magnetic potential and nonlocal nonlinearities by using the method of penalization argument we show that there exists a family of solutions having multiple concentration regions which are concentrate at the minimum points of potential v

refinements of stationary points with applications to noncooperative games and economics it is well known that any continuous function f defined on a nonempty compact and convex set x has a stationary point in many circumstances there may exist multiple stationary points and some of them may be undesirable from the viewpoint of stability in this paper we introduce a new method of eliminating those undesirable stationary points while at the same time retaining some desirable stationary points the main idea of refining the concept of stationary point is to perturb simultaneously both the domain set x by taking a sequence of sets in the relative interior of x converging to x and the solution concept by replacing the concept of stationary point by a coincidence point with some well defined mapping if a stationary point is the limit of a sequence of coincidence points we say that the stationary point is stable with respect to this sequence of subsets of x and the coincidence mapping it is shown that stable stationary points exist for a large class of perturbations a stable point is said to be normal stable if we take the normal cone as the coincidence mapping implying that any coincidence point on a subset in the sequence is a stationary point of f on this subset it is shown that a normal stable stationary point always exists for any sequence of subsets which starts from an interior point and converges to x in a continuous way special cases of normal stability are perfect stationary points and robust stationary points in addition several practical applications of these new concepts are provided

additional criteria in fuzzy two level pattern recognition abstract pattern recognition with fuzzy sets is considered with regard to medical diagnostic problems an alternative interpretation of membership grade is suggested which expresses the degree of illness of a patient with respect to a certain disease a formal statement of a two level fuzzy classifier is presented the main emphasis is put on additional criteria designed to evaluate a fuzzy diagnostic decision for an unknown object two groups of criteria are formed decision making connected and domain dependent the decision making group of criteria can be viewed as related to the soft consensus measure in group decisions theory interconnections between criteria are discussed the notions proposed are illustrated with real medical data from the fields of aviation medicine and neonatology

fuzzy set connectives as combinations of belief structures abstract consonant belief structures provide a representation for fuzzy sets owing to the fact that their plausibility measures are essentially possibility measures we note that two belief structures are equivalent if their plausibility and belief functions are equal this observation leads us to provide a multiple number of equivalent representations for any belief structure commensurate representations can be induced for two different belief structures by forcing the same number of focal elements with the same weights we show that if we represent two consonant belief structures in a commensurate manner then their aggregations are closed with respect to consonance provided that the additional requirement that the underlying probability distributions satisfy a condition of correlation is imposed the results of this work allow us to use belief structure representations for the manipulation of fuzzy subsets under various logical combinations all basic fuzzy set connectives can thus be interpreted in the framework of the theory of evidence

a framework for linguistic modelling a new framework for linguistic reasoning is proposed based on a random set model of the degree of appropriateness of a label labels are assumed to be chosen from a finite predefined set of labels and the set of appropriate labels for a value is defined as a random set valued function from a population of individuals into the set of subsets of labels appropriateness degrees are then evaluated relative to the distribution on this random set where the appropriateness degree of a label corresponds to the probability that it is contained in the set of appropriate labels this interpretation is referred to as label semantics a natural calculus for appropriateness degrees is described which is weakly functional while taking into account the logical structure of expressions given this framework it is shown that a bayesian approach can be adopted in order to infer probability distributions on the underlying variable given constraints both in the form of linguistic expressions and mass assignments in addition two conditional measures are introduced for evaluating the appropriateness of a linguistic expression given other linguistic information

