on viewing fuzzy measures as fuzzy subsets we introduce the concept of a fuzzy measure discuss some of their basic properties and look at some notable examples we introduce a correspondence between a fuzzy measure on x and a fuzzy subset over the power set of x these fuzzy subsets are required to have special properties on the space 2x and we refer to these as measure type fuzzy sets we take advantage of this correspondence between a fuzzy measure and a fuzzy set to suggest the formulation of various set operations on fuzzy measures

exchangeability and convergence for random sets abstract in many statistical analysis some kind of dependency of random variables may be required and exchangeability as an alternative to the random sample with independent identically distributed random variables gives the study of asymptotic properties of random variables on the other hand statistical estimators are expressible of the linear form of random variables which involves possibly random weights and possibly functions of dependent random variables which are permutation invariant with respect to distributions this paper attempts to provide some properties useful to the limit theorems and laws of large numbers for arrays of banach spaced valued fuzzy random sets are examined with emphasis on permutation invariance among the random sets

a strong law of large numbers for fuzzy random sets abstract an slln for fuzzy random variables in rp including other convergence theorems has been proved by klement puri and ralescu the present paper is motivated by the definitions and properties developed by puri and ralescu and concerned with an slln for banach spaced valued fuzzy random sets as an easy application of the previous results

robust characterizations of polynomials withapplications to program testing the study of self testing and self correcting programs leads to the search for robust characterizations of functions here the authors make this notion precise and show such a characterization for polynomials from this characterization the authors get the following applications simple and efficient self testers for polynomial functions are constructed the characterizations provide results in the area of coding theory by giving extremely fast and efficient error detecting schemes for some well known codes this error detection scheme plays a crucial role in subsequent results on the hardness of approximating some np optimization problems

convergence of conditional expectations and strong laws of large numbers for multivalued random variables on etablit des lemmes de fatou et des theoremes de convergence de lebesgue pour des esperances conditionnelles multivaluees de variables aleatoires a valeurs dans des sous ensembles fermes d un espace de banach separable

a note on the conditional probability of fuzzy subsets of a continuous domain abstract the notion of a mass assignment corresponding to a discrete fuzzy set is generalised to include all fuzzy sets the definition is motivated by considering a possible mechanism by which an intelligent agent could extend set theoretic operations to fuzzy sets this same mechanism also suggests an alternative definition for the conditional probability of fuzzy sets it is noted that this definition is probability possibility consistent and it is shown that in the case where the probability distribution on the domain has a density function then the conditional probability of f given g can be expressed as the probability of f calculated relative to a conditional density function dependent on g this result is used to develop an analytical method for calculating conditional probabilities of fuzzy subsets of the reals where the membership functions of f and g and the density function are continuous piecewise linear functions such a method has applications to the development of a general semantic unification in fril

statistical mechanics of stochastic neural networks relationship between the self consistent signal to noise analysis thouless anderson palmer equation and replica symmetric calculation approaches we study the statistical mechanical aspects of stochastic analog neural network models for associative memory with correlation type learning we take three approaches to derive the set of the order parameter equations for investigating statistical properties of retrieval states the self consistent signal to noise analysis scsna the thouless anderson palmer tap equation and the replica symmetric calculation on the basis of the cavity method the scsna can be generalized to deal with stochastic networks we establish the close connection between the tap equation and the scsna to elucidate the relationship between the onsager reaction term of the tap equation and the output proportional term of the scsna that appear in the expressions for the local fields

a hierarchical neural model of matching abstract the issue of matching two fuzzy sets emerges as an essential design aspect in many algorithms of processing fuzzy information including fuzzy controllers pattern classifiers knowledge based systems etc this paper introduces a new model of matching the primordial features of the proposed model involve 1 i matching being carried out with respect to the grades of membership of the discussed fuzzy sets as well as some of their functionals like energy entropy transom 2 ii the concepts of hierarchies of matching in the model leading directly to a straightforward distinction between local and global levels of matching and 3 iii a distributed character of the model that is realized as a logic based neural network with enhanced learning capabilities

