are grades of membership probabilities abstract a small sample is given of the difficulties with present day fuzzy set theory with respect to the requirements of a scientific theory and a summary of some probabilistic interpretations of grades of membership to be found in the literature the tee model for grades of membership is then suggested instead of starting out from mathematical postulates such as the assumed max and min operations for or and and the tee model uses a semantic physio logical psycho logical starting point by investigating the possibilities which a person has for assigning linguistic labels and partial grades of membership in a meaningful way the assignment of partial grades of membership is explained by the assumption that humans have the ability to take errors of observation and intersubject differences into consideration a membership value uex is identified with the subject s estimate of the probability that the label e g tall would be assigned to an object of exact attribute value uex in a natural language situation this situation being a labeling or a yes no one in which at least one of three possible sources of uncertainty is present almost all the accepted formulas of fuzzy set theory as well as the shapes of the grade of membership curves are then derived instead of having to be postulated however the derived formulas for or and and reduce to the max and min operations only in a number of special cases in another special case they reduce to the summation and the zero operation respectively the generally valid formulas are presented

on the notion of concept ii it is well known that classical set theory is not expressive enough to adequately model categorization and prototype theory recent work on compositionality and concept determination showed that the quantitative solution initially offered by classical fuzzy logic also led to important drawbacks several qualitative approaches were thereafter tempted that aimed at modeling membership through ordinal scales or lattice fuzzy sets most of the solutions obtained by these theoretical constructions however are of difficult use in categorization theory we propose a simple qualitative model in which membership relative to a given concept f is represented by a function that takes its value in a finite abstract set a f equipped with a total order this function is recursively built through a stratification of the set of concepts at hand based on a notion of complexity similarly the typicality associated with a concept f will be described using an ordering that takes into account the characteristic features of f once the basic notions of membership and typicality are set the study of compound concepts is possible and leads to interesting results in particular we investigate the internal structure of concepts and obtain the characterization of all smooth subconcepts of a given concept

alternative characterizations for the representation of families of sets by fuzzy sets we investigate the representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets this concept of representation is widely used in the literature for the classification and study of fuzzy structures on the same underlying set negoita and ralescu obtained two theorems both giving necessary and sufficient conditions one called the representation theorem and the second called the generalized representation theorem our aim in this paper is to take a closer look at both theorems hence establishing some equivalent forms and several constructive remarks including some directions for future investigations

a fuzzy set preference model for consumer choice abstract a fuzzy set model for predicting consumer choice is developed by combining the vector conjoint model with fuzzy measurement in a new model that requires only ordinal data fuzzy set measurement of consumer preferences is examined by testing alternative set representations and an interactive set definition method a comparison test of predictive validity using identical data from 139 subjects showed large improvements in rank order prediction validity compared to a non fuzzy approach prediction improvements of 83 are attributable to the use of fuzzy set definitions for subject ratings with the first two of six ordered ranking predicted on average

determining the number of factors in the general dynamic factor model this article develops an information criterion for determining the number q of common shocks in the general dynamic factor model developed by forni et al as opposed to the restricted dynamic model considered by bai and ng and by amengual and watson our criterion is based on the fact that this number q is also the number of diverging eigenvalues of the spectral density matrix of the observations as the number n of series goes to infinity we provide sufficient conditions for consistency of the criterion for large n and t where t is the series length we show how the method can be implemented and provide simulations and empirics illustrating its very good finite sample performance application to real data adds a new empirical facet to an ongoing debate on the number of factors driving the u s economy

asymptotic nonequivalence of garch models and diffusions this paper investigates the statistical relationship of the garch model and its diffusion limit regarding the two types of models as two statistical experiments formed by discrete observations from the models we study their asymptotic equivalence in terms of le cam s deficiency distance to our surprise we are able to show that the garch model and its diffusion limit are asymptotically equivalent only under deterministic volatility with stochastic volatility due to the difference between the structure with respect to noise propagation in their conditional variances their likelihood processes asymptotically behave quite differently and thus they are not asymptotically equivalent this stochastic nonequivalence discredits a general belief that the two types of models are asymptotically equivalent in all respects and warns against the common financial practice that applies statistical inferences derived under the garch model to its diffusion limit

the concept of grade of membership abstract the concept of grade of membership is bound up with the general problem of fuzzy reasoning in this paper an approach to the latter problem is outlined and the consequences for the notion of grade of membership are described particular care is taken from the beginning to base the theory on concepts which have an immediate practical meaning for this reason assertions rather than sentences are taken as fundamental an assertion being characterized in terms of utility and decision theory fuzzy sentences and degrees of truth and then fuzzy sets and grades of membership are defined in terms of assertions the definitions being illustrated with examples the problem of defining unary and binary operations connectives for assertions sentences and fuzzy sets is discussed finally some other aspects of the logic of assertions are outlined and the extent to which the concepts of fuzzy set and grade of membership can be incorporated into the theory is discussed

a fuzzy neural network for pattern classification and feature selection a fuzzy neural network with memory connections for classification and weight connections for selection is introduced thereby solving simultaneously two major problems in pattern recognition pattern classification and feature selection the proposed network attempts to select important features from among the originally given plausible features while maintaining the maximum recognition rate the resulting value of weight connection represents the degree of importance of feature moreover the knowledge acquired by the network can be described as a set of interpretable rules the effectiveness of this new method has been validated by using anderson s iris data the results are first the use of two features selected by our method from among the original four in the proposed network results in virtually identical classifier performance and second the constructed classifier is described by three simple rules that are of if then form