variational inequality problems with a continuum of solutions existence and computation in this paper three sufficient conditions are provided under each of which an upper semicontinuous point to set mapping defined on an arbitrary polytope has a connected set of zero points that connect two distinct faces of the polytope furthermore we obtain an existence theorem of a connected set of solutions to a nonlinear variational inequality problem over arbitrary polytopes these results follow in a constructive way by designing a new simplicial algorithm the algorithm operates on a triangulation of the polytope and generates a piecewise linear path of points connecting two distinct faces of the polytope each point on the path is an approximate zero point as the mesh size of the triangulation goes to zero the path converges to a connected set of zero points linking the two distinct faces as a consequence our results generalize browder s fixed point theorem summa brasiliensis mathematicae 4 1960 pp 183 191 and an earlier result by the authors math oper res 21 1996 pp 675 696 1996 on the n dimensional unit cube an application in economics and some numerical examples are also discussed

a behavioral model for linguistic uncertainty abstract the paper discusses the problem of modelling linguistic uncertainty which is the uncertainty produced by statements in natural language for example the vague statement mary is young produces uncertainty about mary s age we concentrate on simple affirmative statements of the type subject is predicate where the predicate satisfies a special condition called monotonicity for this case we model linguistic uncertainty in terms of upper probabilities which are given a behavioural interpretation as betting rates possibility measures and probability measures are special types of upper probability measure we evaluate zadeh s suggestion that possibility measures should be used to model linguistic uncertainty and the bayesian claim that probability measures should be used our main conclusion is that when the predicate is monotonic possibility measures are appropriate models for linguistic uncertainty we also discuss several assessment strategies for constructing a numerical model

singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities abstract we consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities the dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect we investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function necessary conditions to locate the possible concentration points are also given

a design for single sampling attribute plan based on fuzzy sets theory abstract problems in the design of sampling inspection plans have been studied for a long time as an important subject in quality control especially in the case of a single sampling attribute plan one would like the producer s risk to equal and the consumer s risk to equal exactly but this is usually not possible since the sample size n and the acceptance number c must be integers in this paper a new design procedure for the single sampling attribute plan based on fuzzy sets theory is presented by means of formulating this problem as a fuzzy mathematical programming one

fuzzy reliability assessment of systems with multiple dependent competing degradation processes components are often subject to multiple competing degradation processes for multicomponent systems the degradation dependence within one component or and among components need to be considered physics based models and multistate models are often used for component degradation processes particularly when statistical data are limited in this paper we treat dependence between degradation processes within a piecewise deterministic markov process pdmp modeling framework epistemic subjective uncertainty can arise due to the incomplete or imprecise knowledge about the degradation processes and the governing parameters to take this into account we describe the parameters of the pdmp model as fuzzy numbers then we extend the finite volume method to quantify the fuzzy reliability of the system the proposed method is tested on one subsystem of the residual heat removal system of a nuclear power plant and a comparison is offered with a monte carlo simulation solution the results show that our method can be most efficient

fuzzy sets and membership functions based on probabilities abstract since the introduction of fuzzy set theory in 1965 several attempts to establish the relationship between the grades of membership and the classical probability measures have been made it turns out that there are different sources of fuzziness that must be dealt with differently in the present paper we examine in detail two types of fuzziness namely the fuzziness due to classification in an under or overdimensioned universe and the fuzziness due to the intersubject differences in opinion for the former case the membership function is defined to be equal to the normalized distance from the point to the boundary of the set in a specific metric it is shown that this definition of the membership function is fully consistent with the max min operations for the union intersection however the membership function of the complement is defined differently from the usual one minus rule the fuzziness due to the intersubject differences turns out to be a simple averaging process and the explicit formulas for this case were derived several examples that illustrate the notions of fuzzy intervals and fuzzy numbers are given and the interpretations of the derived membership curves are presented the formulas for calculation of the membership function of a sum of fuzzy numbers and of a product of fuzzy numbers times a constant are derived an extended definition of the measure of fuzziness is presented and applied to the defined membership functions