strong laws of large numbers for arrays of rowwise independent random compact sets and fuzzy random sets in this paper we obtain some strong laws of large numbers sllns for arrays of rowwise independent not necessary identically distributed random compact sets and fuzzy random sets whose underlying spaces are separable banach spaces

learning possibilistic graphical models from data graphical models especially probabilistic networks like bayes networks and markov networks are very popular to make reasoning in high dimensional domains feasible since constructing them manually can be tedious and time consuming a large part of recent research has been devoted to learning them from data however if the dataset to learn from contains imprecise information in the form of sets of alternatives instead of precise values this learning task can pose unpleasant problems in this paper we survey an approach to cope with these problems which is not based on probability theory as the more common approaches like e g expectation maximization but uses the possibility theory as the underlying calculus of a graphical model we provide semantic foundations of possibilistic graphical models explain the rationale of possibilistic decomposition as well as the graphical representation of decompositions of possibility distributions and finally discuss the main approaches to learn possibilistic graphical models from data

a linear regression model using triangular fuzzy number coefficients fuzzy regression analysis using fuzzy linear models with symmetric triangular fuzzy number coefficients has been formulated earlier the goal of this regression is to find the coefficients of a proposed model for all given input output data sets in this paper we extend the results of a fuzzy linear regression model that uses symmetric triangular coefficient to one with non symmetric fuzzy triangular coefficients this work eradicates the inflexibility of existing fuzzy linear regression models

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

the theory of fuzzy stochastic processes abstract in this paper we deal with the general theory of fuzzy stochastic processes we give the suitable definitions of fuzzy random function fuzzy stochastic process and their fall shadow distribution functions the concepts of separability and measurability of a fuzzy random function are introduced and some properties of these new concepts are derived we investigate the continuity of fuzzy sample function of a fuzzy stochastic process

time to first spike in stochastic hodgkin huxley systems abstract the time to first spike is an experimentally observed quantity in laboratory experiments in the auditory somatic and visual sensory modalities the times of first spikes in the corresponding cortical neurons have been implicated as coding much of the information about stimulus properties we describe an analytical approach for determining the time to first spike from a given initial state which may be applied to a general nonlinear stochastic model neuron we illustrate with a standard hodgkin huxley model with a gaussian white noise input current whose drift parameter is and whose variance parameter is partial differential equations pdes of second order are obtained for the first two moments of the time taken for the depolarization to reach a threshold value from rest state as functions of the initial values simulation confirms that for small noise amplitudes a 2 component model is reasonably accurate for small values of the noise parameter including the deterministic case 0 perturbation methods are used to find the moments of the firing time and the results compare favorably with those from simulation the approach is accurate for almost all when is above threshold for action potentials in the absence of noise and over a considerable range of values of when is as small as 2 the same methods may be applied to models similar to hodgkin huxley which involve channels for additional or different ionic currents

law of large numbers for non additive measures our aim is to establish law of large numbers for classes of non additive measures for balanced games we obtain weak and strong law of large numbers for bounded random variables a sharper result is obtained for exact games we also provide an extension to upper envelope measures

probability and fuzziness we discuss some relationships between probability theory and statistics on one hand and the theory of fuzzy sets on the other hand we develop various statistical techniques for the analysis of imprecise data and for inference based on imprecise data we study the estimation of fuzzy parameters testing problems with fuzzy probabilities and confidence estimation under uncertainty finally we indicate some possible applications to the assessment of credibility in expert systems