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

a measure theoretic axiomatization of fuzzy sets abstract using measurement theory this paper examines three empirical structures that underlie the representation of fuzzy sets the fuzzy membership structure the fuzzy component structure and the fuzzy system structure these qualitative structures justify the use of the standard min max system to represent fuzzy sets the results of this study facilitate the development of a sound measurement theoretic axiomatization of fuzzy systems

a review and comparison of six reasoning methods abstract five fuzzy reasoning methods are reviewed and their performance is compared with respect to a fuzzy control system model developed by an objective method based on three sets of real life data it is found from the investigation that the reasoning precision the calculation time and the number of possible input states to which a given reasoning method responds differ according to each reasoning method generally the point valued reasoning methods which are based on the assumption that connectives are crisp give better precision and shorter calculation time when the connectives are assumed to be linguistic reasoning with interval valued fuzzy sets are more appropriate to represent linguistic uncertainty for this reason type ii fuzziness generated with interval valued fuzzy sets are better handled with interval valued reasoning methods

designing fuzzy sets with the use of the parametric principle of justifiable granularity this study is concerned with a design of membership functions of fuzzy sets the membership functions are formed in such a way that they are experimentally justifiable and exhibit a sound semantics these two requirements are articulated through the principle of justifiable granularity the parametric version of the principle is discussed in detail we show linkages with type 2 fuzzy sets which are constructed on a basis of type 1 fuzzy sets several experimental studies are reported which illustrate a behavior of the introduced method

fuzzy expert systems for ie or ms abstract fuzzy expert systems can be developed for the effective use of management within the domains of concern associated with industrial engineering ie operations research or and management science ms these models are designed with 1 expressive powers of representation embedded in linguistic variables and their linguistic values in natural language expressions and 2 improved methods of inference based on fuzzy logic which is a generalization of multi valued logic the results of these fuzzy expert system models are either 1 approximately good in comparison with their classical counterparts i e within a few percentage points of the optimal results of classical models as for example for the cases of production and spare parts planning or 2 much better than their counterparts as for example for the cases of job shop scheduling and consumer preference prediction in market share analysis in order to establish a sound basis for the preceding claims basic issues of representation and inference are first reviewed and then several prototype case studies are presented to support these claims

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

membership function elicitation in possibilistic programming problems in this paper we propose an elicitation method for membership functions in the setting of possibilistic programming each possibility distribution is defined by several possible domains with different possibility degrees taking advantage of the properties of possibility and necessity measures a nested target table for the objective function and for each constraint is elicited from a decision maker to aggregate all the goals an additional nested target table is established through those nested target tables membership functions can be determined it is emphasized that the information of nested target tables is much more important than that of membership values in the setting of possibilistic programming the membership functions in possibilistic programming are not assumed to be cardinal but ordinal in this sense the decision making via possibilistic programming can be regarded as a qualitative one

the measurement of membership by subjective ratio estimation when we perform algebraic operations on membership degrees we must be sure that they are compatible with the kind of scale on which these degrees are measured for example we will not add in a weighted average membership degrees that are measured on an ordinal scale an important question is therefore how to determine on which kind of scale we measure the membership there are several techniques for measuring membership in this paper we present some of them based on stevens technique of ratio estimation and we characterize them thereby providing a sound way to determine the level of measurement we also discuss the problem of the representation of the union or intersection by a t conorm or t norm

modelling dynamics of marathons a mixture model approach abstract in this paper statistical properties of marathon dynamics are studied we find that changes of velocity in marathons follow an unconventional mechanism in which the log change of velocity is highly dependent on current velocity with a complex relationship the conditional distributions of log change of velocity exhibit patterns of varying means variances and skewnesses as such the overall velocity distributions are also found to have departed from gaussian we illustrate the mechanism with a finite mixture of generalized linear regressions with varying weights and skew normal errors and we show that the completion time distribution can be approximated by skewed distributions

the symbol grounding problem there has been much discussion recently about the scope and limits of purely symbolic models of the mind and about the proper role of connectionism in cognitive modeling this paper describes the symbol grounding problem how can the semantic interpretation of a formal symbol system be made intrinsic to the system rather than just parasitic on the meanings in our heads how can the meanings of the meaningless symbol tokens manipulated solely on the basis of their arbitrary shapes be grounded in anything but other meaningless symbols the problem is analogous to trying to learn chinese from a chinese chinese dictionary alone a candidate solution is sketched symbolic representations must be grounded bottom up in nonsymbolic representations of two kinds 1 iconic representations which are analogs of the proximal sensory projections of distal objects and events and 2 categorical representations which are learned and innate feature detectors that pick out the invariant features of object and event categories from their sensory projections elementary symbols are the names of these object and event categories assigned on the basis of their nonsymbolic categorical representations higher order 3 symbolic representations grounded in these elementary symbols consist of symbol strings describing category membership relations e g an x is a y that is z connectionism is one natural candidate for the mechanism that learns the invariant features underlying categorical representations thereby connecting names to the proximal projections of the distal objects they stand for in this way connectionism can be seen as a complementary component in a hybrid nonsymbolic symbolic model of the mind rather than a rival to purely symbolic modeling such a hybrid model would not have an autonomous symbolic module however the symbolic functions would emerge as an intrinsically dedicated symbol system as a consequence of the bottom up grounding of categories names in their sensory representations symbol manipulation would be governed not just by the arbitrary shapes of the symbol tokens but by the nonarbitrary shapes of the icons and category invariants in which they are grounded

probably approximately correct learning in fuzzy classification systems an efficient method for learning trapezoidal membership functions for fuzzy predicates is presented positive and negative examples of one class are given together with a system of classification rules the learned membership functions can be used for the fuzzy predicates occurring in the given rules to classify further examples we show that the obtained classification is approximately correct with high probability this justifies the obtained fuzzy sets within one particular classification problem instead of relying on a subjective meaning of fuzzy predicates as normally done by a domain expert

