combination of fuzzy numbers representing expert opinions abstract expert opinions or imprecise estimates of a physical variable are expressed as fuzzy numbers and five techniques for combining these numbers into a single fuzzy number estimate are developed seven characteristics of the combination technique are defined namely agreement preservation order independence transformation variance possibility conservation possibility interval conservation relationship between uncertainty of individual estimates and overall uncertainty and desirability of resultant estimate the five techniques listed in increasing order of preference are 1 crisp weighting 2 fuzzy weighting 3 minimal fuzzy extension 4 convex fuzzy extension and 5 mixed linear extension an example of estimating nitrate concentration in ground water illustrates the approach the cases of equally versus unequally reliable estimates are distinguished and guidelines for choice of combination technique are provided

a new approach to fuzzy regression models with application to business cycle analysis recently fuzzy regression analysis has been largely applied in the modeling of economic or financial data however those data often exhibit certain kinds of linguistic terms for instance very good a little reclining or stable in the business cycle or the growth rate of gdp etc the goal of this paper is to construct a fuzzy regression model by fuzzy parameters estimation using the fuzzy samples it deals with imprecise measurement of observed variables fuzzy least square estimation and nonparametric methods this is different from the assumptions as well as the estimation techniques of the classical analysis empirical results demonstrate that our new approach is efficient and more realistic than the traditional regression analysis

fuzzy nonparametric regression based on local linear smoothing technique in a great deal of literature on fuzzy regression analysis most of research has focused on some predefined parametric forms of fuzzy regression relationships especially on the fuzzy linear regression models in many practical situations it may be unrealistic to predetermine a fuzzy parametric regression relationship in this paper a fuzzy nonparametric model with crisp input and lr fuzzy output is considered and based on the distance measure for fuzzy numbers suggested by diamond p diamond fuzzy least squares information sciences 46 1988 141 157 the local linear smoothing technique in statistics with the cross validation procedure for selecting the optimal value of the smoothing parameter is fuzzified to fit this model some simulation experiments are conducted to examine the performance of the proposed method and three real world datasets are analyzed to illustrate the application of the proposed method the results demonstrate that the proposed method works quite well not only in producing satisfactory estimate of the fuzzy regression function but also in reducing the boundary effect significantly

estimation of a fuzzy regression model using fuzzy distances regression analysis is a powerful statistical tool that has many applications in different areas the problem of regression analysis under a fuzzy environment has been treated in the literature from different points of view and considering a variety of input output data crisp or fuzzy however we realize that in general most research papers have a conflict between the solution of the fuzzy regression problem using crisp distances minimizing a real error function and the interpretation of fuzzy data as possibility distributions the main aim of this paper is to develop a methodology to solve this problem introducing a fuzzy partial order and a family of fuzzy distance measures on the whole set of fuzzy numbers the new approach allows us to obtain linear and nonlinear models that reach the lowest fuzzy error the estimation process in general can be considered easier to apply in practice and it is not limited to triangular fuzzy numbers numerical examples are provided to illustrate the usefulness and applicability of these results and comparisons with existing methodologies show that the performance of the proposed solution is very satisfactory

modeling temporal functions with granular regression and fuzzy rules modeling of temporal functions has been studied extensively in last few decades temporal functions or time series are found practically in the gamut of applications ranging from engineering analysis to financial and commercial transactions numerous time series models been proposed and they can be found in the relevant time series literature deterministic and stochastic models have dominated much of the research in this area nonetheless these models have their limitations and particularly specific models solve specific problem domains therefore there is no general mechanism that can address the issues of modeling of temporal functions granular information is now being used to address a few of the problems in the theory of time series this paper proposes two models that use the concepts of regression and interleaved information granules intervals and fuzzy sets to identify the structure existing in a given time series in brevity we are attempting to combine regression and fuzzy rules to model temporal systems

