statistical tests optimally meeting certain fuzzy requirements on the power function and on the sample size abstract usually the parameters of statistical tests are determined by crisp constraints on the probabilities and of the errors of type i and of type ii respectively and by the additional condition that the sample size n is to be selected as small as possible in this paper this approach is modified here statistical tests are proposed to be evaluated by the degree to which certain fuzzy requirements on and n are met aggregating these requirements by the minimum operator a procedure is presented how to determine a test maximizing this degree of satisfaction statistical tests with a continuously distributed test statistic and in particular two sided gau tests are discussed in more detail

testing fuzzy hypotheses with crisp data abstract in this paper an approach is presented how statistical tests originally constructed to examine crisp hypotheses can also be applied to fuzzily formulated hypotheses in particular criterions and are proposed generalizing the probabilities of the errors of type i and type ii respectively the general approach is applied to one and two sided gaus tests here diagrams are given to determine the critical values in the most popular cases of 0 01 and 0 05 if in addition the value of is fixed in advance the sample size of a one or two sided gaus test can be obtained using supplementary graphs

positive filtering for continuous time positive systems under l 1 performance in this paper we address the positive filtering problem for positive continuous time systems under the l1 induced performance a pair of positive filters with error bounding feature is proposed to estimate the output of positive systems a novel characterisation is first obtained to ensure that the filtering error system is asymptotically stable with a prescribed l1 induced performance then necessary and sufficient conditions for the existence of required filters are presented and the obtained results are expressed in terms of linear programming problems which can be easily checked by standard software finally a numerical example is given to illustrate the effectiveness of the proposed design procedures

on the spectral and combinatorial structure of 2d positive systems abstract the dynamics of a 2d positive system depends on the pair of nonnegative square matrices that provide the updating of its local states in this paper several spectral properties such as finite memory separability and property l which depend on the characteristic polynomial of the pair are investigated under the nonnegativity constraint and in connection with the combinatorial structure of the matrices some aspects of the perron frobenius theory are extended to the 2d case in particular conditions are provided guaranteeing the existence of a common maximal eigenvector for two nonnegative matrices with irreducible sum finally some results on 2d positive realizations are presented

a certain one parameter additive family of operators defined on analytic functions abstract y komatu proved several interesting results for a certain one parameter additive family of operators defined on analytic functions regular in the unit disk and presented two conjectures for univalent functions in the unit disk it is the purpose of this paper to prove a number of results relevant to komatu s conjectures for certain classes of univalent functions with negative coefficients in the unit disk

combining ambiguous evidence with respect to ambiguous a priori knowledge part ii fuzzy logic abstract this paper describes fuzzy conditioned dempster shafer fcds theory a probability based calculus for dealing with possibly imprecise and vague evidence when the underlying a priori knowledge base is itself possibly imprecise and vague fcds has the same general form as existing fuzzy dempster shafer theories fcds however employs a finite level zadeh fuzzy logic and a dempster shafer like combination operator which is conditioned to reflect the influence of any a priori knowledge base which can be modeled by a belief measure on finite level fuzzy sets we show that fcds is grounded in probability theory specifically in the theory of random fuzzy sets we also show that it is a generalization of bayesian theory to the case when both evidence and a priori knowledge are imprecise and vague we show that fcds is consistent with the lattice structure of zadeh s min max fuzzy logic and that in particular it possesses an analog of the familiar mobius transform

the fuzzy decision problem an approach to the problem of testing statistical hypotheses with fuzzy information abstract this paper is devoted to the problems of testing statistical hypotheses about an experiment when the available information from its sampling is vague when the information supplied by the experimental sampling is exact the problems of testing statistical hypotheses about the experiment can be regarded as a particular statistical decision problem in addition decision procedures may be used in problems of testing hypotheses in a similar manner the problem of testing statistical hypotheses about an experiment when the available sample information is vague is approached in this paper as a particular fuzzy decision problem as defined by tanaka okuda and asai this approach assumes that the previous information about the experiment can be expressed by means of certain conditional probabilistic information whereas the present information about it can be expressed by means of fuzzy information the preceding framework allows us to extend the notion of risk function and some nonfuzzy decision procedures to the fuzzy case and particularize them to the problem of testing finally several illustrative examples are presented