conditional probability fuzzy sets and possibility a unifying view we propose an interpretation of fuzzy set theory both from a semantic and a syntactic point of view in terms of conditional events and coherent conditional probabilities during past years a large number of papers has been devoted to support either the thesis that probability theory is all that is required for reasoning about uncertainty or the negative view maintaining that probability is inadequate to capture what is usually treated by fuzzy theory in this paper we emphasize the role of conditioning in a proper framework i e de finetti s coherence to get rid of many controversial aspects moreover we introduce suitable operations between fuzzy subsets looked on as corresponding operations between conditional events endowed with the relevant conditional probability finally we show how the concept of possibility function naturally arises as a coherent conditional probability

fuzzy semi markov migration process in credit risk we explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory we prove that a fuzzy market is viable if and only if an equivalent martingale measure exists from which we construct the forward probability measure and under which the discounted price of a default free bond is a martingale we model the evolution of credit migration of a defaultable bond as an inhomogeneous semi markov process with fuzzy states we study the effects of changing the real probability measure to a forward probability measure in addition we investigate the asymptotic behaviour of the survival probability in each fuzzy state given in the absence of default finally we discuss parameter estimation and calibration of the inhomogeneous markov chain with fuzzy states

a behavioral foundation for fuzzy measures in savage 41 a behavioral foundation was given for subjective probabilities to be used in the maximization of expected utility this paper analogously gives a behavioral foundation for fuzzy measures to be used in the maximization of choquet expected utility this opens the way to empirical verification or falsification of fuzzy measures and frees them of their ad hoc character

on direct construction of fuzzy systems abstract in this paper we show that the fuzzy logic system presented by wang and mendel 1992 is a linear expansion of the fuzzy rule base if fuzzy events are represented by triangular basis functions therefore based on linear regression theory we propose a direct approach for constructing a fuzzy logic system from input output data the effectiveness of this approach is demonstrated by simulations

fuzzy logic and the quality of assessment of portfolios abstract in this paper it is argued that competence like height is a fuzzy concept this paper attempts to show the advantages of using fuzzy logic in the domain of assessment of prior learning apl via portfolios of evidence this issue which presently is of great concern to vocational educators highlights the need for defining some procedure in order to rationalize decisions made by assessors dealing with such type of evidence the assessment of portfolios are very much perceived presently as the way forward in modern educational assessment as portfolios do not only address evidence for past achievement but also evidence for current achievement thus highlighting the relevance of fuzzy logic in this area may raise awareness among educators of the benefits of such methodology the paper argues that fuzzy logic could offer useful insight in the attempt to rationalize assessor s deliberations when facing portfolio types of evidence the paper also tried to refine the fuzzy logic methodology perceived as most adequate for portfolio assessment as well as raise some issues that need addressing the examples referred to are based on the british national council for vocational qualifications ncvq apl procedures

fuzzy linear systems a general fuzzy linear system is investigated using the embedding approach conditions for the existence of a unique fuzzy solution to n n linear system are derived and a numerical procedure for calculating the solution is designed the applicability of the proposed model is illustrated with examples

analytical fuzzy plane geometry iii in this study we attempt to construct fuzzy circles in a fuzzy geometrical plane we provide a comprehensive study where we find a fuzzy number with a predetermined fuzzy distance from a given fuzzy number we formulate fuzzy circles using the conventional basic definitions of crisp circles two different formulations of fuzzy circles are proposed which depend on the information known about a fuzzy circle the interrelations between the evaluated fuzzy circles are investigated we show that the center of a fuzzy circle may not be a fuzzy point however the radius of a fuzzy circle that passes through three given fuzzy points is always a fuzzy number the concepts of same and inverse points and a fuzzy number along a line are used to define and analyze the proposed ideas our discussions and studies are supported by suitable numerical and pictorial examples