fuzzy regression analysis using neural networks abstract in this paper we propose simple but powerful methods for fuzzy regression analysis using neural networks since neural networks have high capability as an approximator of nonlinear mappings the proposed methods can be applied to more complex systems than the existing lp based methods first we propose learning algorithms of neural networks for determining a nonlinear interval model from the given input output patterns a nonlinear interval model whose outputs approximately include all the given patterns can be determined by two neural networks next we show two methods for deriving nonlinear fuzzy models from the interval model determined by the proposed algorithms nonlinear fuzzy models whose h level sets approximately include all the given patterns can be derived last we show an application of the proposed methods to a real problem

linear regression analysis for fuzzy input and output data using the extension principle abstract the method for obtaining the fuzzy least squares estimators with the help of the extension principle in fuzzy sets theory is proposed the membership functions of fuzzy least squares estimators will be constructed according to the usual least squares estimators in order to obtain the membership value of any given value taken from the fuzzy least squares estimator optimization problems have to be solved we also provide the methodology for evaluating the predicted fuzzy output from the given fuzzy input data

online submodular minimization we consider an online decision problem over a discrete space in which the loss function is submodular we give algorithms which are computationally efficient and are hannan consistent in both the full information and partial feedback settings

a variable spread fuzzy linear regression model with higher explanatory power and forecasting accuracy fuzzy regression models have been applied to operational research or applications such as forecasting some of previous studies on fuzzy regression analysis obtain crisp regression coefficients for eliminating the problem of increasing spreads for the estimated fuzzy responses as the magnitude of the independent variable increases however they still cannot cope with the situation of decreasing or variable spreads this paper proposes a three phase method to construct the fuzzy regression model with variable spreads to resolve this problem in the first phase on the basis of the extension principle the membership functions of the least squares estimates of regression coefficients are constructed to conserve completely the fuzziness of observations in the second phase then they are defuzzified by the center of gravity method to obtain crisp regression coefficients in the third phase the error terms of the proposed model are determined by setting each estimated spread equals its corresponding observed spread furthermore the mamdani fuzzy inference system is adopted for improving the accuracy of its forecasts compared to the previous studies the results from five examples and an application example of japanese house prices show that the proposed fuzzy linear regression model has higher explanatory power and forecasting performance

fuzzy regression models using the least squares method based on the concept of distance fuzzy regression models are developed to construct the relationship between explanatory variables and responses in a fuzzy environment in order to increase the explanatory performance of the model the least squares method is applied to determine the numeric coefficients based on the concept of distance unlike most existing approaches the numeric coefficients in the proposed model can have negative values the proposed model minimizes total estimation error in terms of the sum of the average squared distance between the observed and estimated responses based on a few alpha cuts the proposed approach is not limited to triangular fuzzy numbers it can be used to carry out a large number of fuzzy observations efficiently because the model is based on traditional statistical methods comparisons with existing methods show that based on the total estimation error using the mean squared error and kim and bishu s criterion the explanatory performance of the proposed model is satisfactory

development of a cost function for wastewater treatment systems with fuzzy regression abstract a fuzzy linear regression analysis is performed to study the cost function of wastewater that is developed by using a pool of data from 26 municipal wastewater treatment systems in taiwan the choice of a primary or more higher wastewater treatment process is intimately related to financial land availability effluent standards and managerial factors especially financial consideration plays an important role in the planning stage in addition cost estimation is difficult to precisely evaluate in an uncertain environment in the conventional regression model deviations between the observed values and the estimated values are supposed to be due to measurement errors here taking a different perspective these deviations are regarded as the fuzziness of the system s parameters thus these deviations are reflected in a linear function with fuzzy parameters using linear programming algorithm this fuzzy linear regression model might be very convenient and useful for finding a fuzzy structure in an evaluation system in this paper the details of the fuzzy linear regression concept and its applications to the cost function in an uncertain environment are shown and discussed

a fuzzy systems approach in data envelopment analysis abstract the use of fuzzy set theoretic measures is explored here in the context of data envelopment analysis which utilizes a nonparametric approach to measure efficiency three types of fuzzy statics are employed here e g fuzzy mathematical programme fuzzy regression and fuzzy entropy to illustrate the types of decisions and solutions that are achievable when the data are vague and prior information is inexact and imprecise