dealing with uncertainties in engineering problems using only available data abstract this paper is the second in the series of three addressing the problem of how to deal with uncertainties in the modelings of engineering problems using only available data the problem of the heat exchanger was selected as the example because it is sufficiently complex but still simple so that the difficulties and ideas will be apparent there are various theories on how to treat the uncertainties in the first paper in the series we use only the fuzzy set respectively the possibility theory in this paper we use the fuzzy approach for some model parameters and the probability approach for the others the third paper will use the bayesian theory

a linear framework on the distributed model predictive control of positive systems abstract this paper investigates the distributed model predictive control dmpc for positive systems with interval and polytopic uncertainties respectively different from the traditional quadratic dmpc a new linear dmpc framework is established for positive systems first a linear cooperative cost function performance index function is introduced for the dmpc of positive systems accordingly some linear constraint conditions are imposed on the systems then a linear co positive lyapunov function is constructed using the linear lyapunov function the dmpc controllers are designed for interval positive systems and polytopic positive systems respectively the controller of each subsystem contains not only the state information of the subsystem but also the state information of other correlated subsystems linear programming based optimization algorithm is addressed to compute the controller parameters to guarantee the feasibility and stability of the systems a cone invariant set is proposed finally two case studies are provided to illustrate the effectiveness of the theoretical findings

l1 induced norm and controller synthesis of positive systems in this paper the problem of 1 1 induced controller design for discrete time positive systems is investigated with the use of linear lyapunov function an analytical method to compute the exact value of 1 1 induced norm is first presented then a novel characterization for stability and 1 1 induced performance is proposed based on the characterization a necessary and sufficient condition for the existence of desired controllers is derived and an iterative convex optimization approach is developed to solve the condition in addition the synthesis of the state feedback controller for single input multiple output simo positive systems is investigated for this special case an analytic solution is established to show how the optimal 1 1 induced controller can be designed and some links to the spectral radius of the closed loop systems are provided finally the theoretical results are illustrated through a numerical example

remarks on excitability stability and sign of equilibria in cooperative systems the strict positivity of equilibria is known to be equivalent to asymptotic stability in excitable positive linear systems in contrast with intuition this property does not hold for nonlinear cooperative systems however weaker results are pointed out and proved to be effective in the analysis of several nonlinear cooperative systems

on fuzzy bayesian inference abstract the paper combines methods from bayesian statistics with ideas from fuzzy set theory to generalize bayesian methods both for samples of fuzzy data and for prior distributions with non precise parameters applying the principle of propagation of fuzziness we introduce a fuzzy valued likelihood function bayes theorem for both fuzzy data and fuzzy priors a fuzzy bayes estimator fuzzy predictive densities and distributions and fuzzy h p d regions

bayesian reliability analysis for fuzzy lifetime data lifetime data are important in reliability analysis classical reliability estimation is based on precise lifetime data it is usually assumed that observed lifetime data are precise real numbers however some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers thus it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers bayesian methods have proved to be very useful when the sample size is small there is little study on bayesian reliability estimation based on fuzzy lifetime data most of the reported works in this area is limited to single parameter lifetime distributions in this paper we propose a new method to determine the membership function of the estimates of the parameters and the reliability function of multi parameter lifetime distributions an artificial neural network is used to approximate the calculation process of parameter estimation and reliability prediction the genetic algorithm is used to find the boundary values of the membership function of the estimate of interest at any cut level this method can be used to determine the membership functions of the bayesian estimates of multi parameter distributions the effectiveness of the proposed method is illustrated with normal and weibull distributions

the fuzzy estimators of fuzzy parameters based on fuzzy random variables abstract the concept for fuzzy estimators of fuzzy parameters in fuzzy random variables is proposed in this paper due to the resolution identity the construction of a fuzzy number from a family of closed intervals is developed the construction of the fuzzy estimators of fuzzy parameters from the known conventional estimators inherits evidently from the aforementioned technique in the meantime we prove that the fuzzy estimator is a fuzzy random variable in order to provide the notion of fuzzy expectation and variance of fuzzy estimator the computational procedure is also provided by solving the related nonlinear programming problems that are transformed from the original problem for the purpose of evaluation of the membership values of the fuzzy estimates

self organizing maps for imprecise data self organizing maps soms consist of a set of neurons arranged in such a way that there are neighbourhood relationships among neurons following an unsupervised learning procedure the input space is divided into regions with common nearest neuron vector quantization allowing clustering of the input vectors in this paper we propose an extension of the soms for data imprecisely observed self organizing maps for imprecise data soms id the learning algorithm is based on two distances for imprecise data in order to illustrate the main features and to compare the performances of the proposed method we provide a simulation study and different substantive applications