measurement of membership functions and their acquisition abstract three basic views of the representation of membership functions are reviewed together with fundamental measurement of linguistic terms of linguistic variables the conclusion is that such measurements are either on an ordinal or an interval scale based on whether the appropriate axioms are validated by the empirical data with an allowance for stochastic variation the conjoint measurement is introduced for the case of multi dimensional linguistic variables whose linguistic terms are compositions of two or finitely many more component linguistic terms of distinct independent linguistic variables it is shown that any composition of distinct independent component linguistic terms of component linguistic variables by any t norm or s norm or any finite convext linear combination preserves the monotonic weak order property of the components in the composite once the scale properties of the measurement values of particular terms are validated composition procedures may be applied to the experimental data to obtain compound membership functions of fuzzy sets induced by meaningful representations of compositions of linguistic terms of linguistic variables finally four methods of membership acquisition and construction are reviewed from the perspective of fundamental measurement

interpretative versus prescriptive fuzzy set theory both traditional fuzzy set theory and the theory of subjective probabilities postulate their formulas the former because it does not accept a probabilistic interpretation of grades of membership the latter because it makes no connection between subjective probabilities and relative frequencies the interpretational theory of the tee model ascribes a well defined meaning to a membership value spl mu sub spl lambda in terms of probabilities which are limits of frequencies spl mu sub spl lambda is interpreted as the estimate by the subject of the probability that a given object would be assigned the label x in an everyday situation of uncertainty in contrast to the max min formulas used in many applications of fuzzy sets the postulated summation to 1 formula of all fuzzy clustering algorithms follows from the interpretative tee model this agrees with the author s view that fuzzy set theory should be defined as a theory which allows partial membership values of an object in a class or cluster not as a theory which uses specific mathematical operators the operators must be derived from the well defined interpretation of partial membership values

existence and concentration result for the kirchhoff type equations with general nonlinearities in this paper we study the existence and concentration behaviors of positive solutions to the kirchhoff type equations varepsilon 2 m left varepsilon 2 n int mathbf r n nabla u 2 mathrm d x right delta u v x u f u quad rm in mathbf r n quad u in h 1 mathbf r n n geqq 1 where m and v are continuous functions under suitable conditions on m and general conditions on f we construct a family of positive solutions u varepsilon varepsilon in 0 tilde varepsilon which concentrates at a local minimum of v after extracting a subsequence ek

interval regression analysis by quadratic programming approach when we use linear programming in possibilistic regression analysis some coefficients tend to become crisp because of the characteristic of linear programming on the other hand a quadratic programming approach gives more diverse spread coefficients than a linear programming one therefore to overcome the crisp characteristic of linear programming we propose interval regression analysis based on a quadratic programming approach another advantage of adopting a quadratic programming approach is to be able to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression by changing the weights of the quadratic function we can analyze the given data from different viewpoints for data with crisp inputs and interval outputs the possibility and necessity models can be considered therefore the unified quadratic programming approach obtaining the possibility and necessity regression models simultaneously is proposed even though there always exist possibility estimation models the existence of necessity estimation models is not guaranteed if we fail to assume a proper function fitting to the given data as a regression model thus we consider polynomials as regression models since any curve can be represented by the polynomial approximation using polynomials we discuss how to obtain approximation models which fit well to the given data where the measure of fitness is newly defined to gauge the similarity between the possibility and the necessity models furthermore from the obtained possibility and necessity regression models a trapezoidal fuzzy output can be constructed

the efficacy of fuzzy representations of uncertainty advocates of the theory of fuzzy sets as a system for representing uncertainty have based their case on five basic arguments these are 1 the reality hypothesis which holds that imprecision is an inherent property of the world external to an observer 2 the subjectivity hypothesis which holds that probability is an exclusively objective measure of uncertainty and that therefore subjective uncertainty can only be represented with fuzzy sets 3 the behaviorist hypothesis which claims that uncertainty systems should emulate rather than prescribe human behavior in the face of uncertainty 4 the probability as fiction hypothesis which claims that probability does not comprise a field of study in its own right and 5 the superset hypothesis which holds that fuzzy set theory includes probability as a special case and thus provides a richer uncertainty modeling environment we discuss and criticize all five we then criticize the argument that fuzziness represents a type of uncertainty distinct from probability and also the inordinate complexity of fuzzy methods we present a method for assessing the efficacy of fuzzy representations of uncertainty and apply this method in three examples

