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

a geometrical fuzzy clustering based solution to glottal wave estimation accurate estimation of the glottal waveform gw is required for purposes such as natural speech synthesis speaker recognition physiological speech processing etc most methods available for gw estimation are based on inverse filtering of the speech signal through the vocal tract and they all suffer from inaccuracies due to incorrect assumptions the method for gw estimation developed in the present study is based on fuzzy clustering of quasi linear geometrical substructures represented within the signal shifts hyperspace algorithms for estimation of the driving function to the vocal tract are presented and evaluated on simulated and real data comparison of the fuzzy clustering based method with the psiaif and wong s closed phase algorithms shows that the present method is superior with respect to both the gw estimation and determination of gw event time instants

a study of parameter values for a mahalanobis distance fuzzy classifier a supervised mahalanobis distance fuzzy classifier and the related fuzzy c means clustering algorithm requires the a priori selection of a weighting parameter called the fuzzy exponent guidance in the existing literature on an appropriate value is not definitive this paper attempts to rigorously justify previous experimental findings on suitable values for this fuzzy exponent using the criterion that fuzzy set memberships reflect class proportions in the mixed pixels of a remotely sensed image

technical communique identification of dynamical systems with a robust interval fuzzy model in this paper we present a new method of interval fuzzy model identification the method combines a fuzzy identification methodology with some ideas from linear programming theory on a finite set of measured data an optimality criterion that minimizes the maximal estimation error between the data and the proposed fuzzy model output is used the idea is then extended to modelling the optimal lower and upper bound functions that define the band that contains all the measurement values this results in a lower and an upper fuzzy model or a fuzzy model with a set of lower and upper parameters the model is called the interval fuzzy model infumo the method can be used when describing a family of uncertain nonlinear functions or when the systems with uncertain physical parameters are observed we believe that the fuzzy interval model can be very efficiently used especially in fault detection and in robust control design

fuzzy shell clustering algorithms in image processing fuzzy c rectangular and 2 rectangular shells objective function based clustering has been generalized recently to detect contours of circles and ellipses or even hyperbolas in a set of binary data vectors although there are special algorithms to discover lines the detection of rectangles needs further treatment a simple line detection algorithm is not sufficient for rectangles since for identifying four lines as one rectangle additional information such as the length of the lines and whether they are parallel or meet at a right angle is necessary in this paper a special fuzzy shell clustering algorithm for rectangular contours is developed the principal idea behind it can be generalized for other polygons so we also derive an algorithm that is capable of detecting rectangles and other polygons as well as approximating circles ellipses and lines

games with discontinuous payoffs a strengthening of reny s existence theorem we provide a pure nash equilibrium existence theorem for games with discontinuous payoffs whose hypotheses are in a number of ways weaker than those of the theorem of reny 1999 in comparison with reny s argument our proof is brief our result subsumes a prior existence result of reny 1999 that is not covered by his theorem we use the main result to prove the existence of pure nash equilibrium in a class of finite games in which agents pure strategies are subsets of a given set and in turn use this to prove the existence of stable configurations for games similar to those used by schelling 1971 1972 to study residential segregation in which agents choose locations

a parametric model for fusing heterogeneous fuzzy data presented is a model that integrates three data types numbers intervals and linguistic assessments data of these three types come from a variety of sensors one objective of sensor fusion models is to provide a common framework for data integration processing and interpretation that is what our model does we use a small set of artificial data to illustrate how problems as diverse as feature analysis clustering cluster validity and prototype classifier design can all be formulated and attacked with standard methods once the data are converted to the generalized coordinates of our model the effects of reparameterization on computational outputs are discussed numerical examples illustrate that the proposed model affords a natural way to approach problems which involve mixed data types

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

adaptive fuzzy c shells clustering and detection of ellipses several generalizations of the fuzzy c shells fcs algorithm are presented for characterizing and detecting clusters that are hyperellipsoidal shells an earlier generalization the adaptive fuzzy c shells afcs algorithm is examined in detail and is found to have global convergence problems when the shapes to be detected are partial new formulations are considered wherein the norm inducing matrix in the distance metric is unconstrained in contrast to the afcs algorithm the resulting algorithm called the afcs u algorithm performs better for partial shapes another formulation based on the second order quadrics equation is considered these algorithms can detect ellipses and circles in 2d data they are compared with the hough transform ht based methods for ellipse detection existing ht based methods for ellipse detection are evaluated and a multistage method incorporating the good features of all the methods is used for comparison numerical examples of real image data show that the afcs algorithm requires less memory than the ht based methods and it is at least an order of magnitude faster than the ht approach

