on the variance of fuzzy random variables abstract this paper deals with an expectation and a real valued variance of fuzzy random variables the expectation and the variance of a fuzzy random variable is characterized by frechet s principle in a metric space we study properties of the variance of a fuzzy random variable and compare it with the common variance of real valued random variables using the expectation and the variance of fuzzy random variables we consider a linear regression problem and limit theorems

brief paper strong γk γclh stabilization with a new slack variable approach this paper deals with the strong c email protected c lh stabilization problem a new bounded real lemma is provided by adding two slack variables which provides extra freedom to include more control constraints in handling robust control problems with the extended bounded real lemma a new sufficient condition is proposed for nth order strong c email protected c lh stabilization it is also proved that the design of an lxnth order l 2 strong c email protected c lh controller can be transformed into that of an nth order controller for an associated plant which tackles the possible order problem furthermore the h stable stabilization and strong stabilization problems can also be solved by relaxing certain constraints accordingly despite the bmi constraints the path following and alternative iteration methods are adopted to propose procedures for both nth and lxnth order controller designs the effectiveness is indicated by several numerical examples

higher order models for fuzzy random variables a fuzzy random variable is viewed as the imprecise observation of the outcomes in a random experiment since randomness and vagueness coexist in the same framework it seems reasonable to integrate fuzzy random variables into imprecise probabilities theory nevertheless fuzzy random variables are commonly presented in the literature as classical measurable functions associated to a classical probability measure we present here a higher order possibility model that represents the imprecise information provided by a fuzzy random variable we compare it with previous classical models in the literature first some aspects about the acceptability function associated to a fuzzy random variable are investigated secondly we present three different higher order possibility models all of them arising in a natural way we investigate their similarities and differences and observe that the first one the fuzzy probability envelope is the most informative finally we compare the fuzzy probability envelope with the classical probability measure induced by the fuzzy random variable we conclude that the classical probability measure does not always contain all relevant information provided by a fuzzy random variable

brief a parametric optimization approach to h and h2 strong stabilization this paper presents an approach for designing stable mimo h and h 2 controllers by directly computing the norm constrained stable transfer matrices q in the h and h 2 suboptimal controller parameterizations this is done by first converting the h 2 and h strong stabilization problems into some nonlinear unconstrained optimization problems through explicit parameterization of the norm constrained q s for any fixed order then a two stage numerical search is carried out by using a combination of a genetic algorithm and a quasi newton algorithm in order to reach an optimal solution the effectiveness of the proposed algorithms is illustrated through some benchmark numerical examples

ostrowski type inequalities on h type groups the main aim of this paper is to establish an ostrowski type inequality on h type groups using the l norm of the horizontal gradient the work has been motivated by the work of anastassiou and goldstein in g a anastassiou j a goldstein higher order ostrowski type inequalities over euclidean domains j math anal appl 337 2008 962 968

higher order ostrowski type inequalities over euclidean domains abstract the classical ostrowski inequality for functions on intervals estimates the value of the function minus its average in terms of the maximum of its first derivative this result is extended to functions on general domains using the l norm of its n th partial derivatives for radial functions on balls the inequality is sharp

on the variability of the concept of variance for fuzzy random variables fuzzy random variables possess several interpretations historically they were proposed either as a tool for handling linguistic label information in statistics or to represent uncertainty about classical random variables accordingly there are two different approaches to the definition of the variance of a fuzzy random variable in the first one the variance of the fuzzy random variable is defined as a crisp number which makes it easier to handle in further processing in the second case the variance is defined as a fuzzy interval thus offering a gradual description of our incomplete knowledge about the variance of an underlying imprecisely observed classical random variable in this paper we also discuss another view of fuzzy random variables which comes down to a set of random variables induced by a fuzzy relation describing an ill known conditional probability this view leads to yet another definition of the variance of a fuzzy random variable in the context of the theory of imprecise probabilities the new variance is a real interval which achieves a compromise between both previous definitions in terms of representation simplicity our main objective is to demonstrate with the help of simple examples the practical significance of these definitions of variance induced by various existing views of fuzzy random variables

some new results on strong gamma_k gamma_ cl h_ infty stabilization the strong gammak gammacl hinfin stabilization problem is addressed in this paper based on the bounded real lemma an improved sufficient condition on strong hinfin stabilization as well as its dual form is derived which extends the are constraint in the former results into an inequality constraint two path following algorithms are proposed to solve the associated optimization problems with bmi constraints which lead to full order controllers a numerical example is presented to demonstrate the effectiveness of the result