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

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

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

aggregating constraint satisfaction degrees expressed by possibilistic truth values in information systems one often has to deal with constraints in order to compel the semantics and integrity of the stored information or to express some querying criteria hereby different constraints can be of different importance a method to aggregate the information about the satisfaction of a finite number of constraints for a given data instance is presented central to the proposed method is the use of extended possibilistic truth values to express the degree of satisfaction of a constraint and the use of residual implicators and residual coimplicators to model the impact and relevance of a constraint the proposed method can be applied to any constraint based system a database application is discussed and illustrated

representation similarity measures and aggregation methods using fuzzy sets for content based recommender systems representation of features of items and user feedback and reasoning about their relationships are major problems in recommender systems this is because item features and user feedback are subjective imprecise and vague the paper presents a fuzzy set theoretic method ftm for recommender systems that handles the non stochastic uncertainty induced from subjectivity vagueness and imprecision in the data and the domain knowledge and the task under consideration the research further advances the application of fuzzy modeling for content based recommender systems initially presented by ronald yager the paper defines a representation method similarity measures and aggregation methods as well as empirically evaluates the methods performance through simulation using a benchmark movie data ftm consist of a representation method for items features and user feedback using fuzzy sets and a content based algorithm based on various fuzzy set theoretic similarity measures the fuzzy set extensions of the jaccard index cosine proximity or correlation similarity measures and aggregation methods for computing recommendation confidence scores the maximum minimum or weighted sum fuzzy set theoretic aggregation methods compared to the baseline crisp set based method csm presented the empirical evaluation of the ftm using the movie data and simulation shows an improvement in precision without loss of recall moreover the paper provides a guideline for recommender systems designers that will help in choosing from a combination of one of the fuzzy set theoretic aggregation methods and similarity measures

reliability estimation and design with insufficient data based on possibility theory early in the engineering design cycle it is difficult to quantify product reliability or compliance to performance targets because of insufficient data or information for modeling the uncertainties design decisions are therefore based on fuzzy information that is vague imprecise qualitative linguistic or incomplete the uncertain information is usually available as intervals with lower and upper limits in this work the possibility theory is used to assess design reliability with incomplete information the possibility theory can be viewed as a variant of fuzzy set theory the formal theories to handle uncertainty are first introduced using the theoretical fundamentals of fuzzy measures a computationally efficient and accurate hybrid global local optimization approach is subsequently described for calculating the confidence level of fuzzy response the method combines the advantages of the commonly used vertex and discretization methods subsequently the possibility theory is used in design a possibility based design optimization method is proposed where all design constraints are expressed in a possibilistic way it is shown that the method gives a conservative solution compared with all conventional reliability based designs obtained with different probability distributions finally a general possibility based design optimization method which handles a combination of random and possibilistic design variables is presented several numerical examples demonstrate the application of possibility theory in design

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

computational aspects of the coarsening at random model and the shapley value this paper is concerned with computational aspects of coarse data in information technology a method for calculating the car coarsening at random model of a finite random set is obtained and a formula for computing the shapley value using the distribution of the random set is given the relationship between the car model solution and the shapley value is also discussed several examples are given to illustrate our results

a prototype based rule inference system incorporating linear functions a calculus of appropriateness measures of linguistic expressions is proposed which is based on the prototype theory and random set theory interpretation of vague concepts a prototype based rule inference system is then introduced to incorporate linguistic labels in the rule antecedents and linear functions in the consequents of rules and a rule learning algorithm is developed by combining a new clustering algorithm and a conjugate gradient algorithm the proposed prototype based inference system is then applied to a number of benchmark prediction problems including a nonlinear two dimensional surface the mackey glass time series and the sunspot time series results suggest that the proposed model is very robust and can perform well in high dimensional noisy data

random set theory and problems of modeling the three or four dimensional world in which we live is full of objects to be measured and summarized very often a parsimonious finite collection of measurements is enough for scientific investigation into an object s genesis and evolution there is a growing need however to describe and model objects through their form as well as their size the purpose of this article is to show the potentials and limitations of a probabilistic and statistical approach collections of objects the data are assimilated to a random set the model whose parameters provide description and or explanation

testing whether a digraph contains h free k induced subgraphs a subgraph induced by k vertices is called a k induced subgraph we prove that determining if a digraph g contains h free k induced subgraphs is w n 2 evasive then we construct an e tester to test this property an e tester for a property p is guaranteed to distinguish with probability at least 2 3 between the case of g satisfying p and the case of g being e far from satisfying p the query complexity of the e tester is independent of the size of the input digraph an e d tester for a property p is an e tester for p that is furthermore guaranteed to accept with probability at least 2 3 any input that is d close to satisfying p this paper presents an e d tester for whether a digraph contains h free k induced subgraphs