fuzzy prediction based on regression models abstract we use a linguistic variable to represent imprecise information to be inserted in regression models used for prediction we show how one can obtain probabilistic statements about the forecasted variables

a faster scaling algorithm for minimizing submodular functions combinatorial strongly polynomial algorithms for minimizing submodular functions have been developed by iwata fleischer and fujishige iff and by schrijver the iff algorithm employs a scaling scheme for submodular functions whereas schrijver s algorithm achieves strongly polynomial bound with the aid of distance labeling subsequently fleischer and iwata have described a push relabel version of schrijver s algorithm to improve its time complexity this paper combines the scaling scheme with the push relabel framework to yield a faster combinatorial algorithm for submodular function minimization the resulting algorithm improves over the previously best known bound by essentially a linear factor in the size of the underlying ground set

exponentials in simple z stable c algebras abstract let a be a unital separable simple z stable c algebra which has rational tracial rank at most one and let u u 0 a where u 0 a is the connected component of the unitary group of a containing the identity we show that for any 0 there exists a self adjoint element h a such that 0 1 u exp i h but there is no control of h in general let cu a be the closure of the commutator subgroup of the unitary group of a and let u cu a we prove that there exists a self adjoint element h a such that 0 2 u exp i h and h 2 examples are given that the bound 2 for h is optimal in general for the jiang su algebra z we show that if u u 0 z and 0 there exists a real number t and a self adjoint element h z with h 2 such that e i t u exp i h

least squares estimation in linear regression models with vague concepts the paper is a contribution to parameter estimation in fuzzy regression models with random fuzzy sets here models with crisp parameters and fuzzy observations of the variables are investigated this type of regression models may be understood as an extension of the ordinary single equation linear regression models by integrating additionally the physical vagueness of the involved items so the significance of these regression models is to improve the empirical meaningfulness of the relationship between the items by a more sensitive attention to the fundamental adequacy problem of measurement concerning the parameter estimation the ordinary least squares method is extended the existence of estimators by the suggested method is shown and some of their stochastic properties are surveyed

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

simple nuclear c algebras of tracial topological rank one we give a classification theorem for unital separable nuclear c algebras with tracial rank no more than one let a and b be two unital separable simple nuclear c algebras with tr a tr b 1 which satisfy the universal coefficient theorem we show that a b if and only if there is an order and unit preserving isomorphism 0 1 2 k0 a k0 a 1a k1 a t a k0 b k0 b 1b k1 b t b where 2 1 x 0 x for each x k0 a and t b

a new approach to the identification of a fuzzy model abstract this paper presents an approach which is useful for the identification of a fuzzy model the identification of a fuzzy model using input output data consists of two parts structure identification and parameter identification in this paper algorithms to identify those parameters and structures are suggested to solve the problems of conventional methods given a set of input output data the consequent parameters are identified by the hough transform and clustering method which consider the linearity and continuity respectively for the premise part identification the input space is partitioned by a clustering method the gradient descent algorithm is used to fine tune parameters of a fuzzy model finally it is shown that this method is useful for the identification of a fuzzy model by simulation

fuzzy ranking and quadratic fuzzy regression abstract three different ranking methods namely the overall existence ranking index oeri the approach proposed by diamond 1 and a new two step method based on oeri are used to estimate the distance between two fuzzy numbers this distance parameter is then used in the least square or quadratic regression nonlinear programming is used to solve the resulting quadratic regression equations with constraints and simulation is used to evaluate the performance of the approaches the criterion used to evaluate the performance is the average of the absolute difference between the estimated and the observed values it appears that the two step oeri obtains better results for the case of small sample size and diamond s approach gets better as the sample size increases