supremum preserving upper probabilities abstract we study the relation between possibility measures and the theory of imprecise probabilities and argue that possibility measures have an important part in this theory it is shown that a possibility measure is a coherent upper probability if and only if it is normal a detailed comparison is given between the possibilistic and natural extension of an upper probability both in the general case and for upper probabilities defined on a class of nested sets we prove in particular that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability defined on a class of nested sets we show that possibilistic extension can be interpreted in terms of natural extension we also prove that when either the upper or the lower cumulative distribution function of a random quantity is specified possibility measures very naturally emerge as the corresponding natural extensions next we go from upper probabilities to upper previsions we show that if a coherent upper prevision defined on the convex cone of all non negative gambles is supremum preserving then it must take the form of a shilkret integral associated with a possibility measure but at the same time we show that such a supremum preserving upper prevision is never coherent unless it is the vacuous upper prevision with respect to a non empty subset of the universe of discourse

inner and outer fuzzy approximations of confidence intervals we extend the notion of confidence region to fuzzy data by defining a pair of fuzzy inner and outer confidence regions we show the connection with previous proposals as well as with recent studies on hypothesis testing with low quality data

on the stable spl hscr sub spl infin controller parameterization under sufficient condition suboptimal spl hscr sub spl infin controller is said to internally stabilize the closed loop transfer matrix however the stability of the controller is not assured in the conventional spl hscr sub spl infin control theory an unstable controller can damage the whole system when the sensor fails or actuator saturates this paper presents a stable n dimensional spl hscr sub spl infin controller and its parameterization for the lti system based on the sufficient condition for the existence of a stable spl hscr sub spl infin controller the stability of spl hscr sub spl infin controller and closed loop transfer matrix is guaranteed if the positive semidefinite solutions for the suggested three riccati equations exist

brief on simultaneous h control and strong h stabilization in this note the simultaneous h control problem is addressed it is first shown that the existence of a single controller which simultaneously stabilizes two systems with a given h norm level is equivalent to that of a strong h stabilization controller of an associated system moreover the result is extended to the case involving r 2 systems then a sufficient condition for the strong h stabilization is proposed based on the bounded real lemma finally a simple numerical example is presented to show the effectiveness of the result

fuzzy set valued stochastic lebesgue integral this paper studies lebesgue integral of a fuzzy closed set valued stochastic process with respect to the time t firstly a progressively measurable fuzzy closed set valued stochastic process is discussed and an almost everywhere problem in the former aumann type lebesgue integral of the level set process is pointed out secondly a new definition of the lebesgue integral by decomposable closure is given focusing on aumann representation theorem representation theorem and property of convexity it is proved that the fuzzy closed set valued stochastic lebesgue integral is a fuzzy closed set valued stochastic process which is widely used in the fuzzy world with randomness finally the fuzzy closed set valued stochastic lebesgue integral in l p space is studied especially on an almost everywhere problem

ostrowski type inequalities and hardy inequalities with boundary terms related to the generalized greiner operator in this paper we first show a representation formula related to the generalized greiner operator by an improved method then using the representation formula and l norm of the horizontal gradient we establish ostrowski type inequalities in the ball and general bounded domain related to the generalized greiner operator moreover in the same spirit we show the hardy inequalities with boundary terms keywords generalized greiner operator representation formula ostrowski type inequal ity hardy inequality mr 2010 subject classification 35r45 26d10 35j70 clc number o178 document code a article id 1000 0917 2015 01 0091 11

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

fuzziness in data analysis the first aim is to emphasize the use of fuzziness in data analysis to capture information that has been traditionally disregarded with a cost in the precision of the conclusions fuzziness can be considered in the data analysis process at various stages but the main target in this paper will be fuzziness in the data depending on the nature of the fuzzy data or the aim to which they are handled different approaches should be applied we attempt to contribute to the clarification of such a difference while focusing on the so called ontic approach in contrast to the epistemic approach the second aim is to underline the need of considering robust methods to reduce the misleading impact of outliers in fuzzy data analysis we propose trimming as a general and intuitive method to discard outliers we exemplify this approach with the case of the ontic fuzzy trimmed mean variance and highlight the differences with the epistemic case all the discussions and developments are illustrated by means of a case study concerning the perception of lengths of men and women

ostrowski type inequalities for functions whose higher order derivatives have a single point of non differentiability ostrowski type inequalities for the class of functions whose n 1 th order derivatives are continuous of bounded variation and have a single point of non differentiability are derived special attention is given to functions whose first derivative has a single point of non differentiability improvements of some previously obtained results are provided

a random set characterization of possibility measures several authors have pointed out the relationship between consonant random sets and possibility measures however this relationship has only been proven for the finite case where the inverse mobius of the upper probability induced by the random set simplifies the computations to a great extent in this paper we study the connection between both concepts for arbitrary referential spaces we complete existing results about the lack of an implication in general with necessary and sufficient conditions for the most interesting cases