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

ergodic theorems for random compact sets and fuzzy variables in banach spaces abstract the individual ergodic theorem is proved in two cases first for compact set valued random variable in a banach space h of type p p 1 second for a random variable having values in the family of all fuzzy subsets of h the set representation of fuzzy subsets is used three types of convergence of fuzzy variables are considered

a uniform strong law of large numbers for partial sum processes of fuzzy random variables indexed by sets we consider random intervals as measurable mappings from a probability space into the set of intervals of r and prove a uniform strong law of large numbers for sequences of independent and identically distributed random intervals also we consider fuzzy random variables and prove a uniform strong law of large numbers for sequences of fuzzy random variables our results generalize that of bass and pyke ann probab 12 1984 268

strong laws of large numbers for t norm based addition of fuzzy set valued random variables this paper proposes a convergence theorem of fuzzy set valued random variables under addition which is given by the general triangular normed extension principles applying the result with the known results of the various types for strong laws of large numbers of fuzzy set valued random variables we obtain a few interesting results of the strong laws of large numbers for triangular norm based addition of fuzzy set valued random variables the finding of teran strong law of large numbers for t normed arithmetics fuzzy sets and systems 159 2008 343 360 is a special case of our results

a law of large numbers for random sets abstract we show a law of large numbers for random sets taking values in a class of subsets of a banach space which is at least larger than the class of compact subsets of the space if the banach space is of infinite dimension to see this we first define the hausdorff distance on the family of all the non empty subsets of a banach space and give some properties of it motivated by the fact that a study of fuzzy sets is to study of set valued analysis

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

sampling fuzzy discretization and signal reconstruction abstract the sampling theorem provides the condition under which a continuous time signal can be precisely represented and completely recovered from its instantaneous values equally spaced in time it gives a condition for the signal reconstruction problem the importance of this theorem pertains to the necessity of knowing when a continuous time signal is uniquely represented by its discrete version so that continuous time signals can be properly converted and handled by sampled data systems like digital computers the problem of the equivalence between either continuous time or discrete time signals and their fuzzy descriptions is analyzed here this encompasses the analysis of signal reconstruction when fuzzy systems are used as sampled data systems such as fuzzy filters fuzzy models or fuzzy controllers a necessary and sufficient condition is given for ensuring that the fuzzy system processes the legitimate representations of the signal presented at its interfaces furthermore a sufficient condition suitable for automatic tuning of membership functions is proposed and discussed the discussion encompasses the outline of some requirements that membership functions at fuzzy system interfaces should meet to allow humans to assess their semantic meaning

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

uncertain robot environment modelling using fuzzy numbers abstract in this paper we present a fuzzy oriented methodology to model an uncertain geometric robot environment and to manipulate geometric uncertainties between robot coordinate frames we describe any geometric primitive of robot environment as a parameter vector in parameter space not only ill known values of the parameterized geometric primitives but the uncertain quantities of coordinate transformations are represented by means of fuzzy numbers restricted to appropriate membership functions for consistent interpretation about geometric primitives between different coordinate frames we manipulate these uncertain quantities using fuzzy arithmetic as an application we deal with a 2 dimensional straight line correspondence problem of a mobile robot vision system this example shows the usefulness of the proposed methodology

a generalized strong law of large numbers strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces in this paper a strong law of large numbers which generalizes some previous ones like those for real valued random variables and compact random sets is established this law is an example of a strong law of large numbers for borel measurable nonseparably valued elements of a metric space