on the invertibility of leslie systems the problem of reconstructing the stocking rate of exploited renewable resources from an output time series is considered the resource is described by a linear age structured model of the leslie type and the measured output is a linear and positive combination of the number of individuals in each reproductive class the problem can be formally answered by using a well known result of system theory a linear system is invertible if and only if the zeros of its transfer function are all smaller than one in modulus in the present case this would imply working with a polynomial of degree n 1 n being the maximum age of the population by explicitly computing the transfer function we show that the invertibility of the system can indeed be established by checking if the zeros of a polynomial of degree r 1 r being the number of reproductive age classes are smaller than one in modulus moreover we investigate two particular measurement schemes and we show that the first one which is possibly the most commonly used to sample animal populations gives rise to an invertible system on the contrary the second scheme can generate noninvertible systems although two single conditions for invertibility that we point out in the paper seem to be often satisfied in the case of real populations

bayesian estimation in forest surveys when samples or prior information are fuzzy abstract fuzzy phenomena are frequently encountered in forestry due to the complexity of the systems under investigation in this paper existing inference methods are presented to accommodate situations in which random events take place in fuzzy circumstances fuzzy concepts are incorporated into these commonly accepted approaches to account for uncertainties used by factors other than randomness which traditional statistical approaches deal primarily with based on two extensions of likelihood functions for fuzzy samples bayesian estimates are generalized for use when sample information and prior distribution of parameters are fuzzy the estimators are developed for some typical membership functions for fuzzy samples and fuzzy priors applications are illustrated using examples for forestry

aleatoric and epistemic uncertainty in groundwater flow and transport simulation 1 the characterization of aleatory hydrogeological parameter uncertainty has traditionally been accomplished using probability theory however when consideration is given to epistemic as well as aleatory uncertainty probability theory is not necessarily appropriate this is especially the case where expert opinion is regarded as a suitable source of information when experts opine upon the uncertainty of a parameter value both aleatoric and epistemic uncertainties are introduced and must be modeled appropriately a novel approach to expert provided parameter uncertainty characterization can be defined that bridges an historical gap between probability theory and fuzzy set theory herein a random set a generalization of a random variable is employed to formalize expert knowledge and fuzzy sets are used to propagate this uncertainty to model estimates of contaminant transport the resultant random set based concentration estimates are shown to be more general than the corresponding random variable estimates in some cases the random set based results are shown as upper and lower probabilities that bound the corresponding random variable s cumulative distribution function

the hardy space of analytic functions associated with certain one parameter families of integral operators abstract recently h m srivastava and s owa j math anal appl 118 1986 80 87 proved a number of results relevant to komatu s conjectures cf y komatu bull fac sci engrg chuo univ ser i math 22 1979 1 22 for a certain one parameter additive family of operators defined on analytic functions regular in the open unit disk the object of the present paper is to investigate several one parameter families of integral operators and to prove certain inclusion theorems involving the hardy space h p of analytic functions and the class of functions whose derivative has a positive real part an application involving a generalized hypergeometric function is also considered

positive filtering for positive takagi sugeno fuzzy systems under l 1 performance in this paper the positive filtering problem is addressed for positive takagi sugeno t s fuzzy systems under the 1 induced performance to estimate the output of positive t s fuzzy systems error bounding positive filters are constructed a new performance characterization is first established to guarantee the asymptotic stability of the filtering error system with the 1 induced performance moreover sufficient conditions expressed by linear programming problems are derived to design the required filters finally a numerical example is presented to show the effectiveness of the derived theoretical results

on possibility analysis of fuzzy data this paper addresses the analysis of fuzzy data from a possibilistic perspective the concept of a double fuzzy variable is introduced and related notions are defined and investigated the properties and fuzzy parameter estimation of the possibility distribution are also investigated a maximum possibility likelihood principle is provided to estimate unknown fuzzy parameters a normal possibility distribution is used as an example and estimates of its fuzzy parameters are given the proposed possibility theory of fuzzy data constitutes a new approach to analyzing fuzzy data from a possibilistic point of view