eigenvalue ratio test for the number of factors this paper proposes two new estimators for determining the number of factors r in static approximate factor models we exploit the well known fact that the r largest eigenvalues of the variance matrix of n response variables grow unboundedly as n increases while the other eigenvalues remain bounded the new estimators are obtained simply by maximizing the ratio of two adjacent eigenvalues our simulation results provide promising evidence for the two estimators

scaling properties of marathon races abstract some regularities in popular marathon races are identified in this paper it is found for high performance participants i e racing times in the range 2 15 3 15 h the average velocity as a function of the marathoner s ranking behaves as a power law which may be suggesting the presence of critical phenomena elite marathoners with racing times below 2 15 h can be considered as outliers with respect to this behavior for the main marathon pack i e racing times in the range 3 00 6 00 h the average velocity as a function of the marathoner s ranking behaves linearly for this racing times the interpersonal velocity defined as the difference of velocities between consecutive runners displays a continuum of scaling behavior ranging from uncorrelated noise for small scales to correlated 1 f noise for large scales it is a matter of fact that 1 f noise is characterized by correlations extended over a wide range of scales a clear indication of some sort of cooperative effect

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

robust high dimensional volatility matrix estimation for high frequency factor model high frequency financial data allow us to estimate large volatility matrices with relatively short time horizon many novel statistical methods have been introduced to address large volatility matr

multivariate realised kernels consistent positive semi definite estimators of the covariation of equity prices with noise and non synchronous trading we propose a multivariate realised kernel to estimate the ex post covariation of log prices we show this new consistent estimator is guaranteed to be positive semi definite and is robust to measurement noise of certain types and can also handle non synchronous trading it is the first estimator which has these three properties which are all essential for empirical work in this area we derive the large sample asymptotics of this estimator and assess its accuracy using a monte carlo study we implement the estimator on some us equity data comparing our results to previous work which has used returns measured over 5 or 10 minutes intervals we show the new estimator is substantially more precise

fuzzy sets probability and measurement abstract in recent years the problem of uncertainty modeling has received much attention from scholars in artificial intelligence and decision theory various formal settings including but not restricted to fuzzy sets and possibility measures have been proposed based on different intuitions and dealing with various kinds of uncertain data the two main research directions are upper and lower probabilities which convey the idea of imprecisely estimated probability measures and distorted probabilities for the descriptive assessment of partial belief possibility measures and thereby fuzzy sets stand at the crossroads of these new approaches traditional views interpretive settings and canonical experiments for the measurement of probability such as frequentist approaches betting theories comparative uncertainty relations are currently extended to the generalized uncertainty measures these works shed new light on various interpretations of fuzzy sets and clarify their links with probability theory conversely zadeh s logical point of view on fuzzy sets suggests a set theoretic perspective on uncertainty measures that brings together numerical quantification and logic

approximate reasoning for production planning abstract approximate reasoning for production planning could effectively be based on an interval valued representation of linguistic variables an interval valued representation provides a framework that i incorporates a dispersion about membership values and ii produces a robust result compared to classical results the proposed representation and inference is applied to a set of production problems in prototype studies the result of one such prototype study is summarized

a context model for fuzzy concept analysis based upon modal logic in this paper we present interesting relationships between the context model modal logic and fuzzy concept analysis it has been shown that the context model proposed by gebhardt and kruse int j approx reason 9 1993 283 can be semantically extended and considered as a data model for fuzzy concept analysis within the framework of the meta theory developed by resconi el al in 1990s consequently the context model provides a practical framework for constructing membership functions of fuzzy concepts and gives the basis for a theoretical justification of suitably use of well known t norm based connectives such as min max and product sum rules in applications furthermore an interpretation of mass assignments of fuzzy concepts within the context model is also established

ordinal decomposability and fuzzy connectives the benefit of computing with linguistic terms is now generally accepted fuzzy set theory provides us the conceptual tool for the interpretation and evaluation of linguistic concepts and expressions it constitutes a quantification of the compatibility degree of objects with the associated linguistic concept through a membership function when we make computation using fuzzy membership values such as in the evaluation of fuzzy rules confidence the implicit assumptions are that the membership values have quantitative semantics the extensive scale assumption and that the numeric values are commensurate among the different fuzzy sets generated by the different concepts involved the common scale assumption in most situations these assumptions are difficult to justify and may lead to various anomalies the membership values are more suitably interpreted only as ordinal scales where the numeric representations reflect compatibility orderings in this paper we examine the concept of fuzzy intersection and union from the perspective of decomposability and ordinal conjoint structure in measurement theory we determine conditions under which a weak order induced by a fuzzy set or otherwise can be decomposed into other weak orders we show particular cases of ordinal decomposability which correspond naturally to our concept of fuzzy intersection and union this perspective of fuzzy connectives help us resolve some of the difficulties related to the above assumptions

job search and employment abstract a multistage search method based on fuzzy preference is used to analyze the processes of job search an employment decision based on fuzzy preference results in an exact choice the incorporation of fuzzy interval preference into employment decision however may lead to multiple choices which provides a theoretical justification for the existence of fuzzy supply of labor thus fuzziness in addition to randomness offers an explanation to wage dispersion