spatial neuron model with two parameter ornstein uhlenbeck input current abstract we consider a new and extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation the synaptic input current is also a function of space and time and satisfies a first order linear partial differential equation driven by a two parameter random process a natural choice for these random input processes is to make them two parameter poisson processes for both excitation and inhibition for such inputs the mean subthreshold voltage is found in the case of an infinite cable and of finite cables assuming uniform amplitudes and rates exact expressions are obtained in the case of particular boundary conditions we then consider a diffusion approximation and show that the membrane current is a two parameter ornstein uhlenbeck process whose statistical properties are derived using representations for the voltage in terms of stochastic integrals in the plane we find in the case of finite space intervals the mean variance and covariance of the subthreshold voltage for large times the voltage process is shown to be covariance stationary in time and the corresponding spectral density is found and compared with the result for a purely two parameter white noise driven cable the limiting white noise case is obtained from the extended model as the decay parameter becomes infinite finally we develop useful simulation methods for the solution of the extended spatial model using properties of stochastic integrals involving eigenfunctions to obtain one dimensional representations which are easily implemented

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

strong laws of large numbers for independent fuzzy set valued random variables abstract in this paper we shall present strong laws of large numbers slln s for independent not necessary identically distributed fuzzy set valued random variables whose base space is a separable banach space or an euclidean space in the sense of the extended hausdorff metric d h we apply the method to the sequence of independent identically distributed fuzzy set valued random variables to give a simple proof of slln s

efficiencies of two stage systems with fuzzy data two stage dea data envelopment analysis models show the performance of individual processes and thus are more informative than the conventional one stage models for making decisions this paper extends this approach from deterministic to uncertain situations where the observations are represented by fuzzy numbers the extension principle is utilized to develop a pair of two level mathematical programs to calculate the lower and upper bounds of the a cut of the fuzzy efficiency by enumerating various values of a the membership functions of fuzzy efficiencies are constructed numerically it is found that the property of the system efficiency being equal to the product of the two process efficiencies which holds for the deterministic case also holds for the fuzzy case this property can be generalized to series systems with more than two processes an example of non life insurance companies in taiwan is used to explain how to calculate the system and process efficiencies and how to derive their relationship when the data is fuzzy

strong limit theorems for random sets and fuzzy random sets with slowly varying weights theories of random sets and fuzzy random sets are useful concepts which are frequently applied in scientific areas including information science probability and statistics in this paper strong limit theorems are derived for random sets and fuzzy random sets with slowly varying weights in separable banach spaces both independent and dependent cases are covered to provide a wide range of applications

on some optimization problems under uncertainty abstract the last ten years show a diversification of the approaches for handling uncertainties in optimization problems by certain relaxations or on the contrary seeking a more conservative solution in this paper we present some new results and points of view in nonlinear inexact and fuzzy mathematical programming possible applications to capital budgeting of interrelated projects and to production planning are also discussed

characterization and comparison of sugeno and choquet integrals abstract starting from the concept of equiordered functions we characterize two types of fuzzy integrals that can be defined on any kind of fuzzy measure sugeno and choquet integrals the characterization theorems that we obtain allow us to carry out a comparative study of these two fuzzy integrals pointing out their formal similarities and their conceptual differences

convergence of set valued sub and supermartingales in the kuratowski mosco sense the purpose of this paper is to prove some convergence theorems of closed and convex set valued sub and supermartingales in the kuratowski mosco sense to get submartingale convergence theorems we give sufficient conditions for the kudo aumann integral and hiai umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables we also give an example of a bounded closed convex set valued random variable whose kudo aumann integral is not closed

random fuzzy sets and fuzzy martingales abstract the present paper is motivated by the definitions and properties developed by puri and ralescu 1986 we study the random fuzzy sets and their relationship to fuzzy set valued measures in a separable banach space using the conditional expectations of random fuzzy sets we introduce the concept of fuzzy martingales some properties and convergence theorems of fuzzy martingales are investigated