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

the probability monopoly probability is a very special case of fuzziness it always faces two limits first it works with bivalent sets a second probability measures need small infinities a probability measure maps the sets in a sigma algebra to the unit interval spl lsqb o 1 spl rsqb fuzzy theory challenges the probability monopoly probabilists have attacked it with gusto to keep their monopoly status to have the only uncertainty theory in the unit interval spl lsqb o 1 spl rsqb but the fuzzy math is sound its world view of shades of gray has a deep intuitive ring and the new fuzzy products have come into their own in the marketplace

comparison of empirical and computed values of fuzzy conjunction abstract the necessity for further experiments such as in 4 for the foundation of fuzzy sets theory is shown generality of the experimental results on fuzzy conjunction in the paper by thole zimmermann and zisno 1979 4 for the other objects and language is verified for other conditions usually the minimum the product and the restricted sum operators are used as an intersection conjunction of fuzzy sets in fuzzy optimization f optimization on the basis of experiments it is shown that these operators are correct for f optimization their correctness is founded on the stability of the extremal point a decision in bellman zadeh s scheme under a positive linear mapping it is shown that these mappings exist between empirical conjunction and considered operators as a statistical result of linear regression analysis

extended support vector interval regression networks for interval input output data in many applications it is natural to use interval data because of uncertainty existence in the measurements variability in defining terms such as the temperature during a given day description for extremely behavior such as the maximum wind speed in a given area etc in order to handle such interval data a novel approach called the interval support vector interval regression networks isvirns is proposed the isvirns is extended from our previous work the support vector interval regression networks svirns it is easy to find that svirns can handle interval output data but for input data they must be crisp in this study the hausdorff distance is employed as the distance measure of interval data and is incorporated into the kernel functions of svirns to determine the initial structure of isvirns because the proposed approach can provide a better initial structure for isvirns it can have a faster convergent speed the experimental results with real data sets show the validity of the proposed isvirns

uncertainty modelling for vague concepts a prototype theory approach an epistemic model of the uncertainty associated with vague concepts is introduced label semantics theory is proposed as a framework for quantifying an agent s uncertainty concerning what labels are appropriate to describe a given example an interpretation of label semantics is then proposed which incorporates prototype theory by introducing uncertain thresholds on the distance between elements and prototypes for description labels this interpretation naturally generates a functional calculus for appropriateness measures a more general model with distinct threshold variables for different labels is discussed and we show how different kinds of semantic dependence can be captured in this model

structural and functional quantization of vagueness abstract this paper proposes two methods of quantizing vague categories in terms of quantities whose identities have been either widely recognized in the fuzzy set theory literature or established by bayesian decision theory the methods are called structural quantization and functional quantization respectively the idea behind structural quantization is that for a comparative vague category like good scores there exists a preference ordering among the possible scores and how good a particular score is depends on how large the portion of examinees whose scores are lower than that score is functional quantization adopts the conclusion of bayesian decision theory that a rational agent s belief can be represented by a probability to quantize the term good score the method first assesses the two probabilities that respectively represent a rational agent s beliefs about john s score before and after learning that john has got a good score it then derives a membership function by comparing those two probabilities the equivalence of the two quantization methods is hypothesized the logic connectives not and and or are discussed in the setting of structural quantization while aggregation of information is addressed in the setting of functional quantization

higher level fuzzy numbers arising from fuzzy regression consider a fuzzy random variable y with expectation symbol b x where b is an unknown fuzzy number and an unknown real number for n observations yi xi there is a model yi b xi ei i 1 2 n where ei are fuzzy valued errors independently and identically distributed in some sense the aim is to obtain estimates of b when all fuzzy numbers are triangular and the ei are uniformly distributed maximum likelihood estimators emerge naturally as a special form of higher level fuzzy number thus mle can be interpreted as an estimation procedure which transforms randomness into higher levels of fuzziness