a nonparametric eigenvalue regularized integrated covariance matrix estimator for asset return data abstract in high frequency data analysis the extreme eigenvalues of a realized covariance matrix are biased when its dimension p is large relative to the sample size n furthermore with non synchronous trading and contamination of microstructure noise we propose a nonparametrically eigenvalue regularized integrated covariance matrix estimator nerive which does not assume specific structures for the underlying integrated covariance matrix we show that nerive is positive definite in probability with extreme eigenvalues shrunk nonlinearly under the high dimensional framework p n c 0 we also prove that in portfolio allocation the minimum variance optimal weight vector constructed using nerive has maximum exposure and actual risk upper bounds of order p 1 2 incidentally the same maximum exposure bound is also satisfied by the theoretical minimum variance portfolio weights all these results hold true also under a jump diffusion model for the log price processes with jumps removed using the wavelet method proposed in fan and wang 2007 they are further extended to accommodate the existence of pervasive factors such as a market factor under the setting p 3 2 n c 0 the practical performance of nerive is illustrated by comparing to the usual two scale realized covariance matrix as well as some other nonparametric alternatives using different simulation settings and a real data set

measurement of membership functions and their acquisition abstract three basic views of the representation of membership functions are reviewed together with fundamental measurement of linguistic terms of linguistic variables the conclusion is that such measurements are either on an ordinal or an interval scale based on whether the appropriate axioms are validated by the empirical data with an allowance for stochastic variation the conjoint measurement is introduced for the case of multi dimensional linguistic variables whose linguistic terms are compositions of two or finitely many more component linguistic terms of distinct independent linguistic variables it is shown that any composition of distinct independent component linguistic terms of component linguistic variables by any t norm or s norm or any finite convext linear combination preserves the monotonic weak order property of the components in the composite once the scale properties of the measurement values of particular terms are validated composition procedures may be applied to the experimental data to obtain compound membership functions of fuzzy sets induced by meaningful representations of compositions of linguistic terms of linguistic variables finally four methods of membership acquisition and construction are reviewed from the perspective of fundamental measurement

a two step estimator for large approximate dynamic factor models based on kalman filtering this paper shows consistency of a two step estimation of the factors in a dynamic approximate factor model when the panel of time series is large n large in the first step the parameters of the model are estimated from an ols on principal components in the second step the factors are estimated via the kalman smoother the analysis develops the theory for the estimator considered in giannone et al 2004 and giannone et al 2008 and for the many empirical papers using this framework for nowcasting

properties and estimation of asymmetric exponential power distribution the new distribution class asymmetric exponential power distribution aepd proposed in this paper generalizes the class of skewed exponential power distributions sepd in a way that in addition to skewness introduces different decay rates of density in the left and right tails our parametrization provides an interpretable role for each parameter we derive moments and moment based measures skewness kurtosis expected shortfall it is demonstrated that a maximum entropy property holds for the aepd distributions we establish consistency asymptotic normality and efficiency of the maximum likelihood estimators over a large part of the parameter space by dealing with the problems created by non smooth likelihood function and derive explicit analytical expressions of the asymptotic covariance matrix where the results apply to the sepd class they enlarge on the current literature also we give a convenient stochastic representation of the distribution our monte carlo study illustrates the theoretical results we also provide some empirical evidence for the usefulness of employing aepd errors in garch type models for predicting downside market risk of financial assets

estimating the quadratic covariation matrix from noisy observations local method of moments and efficiency an efficient estimator is constructed for the quadratic covariation or integrated covolatility matrix of a multivariate continuous martingale based on noisy and non synchronous observations under high frequency asymptotics our approach relies on an asymptotically equivalent continuous time observation model where a local generalised method of moments in the spectral domain turns out to be optimal asymptotic semiparametric efficiency is established in the cramer rao sense main findings are that non synchronicity of observation times has no impact on the asymptotics and that major efficiency gains are possible under correlation simulations illustrate the finite sample behaviour

iterative refinement of possibility distributions by learning for pixel based classification this paper proposes an approach referred as iterative refinement of possibility distributions by learning irpdl for pixel based image classification the irpdl approach is based on the use of possibilistic reasoning concepts exploiting expert knowledge sources as well as ground possibilistic seeds learning the set of seeds is constructed by incrementally updating and refining the possibility distributions synthetic images as well as real images from the rider breast mri database are being used to evaluate the irpdl performance its performance is compared with three relevant reference methods region growing semi supervised fuzzy pattern matching and markov random fields the irdpl performance in terms of recognition rate 87 3 is close to the markovian method 88 8 that is considered to be the reference in pixel based image classification irpdl outperforms the other two methods respectively at the recognition rates of 83 9 and 84 7 in addition the proposed irpdl requires fewer parameters for the mathematical representation and presents a reduced computational complexity

factor garch ito models for high frequency data with application to large volatility matrix prediction abstract several novel large volatility matrix estimation methods have been developed based on the high frequency financial data they often employ the approximate factor model that leads to a low rank plus sparse structure for the integrated volatility matrix and facilitates estimation of large volatility matrices however for predicting future volatility matrices these nonparametric estimators do not have a dynamic structure to implement in this paper we introduce a novel ito diffusion process based on the approximate factor models and call it a factor garch ito model we then investigate its properties and propose a quasi maximum likelihood estimation method for the parameter of the factor garch ito model we also apply it to estimating conditional expected large volatility matrices and establish their asymptotic properties simulation studies are conducted to validate the finite sample performance of the proposed estimation methods the proposed method is also illustrated by using data from the constituents of the s p 500 index and an application to constructing the minimum variance portfolio with gross exposure constraints

equivalence and compositions of fuzzy rationality measures an axiomatic basis for fuzzy rationality measures has already been introduced by the authors in a previous paper 5 formalizing the fact that there exist degrees of consistency when preferences over a fixed set of alternatives are expressed in terms of fuzzy binary preference relations this paper deals with some practical consequences on the one hand similarities and compositions of fuzzy rationality measures are considered showing natural ways of deriving new measures on the other if basic stability properties are introduced in order to assure that small intensity measurement errors never lead to big changes in the associate rationality value it is shown that crisp i e binary rationality measures present serious difficulties when applied to fuzzy preference relations