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

possibility and necessity in optimization abstract we employ possibility and necessity in fuzzy programming and in other optimization problems that traditionally use random variables the possibility approach to multiobjective programming produces the same results as fuzzy programming a detailed example is presented to illustrate the use of possibility and necessity in optimization problems that usually employ random variables to describe uncertainty

granular knowledge representation and inference using labels and label expressions this paper is a review of the label semantics framework as an epistemic approach to modeling granular information represented by linguistic labels and label expressions the focus of label semantics is on the decision making process that a rational communicating agent must undertake in order to establish which available labels can be appropriately used to describe their perceptual information in such a way as they are consistent with the linguistic conventions of the population as such it provides an approach to characterizing the relationship between labels and the underlying perceptual domain which we propose lies at the heart of what is meant by information granules furthermore it is then shown that there is an intuitive relationship between label semantics and prototype theory which provides a clear link with zadeh s original conception of information granularity for information propagation linguistic mappings are introduced which provide a mechanism to infer labeling information about a decision variable from the available labeling information about a set of input variables finally a decision making process is outlined whereby from linguistic descriptions of input variables we can infer a linguistic description of the decision variable and where required select a single expression describing that variable or a single estimated value

on modeling of linguistic information using random sets this paper discusses the formal connection between possibility distributions zadeh 21 and the theory of random sets via choquet s theorem based upon these relationships it is suggested that plausible inferences and modeling of common sense can be derived from the statistics of random sets the analysis of subjectivity in meaning representation of natural languages can be carried out by taking account of covariates of individuals as in the statistical analysis of survival data

fuzzy integrals on l fuzzy sets abstract the fuzzy measures and fuzzy integrals for zadeh s fuzzy sets were studied by qiao and wang 5 6 7 12 in this paper we consider the real valued fuzzy measures and fuzzy integrals for l fuzzy sets where l is a pseudocomplemented infinitely distributive complete lattice on a fuzzy algebra of l fuzzy sets the concepts of fuzzy measure and fuzzy integral are introduced some equivalent forms of fuzzy integrals on l fuzzy sets are given moreover we discuss several properties of fuzzy integrals on l fuzzy sets and prove some convergence theorems for a sequence of fuzzy integrals on l fuzzy sets

network of probabilities associated with a capacity of order 2 abstract this paper studies capacities of order 2 from a geometric point of view that is from convex polytope of probabilities defined by them this convex polytope is characterized by means of a network of probabilities representing relationships among vertices defined by the edges interesting properties may be obtained about the capacities from this network

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

advanced fuzzy inference engines in situation aware computing we focus on the very important family of context aware applications context aware computing relies on tasks like capturing sensing environmental parameters e g lightness location classifying context and inferring further knowledge about that context determine the situation the user is currently in however the relevant applications have to deal with the inherent imperfection of context sensing for decision making we propose an extension of context representation and inference for situation aware applications our model relies on fuzzy set theory to accommodate the imperfection of sensed context based on this model we have developed three fuzzy inference engines which rely on advanced semantics specialization mereological and compatibility relations we have evaluated the proposed engines through a series of experiments involving real users our findings indicate the strong points of the proposed context classification and inference processes

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

user preferences discovery using fuzzy models user preferences discovery aims to learn the patterns of user preferences for various services or items such as movies preferences discovery is essential to the development of intelligent personalization applications based on decision and utility theories traditional approaches to preferences discovery explicitly query users about the behavior of value function or utility of every outcome with respect to each decision criterion consequently these approaches are generally error prone and labor intensive although implicit elicitation approaches have been proposed to address the above limitations extent approaches largely ignore multi valued nature of item features and uncertainty associated with item features and user preferences to address uncertainty due to vagueness and imprecision this research proposed a general framework for preferences discovery based on fuzzy set theories in addition new fuzzy models were created for preferences discovery and representation further an algorithm was developed to predict user preferences with uncertainty and visualization of item features user feedback and the discovered preferences helped improve the interpretation of the discovered knowledge the results of the simulation evaluation using a benchmark movie dataset revealed that the proposed preference discovery method 1 doubled the accuracy of preference discovery as compared to random prediction and 2 outperformed conventional techniques in making movie recommendation these findings suggest that fuzzy models are effective for preferences patterns discovery and personalized recommendation application