measure free conditioning and extensions of additive measures on finite mv algebras using the well known representation of any finite mv algebra as a product of finite mv chains as factors we obtain a representation of its canonical extension as a girard algebra product of the canonical extensions of the mv chain factors based on this representation and using the results from our last paper we characterize the additive measures on any finite mv algebra resp the weakly and the strongly additive measures on its canonical girard algebra extension and that as convex combinations of the corresponding measures on the respective factors after that we apply the results to measure free defined conditional events which for this reason are considered as elements of the canonical extension of a finite mv algebra of events this definition for conditional events is naturally equivalent to the well known definition as lattice intervals of events here the weak additivity is considered as a reasonable notion for a global additivity property of a measure extension these extensions are essentially different from the measure extension which has a form analogous to the classical conditional probability but which is only levelwise additive moreover the measure free approach of this paper is completely different from the classical approach in which conditional measures of events given a fixed condition event are considered based on additive measures states of events without having defined conditional events

generalized fuzzy c shells clustering and detection of circular and elliptical boundaries abstract the fuzzy c shells fcs algorithm and its adaptive generalization called the adaptive fuzzy c shells afcs algorithm are considered for detection of curved boundaries specifically circular and elliptical the fcs algorithms utilize hyper spherical shells as cluster prototypes thus in two dimensions the prototypes are circles the afcs algorithms consider hyper ellipsoidal shells as prototypes hence the ability to characterize elliptical boundaries the generalization is achieved by allowing the distances to be measured through a norm inducing matrix that is symmetric positive definite each cluster is allowed to have a different matrix which is made a variable of optimization the ability of the algorithms to detect circular and elliptical boundaries in two dimensional data is illustrated through several examples

interval fuzzy model identification using l_infty norm in this paper we present a new method of interval fuzzy model identification the method combines a fuzzy identification methodology with some ideas from linear programming theory we consider a finite set of measured data and we use an optimality criterion that minimizes the maximum estimation error between the data and the proposed fuzzy model output the idea is then extended to modeling the optimal lower and upper bound functions that define the band which contains all the measurement values this results in lower and upper fuzzy models or a fuzzy model with a set of lower and upper parameters the model is called the interval fuzzy model infumo we also showed that the proposed structure uniformly approximates the band of any nonlinear function the interval fuzzy model identification is a methodology to approximate functions by taking into account a finite set of input and output measurements this approach can also be used to compress information in the case of large amount of data and in the case of robust system identification the method can be efficiently used in the case of the approximation of the nonlinear functions family if the family is defined by a band containing the whole measurement set the interval of parameters is obtained as the result this is of great importance in the case of nonlinear circuits modeling especially when the parameters of the circuits vary within certain tolerance bands

detection and separation of ring shaped clusters using fuzzy clustering a new fuzzy clustering algorithm designed to detect and characterize ring shaped clusters and combinations of ring shaped and compact spherical clusters has been developed this fkr algorithm includes automatic search for proper initial conditions in the two cases of concentric and excentric intersected combinations of clusters validity criteria based on total fuzzy area and fuzzy density are used to estimate the optimal number of substructures in the data set the fkr algorithm has been tested on a variety of simulated combinations of ring shaped and compact spherical clusters and its performance proved to be very good both in identifying the input shapes and in recovering the input parameters application of the fkr algorithm to an mri image of the heart s left ventricle was used to investigate the possibility of using this algorithm as an aid in image processing

geometrical fuzzy clustering algorithms fuzzy clustering algorithms are a basic tool for cluster analysis among these the geometrical fuzzy clustering algorithms are used when the clustering problem can be viewed as trying to find linear or ellipsoidal concentrations in data this paper provides a theoretical framework in which currently used geometrical fuzzy clustering algorithms become special cases also a family of functions called feasible are defined which can be used to construct such algorithms and convergence results are obtained

linear fuzzy clustering techniques with missing values and their application to local principal component analysis in this paper we propose two methods for partitioning an incomplete data set with missing values into several linear fuzzy clusters by extracting local principal components one is an extension of fuzzy c varieties clustering that can be regarded as the algorithm for the local principal component analysis of fuzzy covariance matrices the other is a simultaneous application of fuzzy clustering and principal component analysis of fuzzy correlation matrices both methods estimate prototypes ignoring only missing values and they need no preprocessing of data such as the elimination of samples with missing values or the imputation of missing elements numerical examples show that the methods provide useful tools for interpretation of the local structures of a database