vagueness and bayesian probability this paper is a response to michael laviolette and john w seaman jr s ibid vol 2 no 1 p 4 1994 position paper the efficacy of fuzzy representations of uncertainty which criticizes fuzzy representations of uncertainty and suggests that bayesian probability can do better the commenter argues that the author s make some misleading comments about bayesian probability and he briefly discusses the problem of giving a satisfactory interpretation of membership functions

pseudo additive measures and integrals abstract we suggest pseudo additive measures based on a pseudo addition and discuss integrals with respect to pseudo additive measures a pseudo additive measure is a special type of fuzzy measures to define an integral a multiplication corresponding to a pseudo addition is introduced the resulting integral is an extension of the lebesgue integral in this context radon nikodym like theorems are shown

uniqueness results in the representation of families of sets by fuzzy sets the important concept of identifying a fuzzy subset of a set r with its corresponding chain of level sets is widely used in the literature we investigate in this paper the problem of characterization of all fuzzy subsets of r that can be identified with a given arbitrary family c of subsets of r together with a given arbitrary subset s of 0 1 although our main focus is on the problem of uniqueness we start by giving necessary and sufficient conditions for both existence and uniqueness of such fuzzy subsets we continue by obtaining several algebraic and topological properties under which uniqueness is guaranteed to further support our results we present some examples which we hope will shed more light on the above mentioned problem of identification we finish by raising a few natural questions that are left as open problems for further investigation

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

fuzzy sets spl minus a convenient fiction for modeling vagueness and possibility this paper is a reply to laviolette and seaman s critical discussion of fuzzy set theory rather than questioning the interest of the bayesian approach to uncertainty some reasons why bayesian find the idea of a fuzzy set not palatable are laid bare some links between fuzzy sets and probability that laviolette and seaman seem not to be aware of are pointed out these links suggest that contrary to the claim sometimes found in the literature probability theory is not a special case of fuzzy set theory the major objection to laviolette and seaman is that they found their critique on as very limited view of fuzzy sets including debatable papers while they fail to account for significant works pertaining to axiomatic derivation of fuzzy set connectives possibility theory fuzzy random variables among others

quasi maximum likelihood estimation of volatility with high frequency data this paper investigates the properties of the well known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise by extending the classic asymptotic results of quasi maximum likelihood estimation when trying to estimate the integrated volatility and the variance of noise this parametric approach remains consistent efficient and robust as a quasi estimator under misspecified assumptions moreover it shares the model free feature with nonparametric alternatives for instance realized kernels while being advantageous over them in terms of finite sample performance in light of quadratic representation this estimator behaves like an iterative exponential realized kernel asymptotically comparisons with a variety of implementations of the tukey hanning2 kernel are provided using monte carlo simulations and an empirical study with the euro us dollar future illustrates its application in practice

designing realised kernels to measure the ex post variation of equity prices in the presence of noise this paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions the issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed gaussian limit theorems the weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem realised kernels can also be selected to i be analysed using endogenously spaced data such as that in databases on transactions ii allow for market frictions which are endogenous iii allow for temporally dependent noise the finite sample performance of our estimators is studied using simulation while empirical work illustrates their use in practice

a measurement theoretic axiomatization of trapezoidal membership functions in many applications of fuzzy set theory the membership of an object is not defined directly one of its attributes like height age weight etc is first mapped to a real number often by means of a physical instrument and a parametric function then maps this real number to a membership degree in some fuzzy set like tall old heavy etc a very common parametric function is the trapezoidal one this paper presents some conditions guaranteeing the existence of such a trapezoidal membership function representing the knowledge of an expert further experimental research is needed for testing whether these conditions are satisfied by human agents

analysis of critical paths in a project network with fuzzy activity times abstract this paper proposes an approach to critical path analysis for a project network with activity times being fuzzy numbers in that the membership function of the fuzzy total duration time is constructed the basic idea is based on the extension principle and linear programming formulation a pair of linear programs parameterized by possibility level is formulated to calculate the lower and upper bounds of the fuzzy total duration time at by enumerating different values of the membership function of the fuzzy total duration time is constructed and the fuzzy critical paths are identified at the same time moreover by applying the yager ranking method definitions of the most critical path and the relative degree of criticality of paths are developed and these definitions are theoretically sound and easy to use in practice two examples with activity times being fuzzy numbers of l r and l l types discussed in previous studies are solved successfully to demonstrate the validity of the proposed approach since the total duration time is completely expressed by a membership function rather than by a crisp value the fuzziness of activity times is conserved completely and more information is provided for critical path analysis

induction of fuzzy decision trees abstract most decision tree induction methods used for extracting knowledge in classification problems do not deal with cognitive uncertainties such as vagueness and ambiguity associated with human thinking and perception in this paper cognitive uncertainties involved in classification problems are explicitly represented measured and incorporated into the knowledge induction process a fuzzy decision tree induction method which is based on the reduction of classification ambiguity with fuzzy evidence is developed fuzzy decision trees represent classification knowledge more naturally to the way of human thinking and are more robust in tolerating imprecise conflict and missing information

pre averaging estimators of the ex post covariance matrix in noisy diffusion models with non synchronous data we show how pre averaging can be applied to the problem of measuring the ex post covariance of financial asset returns under microstructure noise and non synchronous trading a pre averaged realised covariance is proposed and we present an asymptotic theory for this new estimator which can be configured to possess an optimal convergence rate or to ensure positive semi definite covariance matrix estimates we also derive a noise robust hayashi yoshida estimator that can be implemented on the original data without prior alignment of prices we uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high frequency equity data