clustering of clusters abstract an algorithm for the clustering of existing clusters is introduced in this paper the algorithm was adopted from fuzzy c mean and modifications made to take into account the extra information i e some data samples already form clusters partition coefficients together with some other criteria are used for testing cluster validity the method was applied on chinese character recognition and an encouraging result was obtained

differential subordination and superordination results for cho kwon srivastava operator making use of the concepts of differential subordination and superordination many subordination and superordination results are obtained for analytic functions in the open unit disk using the cho kwon srivastava operator by investigating appropriate classes of admissible functions sandwich type results are also obtained

convergence properties of the generalized fuzzy c means clustering algorithms fuzzy c means fcm clustering algorithms have been widely used to solve clustering problems yang and yu 1 extended these to optimization procedures with respect to any probability distribution they showed that the optimal cluster centers are the fixed points of these generalized fcm clustering algorithms the convergence properties of algorithms are the important theoretical issue in this paper we present convergence properties of the generalized fcm clustering algorithms these are global convergence local convergence and its rate of convergence

a robust clustering procedure for fuzzy data in this paper we propose a robust clustering method for handling lr type fuzzy numbers the proposed method based on similarity measures is not necessary to specify the cluster number and initials several numerical examples demonstrate the effectiveness of the proposed robust clustering method especially robust to outliers different cluster shapes and initial guess we then apply this algorithm to three real data sets these are taiwanese tea student data and patient blood pressure data sets because tea evaluation comes under an expert subjective judgment for taiwanese tea the quality levels are ambiguity and imprecision inherent to human perception thus lr type fuzzy numbers are used to describe these quality levels the proposed robust clustering method successfully establishes a performance evaluation system to help consumers better understand and choose taiwanese tea similarly lr type fuzzy numbers are also used to describe data types for student and patient blood pressure data the proposed method actually presents good clustering results for these real data sets

change point analysis of imprecise time series abstract in this paper we describe how to conduct a change point analysis when dealing with time series imprecisely or vaguely observed i e time ordered observations whose values are not known exactly such as interval or ordinal time series imprecise time series in order to treat such time series we propose to employ a fuzzy approach i e data are parameterized in the form of fuzzy variables then to detect the number and location of change points we employ a deviation measure for fuzzy variables in the framework of atheoretical regression trees art we present simulation results pertaining to the behavior of the proposed approach as well as two empirical applications to real imprecise time series

on a class of fuzzy classification maximum likelihood procedures classification maximum likelihood cml procedure is a remarkable mixture of maximum likelihood approach to clustering this has been well documented in the book of mclachlan and basford in this paper we make the fuzzy extension of the cml procedure based on this class of fuzzy cml procedures we extend the fuzzy clustering algorithms of trauwaert kaufman and rousseeuw by adding a penalty term especially we derive a generalized type of fuzzy c means fcm clustering algorithms called the penalized fcm clustering algorithms then we create some asymptotic behaviors of these penalized fcm procedures by doing some numerical examples we find that the penalized fcm is more meaningful and effective than fcm

fuzzy portfolio allocation models through a new risk measure and fuzzy sharpe ratio a new portfolio risk measure that is the uncertainty of portfolio fuzzy return is introduced in this paper beyond the well known sharpe ratio i e the reward to variability ratio in modern portfolio theory we initiate the so called fuzzy sharpe ratio in the fuzzy modeling context in addition to the introduction of the new risk measure we also put forward the reward to uncertainty ratio to assess the portfolio performance in fuzzy modeling corresponding to two approaches based on rm t rm m and rm t rm w fuzzy arithmetic two portfolio optimization models are formulated in which the uncertainty of portfolio fuzzy returns is minimized while the fuzzy sharpe ratio is maximized these models are solved by the fuzzy approach or by the genetic algorithm ga solutions of the two proposed models are shown to be dominant in terms of portfolio return uncertainty compared with those of the conventional mean variance optimization mvo model used prevalently in the financial literature in terms of portfolio performance evaluated by the fuzzy sharpe ratio and the reward to uncertainty ratio the model using rm t rm w fuzzy arithmetic results in higher performance portfolios than those obtained by both the mvo and the fuzzy model which employs rm t rm m fuzzy arithmetic we also find that using the fuzzy approach for solving multiobjective problems appears to achieve more optimal solutions than using ga although ga can offer a series of well diversified portfolio solutions diagrammed in a pareto frontier

an application of certain integral operators the integral operator i n f z was introduced by salagean for func tions of the form f z z 1 p k 2 a kz k which are analytic in the unit disc