some complete metrics on spaces of fuzzy subsets the classes of the lp and lp metrics play an important role to develop a probability theory in fuzzy sample spaces all of these metrics are known to be separable but not complete the classes are closely related as for each lp metric there exists some lp metric which induces the same topology this paper deals with the completion of the lp and lp metrics we can also show that the relationship between the classes of lp and lp metrics still holds for the obtained respective classes of their completions

a class of fuzzy random optimization expected value models fuzzy random variable is a measurable mapping from a probability space to a collection of fuzzy variables this concept may be regard as an extension of both random variable and fuzzy variable in this paper the linearity of a scalar value expected value operator of fuzzy random variable is discussed and a fuzzy random simulation approach is suggested to evaluate the expected value of a fuzzy random variable in addition three types of fuzzy random expected value models are presented to model fuzzy random decision systems moreover a hybrid intelligent algorithm which incorporates simulation neural network and genetic algorithm is designed in order to solve general fuzzy random expected value models at the end of this paper the effectiveness of this algorithm is showed via three numerical examples

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

on the variance of random sets on considere la notion de variance d un ensemble aleatoire on donne un algorithme de determination de la variance des ensembles aleatoires prenant leurs valeurs dans des intervalles compacts non vides sur r

upper and lower probabilities induced by a fuzzy random variable we review two existing interpretations of fuzzy random variables in the first one the fuzzy random variable is viewed as a linguistic random variable in the second case it represents some incomplete knowledge about an otherwise standard random variable for each interpretation the information provided by the frv is described by a specific model namely a standard probability model and a second order imprecise model respectively in this paper we deal with an alternative interpretation guided by simple examples we will observe the usefulness of each interpretation when applied to particular situations then we will demonstrate that the new interpretation leads in a natural way to the pair of order capacities furthermore we show that they are formally related to the former models these results can be applied in future works to make inferences from fuzzy sample data the use of upper lower models instead of second order models will enable us in the future to reach crisp decisions in some specific statistical problems without adding any arbitrary information and taking into account the imprecision in data

unifying parameter learning and modelling complex systems with epistemic uncertainty using probability interval knowledge regarding complex systems are heterogeneous and fragmented modelling dynamic complex systems in the framework of dynamic credal networks practical methodology coupling dirichlet distributions with interval probabilities to incrementally build and update model parameters whatever source and format of knowledge enables to take into account 1 stochastic and epistemic uncertainties pertaining to the system 2 the confidence level on the different sources of information illustrate the application of the methodology to the modelling of a simplified industrial case study modeling complex dynamical systems from heterogeneous pieces of knowledge varying in precision and reliability is a challenging task we propose the combination of dynamical bayesian networks and of imprecise probabilities to solve it in order to limit the computational burden and to make interpretation easier we also propose to encode pieces of numerical knowledge as probability intervals which are then used in an imprecise dirichlet model to update our knowledge the idea is to obtain a model flexible enough so that it can easily cope with different uncertainties i e stochastic and epistemic integrate new pieces of knowledge as they arrive and be of limited computational complexity

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

advocating the use of imprecisely observed data in genetic fuzzy systems in our opinion and in accordance with current literature the precise contribution of genetic fuzzy systems to the corpus of the machine learning theory has not been clearly stated yet in particular we question the existence of a set of problems for which the use of fuzzy rules in combination with genetic algorithms produces more robust models or classifiers that are inherently better than those arising from the bayesian point of view we will show that this set of problems actually exists and comprises interval and fuzzy valued datasets but it is not being exploited current genetic fuzzy classifiers deal with crisp classification problems where the role of fuzzy sets is reduced to give a parametric definition of a set of discriminant functions with a convenient linguistic interpretation provided that the customary use of fuzzy sets in statistics is vague data we propose to test genetic fuzzy classifiers over imprecisely measured data and design experiments well suited to these problems the same can be said about genetic fuzzy models the use of a scalar fitness function assumes crisp data where fuzzy models a priori do not have advantages over statistical regression

sensitivity minimization by strongly stabilizing controllers for a class of unstable time delay systems weighted sensitivity minimization is studied within the framework of strongly stabilizing stable hinfin controller design for a class of infinite dimensional systems this problem has been solved by ganesh and pearson for finite dimensional plants using nevanlinna pick interpolation we extend their technique to a class of unstable time delay systems moreover we illustrate suboptimal solutions and their robust implementation

random sets as imprecise random variables given a random set coming from the imprecise observation of a random variable we study how to model the information about the probability distribution of this random variable specifically we investigate whether the information given by the upper and lower probabilities induced by the random set is equivalent to the one given by the class of the probabilities induced by the measurable selections together with sufficient conditions for this we also give examples showing that they are not equivalent in all cases

on the synthesis of stable spl hscr sup spl infin controllers a sufficient condition is derived for the existence of a stable suboptimal spl hscr sup spl infin controller it is shown that if a certain algebraic riccati equation determined from a parameterization of all suboptimal spl hscr sup spl infin controllers has a positive definite stabilizing solution then a stable spl hscr sup spl infin controller of order 2n can be constructed where n is the order of the generalized plant