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

a fuzzy c means variant for the generation of fuzzy term sets a fuzzy c means fcm variant is proposed for the generation of fuzzy term sets with overlap the proposed variant differs from the original mainly in two areas the first modification ensures that two end terms take the maximum and minimum domain values as their centers the second modification prevents the generation of non convex fuzzy terms that often occurs with the original algorithm the optimal number of terms and the optimal shape of the membership function associated with each term are determined based on the mean squared error criterion the exponential weight m used in the algorithm is found to greatly affect the shape of the membership function the effect of data size used for the generation of fuzzy terms is also discussed a generalized shaped function with a tunable parameter along with its complement is developed to fit all term sets generated by the fcm variant using various m values

an empirical test of some measurement theoretic axioms for fuzzy sets in the previous years some authors have been elaborating on the measurement theoretic foundations of fuzzy set theory a well known problem in this approach is the difficult applicability of the deterministically formulated axioms on data obtained from an expert or social science data in general in this paper a statistical method is proposed which can be used for the testing of measurement axioms in a pairwise comparison design consequently we apply this method for the testing of two of these axioms i e transitivity and order of operations used in the context of fuzzy set theory the results clearly indicate that subjects act in agreement with transitivity but for order of operations there is no such vivid evidence which may have repercussions for some of the earlier theoretical work

from fuzzy sets to the decompositions of non rigid sets the concept of identifying a fuzzy subset of a set r with its collection of level sets and its collection of membership values plays an important role in many real life applications such identification is possible only if the fuzzy set can be recaptured this implies that uniqueness of the fuzzy set plays an important role it has been recently shown that in order to have uniqueness it is necessary and sufficient that the collection s of membership values of the fuzzy set be a rigid set with this in mind the authors went on to show that if s has the min max property then s is a rigid set and left the converse as an open problem we start this paper by constructing a counterexample that shows that rigid sets do not have to have the min max property we then take a closer look at non rigid sets by decomposing them into isomorphism invariant components and into s connected components in the hope of getting a characterization of non rigid sets which will lead to a characterization of rigid sets and hence of uniqueness of the fuzzy set we also investigate in the case of nonuniqueness the number of fuzzy sets corresponding to a collection of level sets and a collection of membership values

measurement theoretic justification of connectives in fuzzy set theory abstract the problem of representing intersection and union in fuzzy set theory is considered there are various proposals in the literature to model these concepts the possibility of using continuous triangular norms and conorms including min and max are taken up in a measurement theoretic setting the conditions are laid out to arrive at cardinal scales on which addition and multiplication are meaningful and critically discussed these conditions must either be accepted on normative grounds or must be empirically verified before the modeling process in order to see which operations are meaningful it is emphasized that the archimedean axiom and the existence of natural bounds are crucial in arriving at ratio and absolute scale representations

factor modeling for high dimensional time series inference for the number of factors this paper deals with the factor modeling for high dimensional time series based on a dimension reduction viewpoint under stationary settings the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix and is therefore applicable when the dimension of time series is on the order of a few thousands asymptotic properties of the proposed method are investigated under two settings i the sample size goes to infinity while the dimension of time series is fixed and ii both the sample size and the dimension of time series go to infinity together in particular our estimators for zero eigenvalues enjoy faster convergence or slower divergence rates hence making the estimation for the number of factors easier in particular when the sample size and the dimension of time series go to infinity together the estimators for the eigenvalues are no longer consistent however our estimator for the number of the factors which is based on the ratios of the estimated eigenvalues still works fine furthermore this estimation shows the so called blessing of dimensionality property in the sense that the performance of the estimation may improve when the dimension of time series increases a two step procedure is investigated when the factors are of different degrees of strength numerical illustration with both simulated and real data is also reported

on choosing rationally when preferences are fuzzy abstract in real life exact choices are induced by fuzzy preferences these choices eventually satisfy certain plausible rationality properties several alternative rules for generating exact choices from fuzzy weak preference relations are introduced and the extent to which exact choice sets generated by these rules satisfy fairly weak rationality conditions is studied max min transitivity is shown to be crucial for obtaining a positive result

optimal sparse volatility matrix estimation for high dimensional ito processes with measurement errors stochastic processes are often used to model complex scientiflc problems in flelds ranging from biology and flnance to engineering and physical science this paper investigates rate optimal estimation of the volatility matrix of a high dimensional it o process observed with measurement errors at discrete time points the minimax rate of convergence is established for estimating sparse volatility matrices by combining the multi scale and threshold approaches we construct a volatility matrix estimator to achieve the optimal convergence rate the minimax lower bound is derived by considering a subclass of it o processes for which the minimax lower bound is obtained through a novel equivalent model of covariance matrix estimation for independent but non identically distributed observations and through a delicate construction of the least favorable parameters in addition a simulation study was conducted to test the flnite sample performance of the optimal estimator and the simulation results were found to support the established asymptotic theory

ordinal fuzzy sets fuzzy set theory has been used as a framework for interpreting imprecise linguistic expressions in general a linguistic term is described by the compatibility ordering induced in some universe of discourse uod a membership function in fuzzy set theory serves to reflect this ordering by assignment of values in 0 1 for objects in uod when we compute the meaning of a linguistic expression such as young and tall using fuzzy membership functions two implicit assumptions are made first we assume the membership values have quantitative meaning so that they can be quantitatively manipulated for example by adding or subtracting the extensive scale assumption second we assume that the scales of the membership values used in describing the different linguistic terms are comparable and the same the common scale assumption in many cases these assumptions cannot be justified some proposals have been made to address the first issue by using ordinal scale in defining fuzzy membership functions however the second issue has not been properly investigated in this paper we propose a framework that does not depend on both of these assumptions such framework will facilitate our understanding and investigation of qualitative reasoning without the extensive scale and common scale assumptions