the fuzzy decision problem an approach to the point estimation problem with fuzzy information abstract this paper is devoted to the extension of the bayesian method for the point estimation when the available information is vague in the nonfuzzy case the parametric estimation can be approached as a particularization in the statistical decision problem this motivates us to accomplish the mentioned extension by looking at the parametric estimation in the fuzzy case as a special situation in the fuzzy decision problem defined by tanaka okuda and asia in this way concepts in the fuzzy decision problem are first expressed in the estimation terminology then on the basis of these concepts we shall introduce some notions and state some interesting results finally several illustrative examples will be exposed

fuzzy clustering procedures for conical fuzzy vector data abstract this note addresses fuzzy cluster analysis of conical fuzzy vector data a fuzzy vector represents fuzzy data in a high dimension in this note a robust type of fuzzy clustering algorithm with a noise cluster is proposed the proposed algorithm is robust with respect to noise and can also detect outliers for conical fuzzy vector data finally some numerical examples are presented

fuzzy clustering algorithms for mixed feature variables this paper presents fuzzy clustering algorithms for mixed features of symbolic and fuzzy data el sonbaty and ismail proposed fuzzy c means fcm clustering for symbolic data and hathaway et al proposed fcm for fuzzy data in this paper we give a modified dissimilarity measure for symbolic and fuzzy data and then give fcm clustering algorithms for these mixed data types numerical examples and comparisons are also given numerical examples illustrate that the modified dissimilarity gives better results finally the proposed clustering algorithm is applied to real data with mixed feature variables of symbolic and fuzzy data

a study of defuzzification with experts knowledge for deteriorating repairable systems abstract this paper presents ideas on the applications of fuzzy concepts to decision making for deteriorating repairable systems a non homogeneous poisson process nhpp with a power law intensity function is used in this study in general classical bayesian decision methods presume that future states of nature can be characterized as probability events however we do not know what the future will entail probabilistically so we devise a method to consider experts opinions which are usually the absence of sharply defined criteria and to develop a fuzzy bayesian decision process for dealing with such situations two cases of the discrimination problem with deteriorating repairable systems are studied 1 fuzzy states and exact information and 2 fuzzy states and fuzzy information the fuzzy decomposition and arithmetic derivation of the experts opinion are presented to facilitate the development of the bayesian decision process for deteriorating repairable systems

fuzzy logic and meteorological variables a case study of solar irradiance solar irradiance is an extreme case of an uncertain variable when measured on an hourly or shorter time interval in this paper solar irradiance has been considered as a case study for physical fuzzy modelling of a climate variable the uncertainty of the solar irradiance is treated as a fuzzy uncertainty whilst other variables are considered crisp the approach is robust as it does not rely on statistical assumptions and it is a possible alternative to modelling complex systems to our knowledge this is the first time that a physical model of a meteorological variable based on fuzzy numbers has been proposed previous rule based fuzzy meteorological models are only descriptive and cannot be extrapolated to non measured cases when compared with previous non fuzzy models of solar irradiance the fuzzy model shows an improved performance and when compared with experimental data the performance can be evaluated by fuzzy indices that take into account the uncertainty of the data and the model output

fuzzy sets and statistical data abstract specific features of probability and possibility theories are discussed with emphasis on semantical aspects instead of putting forward the acknowledged usefulness of possibility theory for the non statistical modelling of subjective categories we try to figure out how statistical data and possibility theory could be matched as a result procedures for constructing weak possibilistic substitutes of probability measures and for processing imprecise statistical data are outlined they provide new insights on relationship between fuzzy sets and probability theories

bayesian sequential test for fuzzy parametric hypotheses from fuzzy information abstract this paper is devoted to the sequential problem of testing fuzzy hypotheses concerning an experiment the outcomes of which provide fuzzy information this problem will be formalized as a special fuzzy sequential decision problem to which it is possible to extend the bayes nonfuzzy decision procedures when the bayesian framework is assumed

ridge estimation for regression models with crisp inputs and gaussian fuzzy output this paper deals with ridge estimation of fuzzy multiple linear and nonlinear regression models with crisp inputs and gaussian fuzzy output using ridge regression learning algorithm in the lagrangian dual space we describe a ridge estimation of fuzzy multiple linear regression model of xu and li fuzzy sets and systems 119 2001 215 it allows us to perform nonlinear regression for xu and li s model by constructing a fuzzy linear regression function in a high dimensional feature space experimental results are then presented which indicate the performance of this algorithm