probabilities and fuzzy events an operational approach abstract this paper tries to give a redesign of a well established research area of traditional fuzzy set theory the notion of a probability of a fuzzy event as well as related results are reexamined and new definitions for them are given which are based on the operational viewpoint of f set theory and on some concepts of operational statistics a critical survey of existing concepts will show some advantages of the operational approach

unity and diversity of fuzziness spl minus from a probability viewpoint like the fields of probability and statistics fuzzy set theory is characterized by a variety of viewpoints adherents of fuzzy methods however consistently maintain that probability is not necessarily the optimal representation of uncertainty we rebut this view

a fuzzy c means variant for the generation of fuzzy term sets a fuzzy c means fcm variant is proposed for the generation of fuzzy term sets with overlap the proposed variant differs from the original mainly in two areas the first modification ensures that two end terms take the maximum and minimum domain values as their centers the second modification prevents the generation of non convex fuzzy terms that often occurs with the original algorithm the optimal number of terms and the optimal shape of the membership function associated with each term are determined based on the mean squared error criterion the exponential weight m used in the algorithm is found to greatly affect the shape of the membership function the effect of data size used for the generation of fuzzy terms is also discussed a generalized shaped function with a tunable parameter along with its complement is developed to fit all term sets generated by the fcm variant using various m values

a semantics for possibility theory based on likelihoods abstract in this paper a semantic basis for possibility theory based on likelihood functions is presented in some cases possibilities have been considered as approximations of shafer plausibility measures this approximation exchanges exactness of plausibility values for the simplicity of use of possibility values in this paper a different direction is followed possibility measures are considered as the supremum of a family of likelihood functions this is an exact interpretation not an approximation the minimum rule to combine possibility distributions is justified in this framework under general conditions conditions under which other rules can be applied are also studied

the generalized modus ponens considered as a fuzzy relation abstract the generalized modus ponens gmp is an inference rule in fuzzy logic with the following scheme if s1 is p1 then s2 is p2 but s1 is q1 hence s2 is q2 the objective is to determine q2 given p1 p2 and q1 thepredicate p1 in the antecedent of the if then rule is fuzzy hence the condition s1 is p1 to derive s2 is p2 is not strict therefore it should be possible to render a reasonable approximation of q2 when q1 diverges slightly from p1 a number of criteria for this inference rule will be presented in literature there are three basic assumptions for dealing with the gmp namely 1 the if then rule is implemented by an implication rule from some multivalued logic i e the if then rule is represented by a fuzzy relation 2 the premiss p1 can be strengthened or weakened to obtain q1 by taking a power of the membership function of p1 and 3 the conclusion q2 is derived by means of the max min composition rule it will be shown by a number of theorems that these three items are incompatible with the criteria furthermore it will be shown that it is arduous to construct an implication rule to meet the conditions 2 and 3 a way to restore this is to alter condition 2 instead of taking a power of the membership function it is also possible to change the parameters of this function but this also causes complications hence requirement 3 should be modified into some functional relation it will be shown how this can be accomplished in combination with the functions with parameters changed

evaluation of fuzzy linear regression models abstract fuzzy linear regression provides means for tackling regression problems lacking a significant amount of data for determining regression models and with vague relationships between the dependent variable and independent variables in the paper a two stage construction of a linear regression model is proposed using an enhancement of a minimal vagueness criterion already discussed in fuzzy regression analysis two different experiments are set up and used for evaluating both models the choice of the fuzzy regression model should be made in the context of a practical application