discrete interval type 2 fuzzy system models using uncertainty in learning parameters fuzzy system modeling fsm is one of the most prominent tools that can be used to identify the behavior of highly nonlinear systems with uncertainty conventional fsm techniques utilize type 1 fuzzy sets in order to capture the uncertainty in the system however since type 1 fuzzy sets express the belongingness of a crisp value x of a base variable x in a fuzzy set a by a crisp membership value mua x they cannot fully capture the uncertainties due to imprecision in identifying membership functions higher types of fuzzy sets can be a remedy to address this issue since the computational complexity of operations on fuzzy sets are increasing with the increasing type of the fuzzy set the use of type 2 fuzzy sets and linguistic logical connectives drew a considerable amount of attention in the realm of fuzzy system modeling in the last two decades in this paper we propose a black box methodology that can identify robust type 2 takagi sugeno mizumoto and linguistic fuzzy system models with high predictive power one of the essential problems of type 2 fuzzy system models is computational complexity in order to remedy this problem discrete interval valued type 2 fuzzy system models are proposed with type reduction in the proposed fuzzy system modeling methods fuzzy c means fcm clustering algorithm is used in order to identify the system structure the proposed discrete interval valued type 2 fuzzy system models are generated by a learning parameter of fcm known as the level of membership and its variation over a specific set of values which generate the uncertainty associated with the system structure

econometric analysis of realized volatility and its use in estimating stochastic volatility models summary the availability of intraday data on the prices of speculative assets means that we can use quadratic variation like measures of activity in financial markets called realized volatility to study the stochastic properties of returns here under the assumption of a rather general stochastic volatility model we derive the moments and the asymptotic distribution of the realized volatility error the difference between realized volatility and the discretized integrated volatility which we call actual volatility these properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation intensive methods

canonical forms of fuzzy truthoods by meta theory based upon modal logic in this paper we redefine with the meta theory based on modal logic the operations between fuzzy sets of verity i e fuzzy sets of truthood with this new approach we can unify the different formulas for the operations and or and not between the fuzzy sets of verity i e truthood the operations between the fuzzy sets of verity become sensitive to the logic value true or false that agents persons sensors assign to the worlds contexts the operations are also sensitive to the difference of the worlds and time of synchronisation it should be pointed out that when we use the logic operations as and or and not we generally assume that the worlds are the same and change their truth value at the same time but it is known that there are cases where this synchronic situation and identity is not always valid that is there exist transformations that change one world for one proposition to another world for another proposition in conclusion the linguistic and or not operations become dependent on the particular truth value of a world on the synchronisation and on the worlds assigned to the two propositions via transformations thus all of these possible changes in the structure of the worlds in the modal logic cause the gradation of the linguistic operations e g and or and not an individual world person agent sensor assigns to an atomic sentence either a true or false value and uses the classical two value logic operations of and or not that is the crisp true or false responses assignments of worlds generate gradation of truthood value the uncertainty in a fuzzy set is represented with sets of worlds in a conflict situation i e the same proposition may be true in one world and false in another consonant or dissonant relations between sets of worlds that depends on the synchronisation and or transformation of worlds cause the generation of gradation by the linguistic operations and or and not that is the linguistic operations change for different concrete situations set of worlds caused by the generation of a representation in each world where the descriptive set membership assignment d may be two or infinite valued but the verity assignment v is two valued in a particular world representation whereas membership values of the fuzzy set of truth verifications associated with the set of worlds are in 0 1 furthermore the combination of fuzzy membership values generates a type ii fuzzy set that is captured by fdcf and fccf formulas in this paper we show that there exist a deeper connection between the set membership assignment and the verity assignment of truthood by modal logic with suitable extensions

an efficient and flexible mechanism for constructing membership functions abstract this paper introduces a bezier curve based mechanism for constructing membership functions of convex normal fuzzy sets the mechanism can fit any given data set with a minimum level of discrepancy in the absence of data the mechanism can be intuitively manipulated by the user to construct membership functions with the desired shape some numerical experiments are included to compare the performance of the proposed mechanism with conventional methods

neuro fuzzy clustering of radiographic tibia image data using type 2 fuzzy sets abstract this paper presents the results of using type 2 fuzzy sets to assist in the pre processing of data for use with neuro fuzzy clustering for classification of sports injuries in the lower leg this research is concerned with the analysis of bone scans from stress related injuries to the tibia of particular interest is whether neural network based clustering techniques can help the consultant in classifying the images the work was motivated by the situation where there is a relatively small amount of relevant data and difficulties are faced by consultants in classifying the various types of injuries for this particular problem the consultant s interpretation of the image lends itself to representation using type 2 fuzzy sets this research sets out to address whether with fuzzy neuro clustering techniques some insights may be provided to the consultant that they can use along with their experience and knowledge the results of this approach indicate that the use of neural clustering using a type 2 representation can improve the classification of shin images

estimating covariation epps effect microstructure noise abstract this paper is about how to estimate the integrated covariance x y t of two assets over a fixed time horizon 0 t when the observations of x and y are contaminated and when such noisy observations are at discrete but not synchronized times we show that the usual previous tick covariance estimator is biased and the size of the bias is more pronounced for less liquid assets this is an analytic characterization of the epps effect we also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms including nonsynchronous trading microstructure noise and time discretization finally a two scales covariance estimator is provided which simultaneously cancels to first order the epps effect and the effect of microstructure noise the gain is demonstrated in data

the measure of the degree of truth and the grade of membership abstract a method is presented to define the meaning of a fuzzy proposition and of a fuzzy set through the construction of fuzzy propositions with non ambiguous intermediate truth values