uncertainty measures decomposability and admissibility abstract uncertainty measures are defined in a general manner as morphism between certain posets which permits a unified treatment of different measures for classical and for fuzzy events admissible measures are discussed in a decision theoretical notation making use of quasilinear entropies connections and differences to decomposable measures are illustrated

a clustering algorithm for fuzzy model identification the fuzzy model proposed by takagi and sugeno can represent highly nonlinear systems and is widely used for the representation of fuzzy rules in this paper the model is firstly modified to make its identification easier based on the fuzzy c partition space four criteria are proposed for optimization of the model parameters following that a clustering algorithm composed of fuzzy c linear functions clustering and like fuzzy c means clustering is developed for minimizing the four criteria an identification scheme for rule s premise and consequence parameters is deduced from the clustering algorithm in succession finally four examples are demonstrated to verify the effectiveness of the proposed algorithm

relational generalizations of cluster validity indices numerous computational schemes have arisen over the years that attempt to learn information about objects based upon the similarity or dissimilarity of one object to another one such scheme clustering looks for self similar groups of objects to use clustering algorithms an investigator must often have a priori knowledge of the number of clusters i e c to search for in the data moreover it is often convenient to have ways to rank the returned results either for a single value of c a range of c s different clustering methods or any combination thereof however the task of assessing the quality of the results so that c may be determined objectively is currently ill defined for object object relationships to bridge this gap we generalize three well known validity indices the modified hubert s gamma xie beni and the generalized dunn s indices to relational data in doing so we develop a framework to convert many other validity indices to a relational form numerical examples on 12 datasets samples from four normal mixtures four real world object datasets and four real world pure relational datasets using the relational duals of the hard fuzzy and possibilistic c means cluster algorithms are offered to illustrate and evaluate the new indices

a note on the equilibrium existence problem in discontinuous games in this note we prove an equilibrium existence theorem for games with discontinuous payoffs and convex and compact strategy spaces it generalizes the classical result of reny 1999 as well as the recent paper of mclennan monteiro and tourky 2011 our conditions are simple and easy to verify importantly examples of spatial location models show that our conditions allow for economically meaningful payoff discontinuities that are not covered by other conditions in the literature

relational duals of cluster validity functions for the c means family clustering aims to identify groups of similar objects to evaluate the results of cluster algorithms an investigator uses cluster validity indices while the theory of cluster validity is well established for vector object data little effort has been made to extend it to relationship based data as such this paper proposes a theory of reformulation for object data validity indices so that they can be used to rank the results produced by the relational means clustering algorithms more specifically we create a class of relational validity indices which is called dual relational indices that are guaranteed under certain but easily met constraints to produce the same results and hence the same cluster counts as their object data counterparts

continuity properties of the nash equilibrium correspondence in a discontinuous setting abstract we introduce a new complete metric space of discontinuous normal form games and prove that the nash equilibrium correspondence is upper semicontinuous with non empty and compact values so using the theorem of fort 1949 we obtain that the correspondence is also lower semicontinuous in a dense subset we introduce new topological assumptions on the payoff functions and a strengthening of standard quasi concavity properties examples show that our results cannot be obtained from the previous ones

local convergence of the fuzzy c means algorithms abstract much understanding has recently been gained concerning global convergence properties of the fuzzy c means fcm family of clustering algorithms these global convergence properties which hold for all iteration sequences guarantee that every fcm iteration sequence converges at least along a subsequence to a stationary point of an fcm objective function in this paper we prove a local convergence property that is a property pertaining to iteration sequences started near a solution specifically a simple result is proved which shows that whenever an fcm algorithm is started sufficiently near a minimizer of the corresponding objective function then the iteration sequence must converge to that particular minimizer the result guarantees that once captured by the local neighborhood of a minimizer the succeeding iterate sequence will not escape thus infinite oscillation of such a sequence cannot occur the rate of convergence of the sequence to such a point is also discussed

estimation of multiple accelerated motions using chirp fourier transform and clustering motion estimation in the spatiotemporal domain has been extensively studied and many methodologies have been proposed which however cannot handle both time varying and multiple motions extending previously published ideas we present an efficient method for estimating multiple linearly time varying motions it is shown that the estimation of accelerated motions is equivalent to the parameter estimation of superpositioned chirp signals from this viewpoint one can exploit established signal processing tools such as the chirp fourier transform it is shown that accelerated motion results in energy concentration along planes in the 4 d space spatial frequencies temporal frequency chirp rate using fuzzy c planes clustering we estimate the plane motion parameters the effectiveness of our method is verified on both synthetic as well as real sequences and its advantages are highlighted