a set of practical experiments to validate the fuzzy kalman filter abstract the fuzzy kalman filter fkf was recently reformulated with a number of basic examples included in order to demonstrate its applicability we now look ahead to its practical implementation introducing deep analysis which includes very detailed simulations and complete real applications this will allow us to not only validate the fkf but also to draw important conclusions which will help users in the adjustment of the fuzzy filter parameters and the choosing of possibilistic distributions

the evidential reasoning approach for mada under both probabilistic and fuzzy uncertainties abstract many multiple attribute decision analysis mada problems are characterised by both quantitative and qualitative attributes with various types of uncertainties incompleteness or ignorance and vagueness or fuzziness are among the most common uncertainties in decision analysis the evidential reasoning er approach has been developed in the 1990s and in the recent years to support the solution of mada problems with ignorance a kind of probabilistic uncertainty in this paper the er approach is further developed to deal with mada problems with both probabilistic and fuzzy uncertainties in this newly developed er approach precise data ignorance and fuzziness are all modelled under the unified framework of a distributed fuzzy belief structure leading to a fuzzy belief decision matrix a utility based grade match method is proposed to transform both numerical data and qualitative fuzzy assessment information of various formats into the fuzzy belief structure a new fuzzy er algorithm is developed to aggregate multiple attributes using the information contained in the fuzzy belief matrix resulting in an aggregated fuzzy distributed assessment for each alternative different from the existing er algorithm that is of a recursive nature the new fuzzy er algorithm provides an analytical means for combining all attributes without iteration thus providing scope and flexibility for sensitivity analysis and optimisation a numerical example is provided to illustrate the detailed implementation process of the new er approach and its validity and wide applicability

fuzzy inference relation based on the theory of falling shadows abstract in this paper we establish a theoretical approach to define a fuzzy inference relation based on the theory of falling shadows the main characteristic of our definition of fuzzy inference relation is that it is semantically dependent in the sense that the formula of our definition will vary according to the correlation of the antecedent and the consequence of the given implication we shall show that the formulae of fuzzy inference relation given by ukasiewicz zadeh and the probability formula are consequences of our definition under three different correlations of the propositions

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

convexification in limit laws of random sets in banach spaces a strong law of large numbers for random compact sets in a banach space with the minkowski addition is provided by puri and ralescu 6 and among other results by gine hahn and zinn 2 and hess 3 this follows the finite dimensional version introduced in 1 the latter however allows nonconvex sets while the generalizations to the banach space setting were proved for convex sets a convexification argument stated in the lemma below shows that the convexity assumption can be dropped the proof reveals information about the nature of the convergence since writing this note we learned that f hiai 4 has independently established this result using a different method some notations let b be a banach space with norm 1111 the minkowski addition of two compact subsets k and l is k l x y x e k y e li and ak is the set tax x e k the closed convex hull of k is denoted cok it is a linear operation the hausdorff distance h k l is defined as max maxxekminyel 11 x y 11 maxyelminxek 11 x y y convergence of sets in this paper is meant in the hausdorff metric

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

random set system identification the paper gives a brief review of the basic mathematical aspects of random set theory concepts such as a random set mapping and its coverage function are introduced in a comprehensive way avoiding too much detail we adapt this theory to system identification and forecasting of time series this is achieved by using the one point coverage function of a random set as a possibility measure of the process which generates such a time series the coverage function of a random set defines a fuzzy set and we thereby establish the relationship between statistical objects and fuzzy systems the possibility measure obtained in this way can be used for either prediction or to evaluate the quality of a model with respect to the training data the technique is adapted to nonlinear time series analysis a practical application of a nonlinear dynamic plant is presented