fuzzy linear regression models for assessing risks of cumulative trauma disorders abstract cumulative trauma disorders ctds are a major problem in industry and today s office environments especially for keyboard users since comprehensive knowledge about individual or a combination of risk factors contributing to ctds of the hand and forearm is still lacking developing methodologies to quantify relationships of risk factors is greatly needed the purpose of this research was to build fuzzy linear regression models to reveal the relationship of ctd risk factors to predict the injuries and to evaluate risk levels of individuals twenty seven keyboard users twenty two for model building and five for model validation and three ctd experts participated in the model building four fuzzy models were built corresponding to four risk categories and a final fuzzy linear model was established using ahp pairwise comparisons multicolinearity effects were addressed and a partial standard deviation scaling method was used to eliminate the effects from a methodological point of view fuzzy linear regression models provide useful insight into risk factors ctd relationship

from a gaussian mixture model to nonadditive fuzzy systems this work explores how a kind of probabilistic system namely the gaussian mixture model gmm can be translated to an additive fuzzy system we will prove the mathematical equivalence between the conditional mean of a gmm and the defuzzified output of a generalized fuzzy model gfm the relationship between a gmm and a gfm and the conditions for gmm to gfm translation will be made explicit in the form of theorems the work will then extend to special cases of the gfm specifically the mamdani larsen and takagi sugeno fuzzy models the possibility of reverse translation that is from a gfm to a gmm will also be discussed finally we will consider the generality of a gmm specifically how it can approximate other distribution functions

a probabilistic and statistical view of fuzzy methods fuzzy set theory has primarily been associated with control theory and with the representation of uncertainty in applications in artificial intelligence more recently fuzzy methods have been proposed as alternatives to traditional statistical methods in statistical quality control linear regression and forecasting among other areas we review some basic concepts of fuzzy methods point out some philosophical and practical problems and offer simpler alternatives based on traditional probability and statistical theory applications in control theory and statistical quality control serve as our primary examples

singular perturbed kirchhoff type problem with critical exponent abstract in the present paper we consider the following kirchhoff type problem with critical exponent e 2 a e b r 3 u 2 d x u v x u f u in r 3 u h 1 r 3 by variational methods we show the existence of the positive solutions concentrating around global minima of the potential v x as e 0 we do not need the monotonicity of the function f 3

erratum standing waves for nonlinear schrodinger equations with a general nonlinearity erratum arch rational mech anal doi 10 1007 s00205 006 0019 3through answering to some questions from joao marcos bezzerra do oon the paper above we found some mistakes in the paper we thank him forhis interest and careful reading here we correct these mistakes first at the beginning of the proof of proposition 4 14 17 line on page193 the correct version is by compactness of s

the fuzzy gmdh algorithm by possibility models and its application abstract it is the purpose of this paper to formulate the group method of data handling gmdh by the possibility model the gmdh formulated by the possibility model is called the fuzzy gmdh to formulate the fuzzy gmdh we discuss possibilistic linear regression by the lp problem called min problem in the fuzzy gmdh the estimated model by using possibilistic linear regression with a multilayer procedure is a non linear system with parameters in the form of fuzzy numbers the estimation by the fuzzy gmdh can be obtained in fuzzy numbers thus this approach can be regarded as fuzzy interval analysis

a lexicographic algebraic theorem and its applications abstract in this paper we prove the following result given any full dimensional simple polytope p x r n a i t x b i i 1 m without redundant constraints and any vector c r n there exists a unique vertex x of p such that the matrix 1 1 1 1 c a i 1 a i 2 a i n 1 exists and is lexicographic positive where x is the solution of the n linear equations a i t t x b i t t 1 n this theorem generalizes a result on the n dimensional cube and can be used to resolve the degeneracy problem for simplicial fixed point algorithms we also discuss several applications for this result

plane answers to complex questions the theory of linear models providing a wide ranging introduction to the use of linear models in analyzing data this text presents a vector space and projections approach to the subject the topics covered include anova estimation hypothesis testing multiple comparison regression analysis and experimental design also covered are testing for lack of fit models with singular covariance matrices variance component estimation best linear prediction colinearity and variable selection