a fuzzy based customer classification method for demand responsive logistical distribution operations in some cases customer classification is important for the development of advanced logistical distribution strategies in response to the growing complexity in business logistical markets this paper presents a new approach that can be employed to cluster customers before executing fleet routing in logistical operations the proposed approach is developed on the basis of fuzzy clustering techniques and involves three sequential mechanisms including 1 binary transformation 2 generation of a fuzzy correlation matrix and 3 customer clustering such a customer clustering method should be performed prior to vehicle dispatching and routing in the process of goods distribution the proposed methodology clusters customers on the basis of their demand attributes rather than the static geographic property which is considered extensively in most published vehicle routing algorithms in addition to methodology development a case study was conducted to demonstrate the potential advantages of the proposed fuzzy clustering based method it is expected that this study can stimulate more research on time based logistics control and management

reciprocal upper semicontinuity and better reply secure games a comment a convex compact and possibly discontinuous better reply secure game has a nash equilibrium we introduce a very weak notion of continuity that can be used to establish that a game is better reply secure and we show that this notion of continuity is satisfied by a large class of games the class of better reply secure games was introduced by reny 1999 who showed the existence of a pure nash equilibrium for games in this class furthermore reny provided two conditions that when combined become suf ficient for a game to be better reply secure the first condition is payoff secu rity which means that given a joint strategy x every player can find a strategy that yields almost the same payoff at x even when the other players slightly deviate from x the second condition is reciprocal upper semicontinuity rusc this condition roughly speaking requires the payoff of one of the players to jump up whenever the payoff of another player jumps down we introduce the notion of weak reciprocal upper semicontinuity wrusc a strict weakening of rusc and prove that a game that is wrusc and payoff secure is better reply secure and therefore has a pure strategy nash equilibrium

equilibria with discontinuous preferences new fixed point theorems abstract we prove new equilibrium existence results for games and economies with discontinuous and non ordered preferences to do so we introduce the notion of continuous inclusion property and prove new fixed point theorems which extend and generalize the results of fan 17 glicksberg 20 browder 7 and gale and mas colell 19 our results also extend the previous work of yannelis 47 and wu and shen 46

detection and characterization of cluster substructure ii fuzzy c varieties and convex combinations thereof in part i siam j appl math 40 1981 pp 339 357 fuzzy c lines was introduced as an algorithm for detection and characterization of linearly clustered data in part ii we address two extensions of the theory in part i specifically we will first generalize the straight line prototype of a cluster developed in part i to any r dimensional linear variety of r s 0 leqq r s secondly we will consider a distance functional which utilizes convex combinations of the distance functionals developed here and in part i all of the notation and symbols used here are unchanged from part i

a complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite mv chain we consider extensions of the unique additive measure on a finite mv chain to uncertainty measures on its canonical girard algebra extension if the underlying mv chain has more than two non trivial elements in a previous paper we have proved the non existence of strongly additive measure extensions where strong additivity is defined as additivity not for all disjoint unions but only restricted to the so called divisible disjoint unions this negative result motivates to look for weakly additive measure extensions which are defined to be additive only on all mv subalgebras of the canonical girard algebra extension we obtain a characterization of all such mv subalgebras which are in fact mv chains and using this result we come up to a characterization of all weakly additive measure extensions moreover we characterize also all the valuation extensions and we prove the incompatibility of these two types of measure extensions except for underlying mv chains with one or two or four non trivial elements

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

on the scoring approach to admissibility of uncertainty measures in expert systems abstract this paper arose from our need to rigorize clarify and address fully the results of lindley s paper scoring rules and the inevitability of probability internat statist rev 50 1982 1 26 herein we develop a calculus of admissibility in a game theoretic setting in the case of an additive aggregation function it is shown that decomposable measures such as those used in fuzzy logics are admissible also the problem of the admissibility of the dempster shafer belief functions is investigated via the concept of random sets it is shown that the class of admissible measures in a scoring framework depends on the assumptions concerning the aggregation function in use in particular for nonadditive aggregation functions an admissible measure may not be transformable to a finitely additive probability measure

topoi and categories of fuzzy sets abstract let h be a completely distributive lattice and hence a heyting algebra goguen s category of fuzzy sets set h eytan s logos fuz h and the topos of sheaves on h sh h are interconnected by pairs of adjoint functors between them each of the categories has a predicate calculus these predicate calculi are related through the functors between the categories change of base lattice gives rise to several functors which preserve or reflect specific kinds of statements in the predicate calculi this paper gives details of the categories the predicate calculi attached to them the functors between the categories and the preservation properties of those functors