random intervals as a model for imprecise information random intervals constitute one of the classes of random sets with a greater number of applications in this paper we regard them as the imprecise observation of a random variable and study how to model the information about the probability distribution of this random variable two possible models are the probability distributions of the measurable selections and those bounded by the upper probability we prove that under some hypotheses the closures of these two sets in the topology of the weak convergence coincide improving results from the literature moreover we provide examples showing that the two models are not equivalent in general and give sufficient conditions for the equality between them finally we comment on the relationship between random intervals and fuzzy numbers

the variance and covariance of fuzzy random variables and their applications the concepts of the variance and covariance of fuzzy random variables and their properties are introduced examples show their computation and applications in statistical estimation of parameters when samples or prior information are fuzzy as their further applications the correlation function and the criterions of mean square calculus for fuzzy stochastic processes are established

stable lqg controllers the optimal lqg controller is known to stabilize the closed loop if certain mild conditions are satisfied the controller itself however may be unstable the paper presents a method of selecting the weighting and covariance matrices such that the optimal controller is internally asymptotically stable the method is very easy to apply and for stable open loop system involves the solution of a single lyapunov equation

genetic learning of fuzzy rules based on low quality data genetic fuzzy systems gfs are based on the use of genetic algorithms for designing fuzzy systems and for providing them with learning and adaptation capabilities in this context fuzzy sets represent linguistic granules of information contained in the antecedents and consequents of the rules whereas the data used in the genetic learning is assumed to be crisp gfs seldom deal with fuzzy valued data in this paper we address this problem and propose a set of techniques that can be incorporated to different gfs in order to learn a knowledge base kb from interval and fuzzy data for regression problems details will be given about the representation of non standard data with fuzzy sets about the needed changes in the reasoning method of the fuzzy rule based system and also about a new generalization of the mean squared error to vague data in addition we will show that the learning process requires a genetic algorithm that must be capable of optimizing a multicriteria fitness function containing both crisp and interval valued criteria lastly we benchmark our procedures with some machine learning related datasets and a real world problem of marketing and the techniques proposed here are shown to improve the generalization properties of other kbs obtained from crisp training data

new sufficient conditions for strong simultaneous stabilization abstract we present sufficient conditions for the existence of a single stable controller to stabilize a set of n siso plants p 1 p 2 p n strong simultaneous stabilization as is well known this is equivalent to the existence of a single controller not necessarily stable to stabilize n 1 plants simultaneous stabilization the basic assumption required in this paper is that all the plants have the same unstable zeros a practical example of where such an assumption holds is when each plant has the same pure delay which is approximated by a non minimum phase rational function the results here are applied to examples of this type

imprecise distribution function associated to a random set some different extensions to random sets of the most common parameters of a random variable share a common rationale a random set represents the imprecise observation of a random variable hence the generalized parameter contains the available information about the respective parameter of the imprecisely observed variable following the same principles in this paper it is proposed a new definition of the distribution function of a random set this definition is simpler in its formulation and it can be used in more general cases than previous proposals the properties of the distribution function defined here are discussed some issues about continuity convergence of the images of the distribution function monotonocity and measurability are studied it is also stated that not all the information conveyed by the random set about the original probability measure the probability measure induced by the imprecisely observed random variable is kept by its distribution function

h sub spl infin strong stabilization this note presents a technique for designing stable h sub spl infin controllers similar to some methods in the existing literature the proposed method also uses the parameterization of all suboptimal h sub spl infin controllers so that the stable h sub spl infin design problem can be conservatively converted into another 2 block standard h sub spl infin problem however a weighting function is introduced in this method to alleviate the conservativeness of the previous formulations it is further shown that the resulting high order controller can be significantly reduced by a two step reduction algorithm numerical examples are presented to demonstrate the effectiveness of the proposed method

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

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

graded sets and points a stratified approach to fuzzy sets and points abstract in this paper we define the concepts of graded set and graded point as well as the usual theoretical set operations and relations and the extension of mappings between graded sets these definitions provide a method to extend concepts of classical set theory this method is related to fuzzy set theory and generalizes some properties of the family of cuts moreover the graded points verify the following two properties which cannot happen with fuzzy points

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

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

a behavioural model for vague probability assessments i present an hierarchical uncertainty model that is able to represent vague probability assessments and to make inferences based on them this model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty and is based on walley s theory of imprecise probabilities it is formally closely related to zadeh s fuzzy probabilities but it has a different interpretation and a different calculus through rationality coherence arguments the hierarchical model is shown to lead to an imprecise first order uncertainty model that can be used in decision making and as a prior in statistical reasoning

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

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

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