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

the three semantics of fuzzy sets abstract three main semantics for membership functions seem to exist in the literature similarity preference and uncertainty each semantics underlies a particular class of applications similarity notions are exploited in clustering analysis and fuzzy controllers uncertainty is captured by fuzzy sets in the framework of possibility theory the membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems this paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance cost and frequency on which scientific traditions exist

a hybrid fuzzy statistical clustering approach for estimating the time of changes in fixed and variable sampling control charts control charts are the most popular statistical process control spc tools used to monitor process changes when a control chart produces an out of control signal it means that the process has changed however control chart signals do not indicate the real time of the process changes which is essential for identifying and removing assignable causes and ultimately improving the process identifying the real time of the process change is known as change point estimation problem most of the traditional change point methods are based on maximum likelihood estimators mle which need strict statistical assumptions in this paper first we introduce clustering as a potential tool for change point estimation next we discuss the challenges of employing clustering methods for change point estimation afterwards based on the concepts of fuzzy clustering and statistical methods we develop a novel hybrid approach which is able to effectively estimate change points in processes with either fixed or variable sample size using extensive simulation studies we also show that the proposed approach performs considerably well in all considered conditions in comparison to powerful statistical methods and popular fuzzy clustering techniques the proposed approach can be employed for processes with either normal or non normal distributions it is also applicable to both phase i and phase ii finally it can estimate the true values of both in and out of control states parameters

semiclassical analysis for pseudo relativistic hartree equations abstract in this paper we study the semiclassical limit for the pseudo relativistic hartree equation e 2 m 2 u v u i u p u p 2 u in r n where m 0 2 p 2 n n 1 v r n r is an external scalar potential i x c n x n is a convolution kernel c n is a positive constant and n 1 p n n for n 3 p 2 our equation becomes the pseudo relativistic hartree equation with coulomb kernel

detection and characterization of cluster substructure i linear structure fuzzy c lines in part i a generalization of the fuzzy c means or fuzzy isodata clustering algorithms is developed necessary conditions for minimization of a generalized total weighted squared orthogonal error objective function lead to a picard iteration scheme which generates simultaneously i c fuzzy clusters in the data ii a set of c prototypical straight lines in feature space which best fit the data in a well defined sense iii a set of c prototpyical centers of mass on the c lines which characterize the core of each linear fuzzy cluster theoretical optimization is achieved using principal components of generalized within cluster fuzzy scatter matrices a convergence theorem for each algorithm in the infinite family is given the algorithms are exemplified by five numerical examples using both real and artificial data sets having essentially linear substructure in part ii the fuzzy c means and fuzzy c lines algorithms are shown to be special cases of a more general class of fuzzy algorithms the

fuzzy probability space defined by means of the fuzzy relation less than abstract in this paper a fuzzy relation less than and fuzzy probability spaces on the real line are defined fuzzy algebras of events are introduced in an analogous way to the classical theory of probability by means of the above relation furthermore specific properties of the fuzzy probability measures presented here are given

approximation of uncertain functional relationships in this paper we consider methods for the linguistic modelling of vague functional relationships where the linguistic model is supposed to be given as a set of rules two approaches for deriving a fuzzy valued function from such models are proposed it is argued that both approaches can be interpreted as a generalization of ordinary methods for the interpolation and approximation of functions mathematical properties of fuzzy functions associated with a set of rules are investigated particularly for the second approach a universal approximation property is shown

supplement to an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i abstract basic idea of part i of an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators 1992 mainly on the conceptual construction of membership functions is straightened out and illustrated by detailed examples the purpose of this article is to clarify unclear points in notations and concepts of mabuchi 1992 as a step toward the extension of the theory from type 1 to three valued fuzzy sets

when upper probabilities are possibility measures abstract a characteristic property is given for a pair of upper and lower probabilities induced by lower probability bounds on a finite set of events to coincide with possibility and necessity measures approximations of upper probabilities by possibility measures are discussed the problem of combining possibility distributions viewed as upper probabilities is investigated and the basic fuzzy set intersections are justified in this framework