non asymptotic fuzzy estimators based on confidence intervals abstract in this paper we study and generalize the method of fuzzy estimation which is based on confidence intervals we provide definitions and the corresponding formulas the method we propose allows new additional ways in constructing fuzzy estimators in particular instead of constructing fuzzy estimators as the exact set of confidence intervals of the parameters to be estimated we generalize them in such a way that we eliminate discontinuities and ensure compact support preserving the statistically derived triangular shape of the estimators our approach is particularly useful in critical situations where subtle fuzzy comparisons between almost equal statistical quantities have to be made an example of this application is provided

parameter estimation for finite mixture distributions abstract in this paper we compare numerically two approaches to the estimation of the parameters of the component densities in a univariate mixture of normal distributions one approach is based on a constrained maximum likelihood ml algorithm the other on the fuzzy c means fcm clustering algorithm our study indicates that i the ml method produces superior estimates when the component densities are well mixed while either algorithm provides good estimates for well separated distributions ii the fcm approach is almost always faster than the ml method and iii initialization of the ml method with the output of fcm almost always improves both the run time and accuracy of the statistical estimates

a bayesian approach to fuzzy hypotheses testing in statistical decisions we may come across imprecise fuzzy concepts one important case is a situation where we are interested in testing the hypotheses that are fuzzy rather than crisp in this paper we consider and analyze the testing of fuzzy hypotheses on the basis of a bayesian approach we illustrate our method by some examples and compare it with previous works on this topic

a fuzzy multiple linear regression based loss formula in electric distribution systems estimation of energy loss is an essential task for both operation and planning in electric distribution systems the energy losses are considered to be uncertain due to measurements a new method based on fuzzy c number clustering and fuzzy multiple linear regression analysis is proposed for developing energy loss formulas to estimate losses in this paper the proposed method is implemented with three stages a part of taipower distribution system in taipei is used for illustrating the performance of the proposed method

excitability stability and sign of equilibria in positive linear systems abstract the relationship between sign and stability of equilibria in positive linear systems is investigated in this paper in particular it is proved that the well known result stability and irreducibility imply strict positivity can be better specified by introducing the notion of excitability in fact stability excitability and strict positivity of the equilibria are three perfectly balanced properties since any one of them is implied by the two others

approximation of functional relationships to fuzzy observations abstract for approximating functional relationships to fuzzy observations there exist several approaches joining some communities the functional relationship is still kept a fuzzy function even if the observations are crisp the case of crisp observations often requires specific considerations in this paper an approximation method using hausdorff like distances is proposed naturally containing the crisp observation case and leading to a crisp function for crisp observations additional the proposed method leads to well known approximation principles from classical theory and can be realized by standard computer software a simple example completes the considerations

on a class of fuzzy c numbers clustering procedures for fuzzy data this paper describes a class of fuzzy clustering procedures for fuzzy data most fuzzy clustering techniques are designed for handling crisp data with their class memberships using the idea of fuzzy set theory here we derive new types of fuzzy clustering procedures in dealing with fuzzy data these procedures are called fuzzy c numbers fcn clusterings specially we construct these fcns for u type triangular trapezoidal and normal fuzzy numbers k jwoy s cluster analysis fuzzy clustering fuzzy data analysis fuzzy numbers fuzzy intervals fuzzy c means clustering fuzzy c numbers clustering

on statistically inference for fuzzy data with applications to descriptive statistics abstract the object of this paper is to discuss how fuzziness of data is propagated when statistical inference for samples of non precise data is carried out first fuzzy data and fuzzy samples are defined subsequently the method of propagation of fuzziness as used in previous work for specific stochastic models is formulated in rather general terms this method may be applied to any statistical method which leads to a result that may be expressed as a function f x 1 x n of the data x 1 x n it is proved that this method is equal to determining the images of a family of compact subsets of the sample space under the function f this equivalence is utilized in order to perform propagation of fuzziness in practice finally this approach is applied to some concepts of descriptive statistics such as fuzzy sample mean fuzzy sample variance and to the empirical distribution function of a fuzzy sample

notes on jung kim srivastava integral operator abstract the object of the present paper is to investigate some properties of certain integral operator q introduced and studied recently by jung kim and srivastava j math anal appl 176 1993 138 147