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

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

linear fuzzy regression abstract this paper presents an approach which is useful for regression analysis in the case of heterogeneity of a set of observations for which regression is evaluated the proposed procedure consists of two stages first for a set of observations fuzzy classification is determined due to this homogenous classes of observations which are of hyperellipsoidal shape are obtained then for each fuzzy class the so called linear fuzzy regression is evaluated in the paper the method of calculating linear fuzzy regression coefficients is given it is a generalized version of the least squares method the formula for the values of coefficients is given some properties of linear fuzzy regression are analyzed it is proved that in one and two dimensional cases the formulae are analogous to those for usual regression a measure of goodness of fit and the method of determination of the number of fuzzy classes are also given presented examples indicate the superiority of fuzzy regression in comparison to usual regression in the case of heterogenous observations

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

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

a fuzzy approach for multiresponse optimization an off line quality engineering problem abstract multiresponse optimization techniques are used to identify settings of process parameters that make the product s performance close to target values in the presence of multiple quality characteristics in this study a fuzzy multiresponse optimization procedure is proposed to search an appropriate combination of process parameter settings based on multiple quality characteristics or responses fuzzy regression models are first used to model the relations between process parameters and responses the possibility distributions of the prediction responses are then obtained the problem is reduced to a multiobjective optimization problem with fuzzy objectives a strategy of optimizing the most possible response values and minimizing the deviations from the most possible values is proposed we consider not only the most possible value but also the imprecision of the prediction responses through a die casting example we show how to use our approach to reach an appropriate machine setting which simultaneously optimize both casting quality and die life

a faster strongly polynomial time algorithm for submodular function minimization we consider the problem of minimizing a submodular function f defined on a set v with n elements we give a combinatorial algorithm that runs in o n 5eo n 6 time where eo is the time to evaluate f s for some s subseteq v this improves the previous best strongly polynomial running time by more than a factor of n we also extend our result to ring families

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

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

the classification of rokhlin flows on mathrm c c algebras we study flows on mathrm c algebras with the rokhlin property we show that every kirchberg algebra carries a unique rokhlin flow up to cocycle conjugacy which confirms a long standing conjecture of kishimoto we moreover present a classification theory for rokhlin flows on mathrm c algebras satisfying certain technical properties which hold for many mathrm c algebras covered by the elliott program as a consequence we obtain the following further classification theorems for rokhlin flows firstly we extend the statement of kishimoto s conjecture to the non simple case up to cocycle conjugacy a rokhlin flow on a separable nuclear mathcal o infty absorbing mathrm c algebra is uniquely determined by its induced action on the prime ideal space secondly we give a complete classification of rokhlin flows on simple classifiable kk contractible mathrm c algebras two rokhlin flows on such a mathrm c algebra are cocycle conjugate if and only if their induced actions on the cone of lower semicontinuous traces are affinely conjugate

exponential rank and exponential length of operators on hilbert c modules in this article we consider unitary operators on the hilbert c module ha associated with a large class of c algebras let l x denote the algebra consisting of all bounded linear operators on a separable hilbert space 7 and let k denote the algebra of all compact operators on r halmos and kakutani proved that every unitary operator on r is a product of four symmetries cf ha 143 haka for the case when x is a compact hausdorff space ringrose r recently considered the c algebra c x l q consisting of all norm continuous maps from x to l x and proved that every unitary element of c x l x can be written as a product of at most three exponentials in other words a norm continuous map u from x to the unitary group of l x forms a product of three norm continuous maps written as exp ihi exp ih2 exp ih3 where hl h2 and h3 are maps from x to the space of self adjoint operators in l x phillips and ringrose phr conjectured that every unitary of c x l x can even be approximated in the norm by products of two exponentials using techniques in the theory of c algebras but by a different approach we generalized ringrose s result to hilbert c modules associated with all a unital in particular all unital c algebras cf z8 we also proved in z8 similar results for the c algebra consisting of norm continuous maps from x to a purely infinite simple c algebra the purpose of this article is to settle the phillips ringrose conjecture in the affirmative and to prove a generalized version of the halmos kakutani theorem in haka we consider the group of unitary operators on hilbert c modules associated with the following two specific classes which together comprise the majority of interesting c algebras