fuzzy clustering on lr type fuzzy numbers with an application in taiwanese tea evaluation this paper presents a fuzzy clustering algorithm called the alternative fuzzy c numbers afcn clustering algorithm for lr type fuzzy numbers based on an exponential type distance function on the basis of the gross error sensitivity and influence function this exponential type distance is claimed to be robust with respect to noise and outliers hence the afcn clustering algorithm is more robust than the fuzzy c numbers fcn clustering algorithm presented by yang and ko fuzzy sets and systems 84 1996 49 some numerical experiments were performed to assess the performance of fcn and afcn numerical results clearly indicate afcn to be superior in performance to fcn finally we apply the fcn and afcn algorithms to real data the experimental results show the superiority of afcn in taiwanese tea evaluation

a unified approach to fuzzy random variables the concept of fuzzy random variable was introduced as an analogous notion to random variables in order to extend statistical analysis to situations when the outcomes of some random experiment are fuzzy sets but in contrary to the classical statistical methods no unique definition has been established yet in this paper a set theoretical concept of fuzzy random variable will be presented this notion provides a useful framework to compare different concepts of fuzzy random variables using methods of general topology and drawing on results from topological measure theory and the theory of analytic spaces as the main result it will be shown that the introduced concept of fuzzy random variable is a unification of the already known ones

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

on possibilistic clustering with repulsion constraints for imprecise data in possibilistic clustering objects are assigned to clusters according to the so called membership degrees taking values in the unit interval differently from fuzzy clustering it is not required that the sum of the membership degrees of an object to all clusters is equal to one this is very helpful in the presence of outliers which are usually assigned to the clusters with membership degrees close to zero unfortunately a drawback of the possibilistic approach is the tendency to produce coincident clusters a remedy is to add a repulsion term among prototypes in the loss function forcing the prototypes to be far enough from each other here a possibilistic clustering algorithm with repulsion constraints for imprecise data managed in terms of fuzzy sets is introduced applications to synthetic and real fuzzy data are considered in order to analyze how the proposed clustering algorithm works in practice

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

application of fuzzy naive bayes and a real valued genetic algorithm in identification of fuzzy model we present a method to identify a fuzzy model from data by using the fuzzy naive bayes and a real valued genetic algorithm the identification of a fuzzy model is comprised of the extraction of if then rules that is followed by the estimation of their parameters the involved parameters include those which determine the membership function of fuzzy sets and the certainty factors of fuzzy if then rules in our method as long as the fuzzy partition in the input output space is given the certainty factor of each rule is computed with the fuzzy conditional probability of the consequent conditioned on the antecedent by using the fuzzy naive bayes which is a generalization of naive bayes the fuzzy model involves the rules characterized by the highest values of certainty factors the certainty factor of each rule is the fuzzy conditional probability and it reflects the inner relationship between the antecedent and the consequent in order to improve the accuracy of the fuzzy model the real valued genetic algorithm is incorporated into our identification process this process concerns the optimization of the membership functions occurring in the rules we just involve the parameters of membership function of the fuzzy sets into the real valued genetic algorithm since the certainty factor of each rule can be computed automatically the performance of the model is shown for the backing truck problem and the prediction of mackey glass time series

equilibria stability and reachability of leslie systems with nonnegative inputs leslie systems a particular class of positive systems used in population dynamics and control are defined and analyzed a simple property relating sign and stability of their equilibria and some geometric characteristics of their reachability set in cases of bounded and unbounded inputs are proved all the properties do not hold in general in other classes of positive systems it is shown that leslie systems have strictly positive equilibria if and only if they are asymptotically stable their reachability set in the case of unbounded inputs is the positive cone generated by the reachability vectors while in the case of bounded inputs it is a polyhedron whose vertices can be easily computed all of these properties follow from the nonpositiveness of the coefficients of the characteristic polynomial of the system

point estimation for the n sizes of random sample with one vague data abstract for a probability space a p a r v x r and random samples x 1 x 2 x n with one vague data associated with p d f f x 0 we tried to define sufficient statistic and unbiased estimator for them and also to test simple and likelihood ratio hypotheses with one vague data we have shown in our examples that using all the data including the vague data we can have the same result as using the n 1 stable data to test the hypotheses in crisp statistics alternatively if the result of our study can be generalized to several vague data it can be seen that we may not need to determine the outliers from our statistical data in the future