treating fuzziness in subjective evaluation data this paper proposes a technique to deal with fuzziness in subjective evaluation data and applies it to principal component analysis and correspondence analysis in the existing method or techniques developed directly from it fuzzy sets are defined from some standpoint on a data space and the fuzzy parameters of the statistical model are identified with linear programming or the method of least squares in this paper we try to map the variation in evaluation data into the parameter space while preserving information as much as possible and thereby define fuzzy sets in the parameter space clearly it is possible to use the obtained fuzzy model to derive things like the principal component scores from the extension principle however with a fuzzy model which uses the extension principle the possibility distribution spreads out as the explanatory variable values increase this does not necessarily make sense for subjective evaluations such as a 5 level evaluation for instance instead of doing so we propose a method for explicitly expressing the vagueness of evaluation using certain quantities related to the eigenvalues of a matrix which specifies the fuzzy parameter spread as a numerical example we present an analysis of subjective evaluation data on local environments

high frequency covariance estimates with noisy and asynchronous financial data this article proposes a consistent and efficient estimator of the high frequency covariance quadratic covariation of two arbitrary assets observed asynchronously with market microstructure noise this estimator is built on the marriage of the quasi maximum likelihood estimator of the quadratic variation and the proposed generalized synchronization scheme and thus is not influenced by the epps effect moreover the estimation procedure is free of tuning parameters or bandwidths and is readily implementable monte carlo simulations show the advantage of this estimator by comparing it with a variety of estimators with specific synchronization methods the empirical studies of six foreign exchange future contracts illustrate the time varying correlations of the currencies during the 2008 global financial crisis demonstrating the similarities and differences in their roles as key currencies in the global market

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

a tale of two time scales it is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns however market microstructure poses challenges to this estimation approach as evidenced by recent empirical studies in finance the present work attempts to lay out theoretical grounds that reconcile continuous time modeling and discrete time samples we propose an estimation approach that takes advantage of the rich sources in tick by tick data while preserving the continuous time assumption on the underlying returns under our framework it becomes clear why and where the usual volatility estimator fails when the returns are sampled at the highest frequencies if the noise is asymptotically small our work provides a way of finding the optimal sampling frequency a better approach the two scales estimator works for any size of the noise

vast volatility matrix estimation for high frequency financial data high frequency data observed on the prices of financial assets are commonly modeled by diffusion processes with micro structure noise and realized volatility based methods are often used to estimate integrated volatility for problems involving a large number of assets the estimation objects we face are volatility matrices of large size the existing volatility estimators work well for a small number of assets but perform poorly when the number of assets is very large in fact they are inconsistent when both the number p of the assets and the average sample size n of the price data on the p assets go to infinity this paper proposes a new type of estimators for the integrated volatility matrix and establishes asymptotic theory for the proposed estimators in the framework that allows both n and p to approach to infinity the theory shows that the proposed estimators achieve high convergence rates under a sparsity assumption on the integrated volatility matrix the numerical studies demonstrate that the proposed estimators perform well for large p and complex price and volatility models the proposed method is applied to real high frequency financial data

uniqueness in the generalized representation by fuzzy sets the concept of representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets is widely used in the literature for the classification and study of fuzzy algebraic structures on the same underlying set the problem of existence of such fuzzy sets was investigated by various authors as far as we know the problem of uniqueness of such fuzzy sets was not studied before we establish among other things necessary and sufficient conditions under which uniqueness is guaranteed in the cases of complete as well as partial representations of the level sets of the fuzzy set

soft sets an ordinal formulation of vagueness with some applications to the theory of choice abstract in the theory of fuzzy sets it is usual to represent the degree of belongingness by a number lying between 0 and 1 this paper develops an alternative framework which dispenses with such cardinal numerical representations of the degree of belongingness and relies instead on an ordinal formulation the ordinal notion of a vague set is called a soft set soft sets are shown to have a tight logical relation with the concept of l sets introduced earlier by goguen the paper also discusses applications of the framework of soft sets to several problems in the theory of choice

hierarchical scheduling based on approximate reasoning a comparison with isis abstract a hierarchical scheduling problem is considered the hierarchy consists of five levels each level has an associated rule base there are six basic input factors 1 the priority of a job 2 the earliness of the due date of a job 3 the likelihood of the availability of resources for the work station w s s 4 the capability of w s s to meet physical constraints 5 the capability of w s s to meet quality requirements and 6 work in progress inventory costs there are five consequents one for each level 1 suggested place of a job in the order of jobs 2 the suggested place of w s s in the order of w s s 3 the current suggested place of a w s s in the order of w s s 4 the next current suggested place of w s s in the order of w s s and 5 the suggested place of the queue arrangements both the input factors antecedents and the output factors consequents take on linguistic terms to express the qualitative assessments of a scheduling expert within the context of the specific factor an approximate reasoning framework based on interval valued fuzzy sets and a search tree heuristic with similarity measures is executed for the analysis of the hierarchical scheduling problem using a fuzzy tool box coded in lisp that interacts with kee for the representation of the knowledge base in a frame structure the software model is tested with real life data used in m fox s isis model a comparison of the two models based on tardiness measure shows that there are slight advantages to fuzzy set representation and approximate reasoning with respect to tardiness measure above and beyond the user friendly nature of knowledge representation available with fuzzy sets

a fuzzy set model for market share and preference prediction abstract this paper presents a new method of modelling preferences using fuzzy sets a fuzzy set conjoint model is developed by incorporating fuzzy measurement of evaluations into the vector preference model a comparison test of rank prediction is conducted on data gathered from consumer subjects by an adaptive software program a methodology is developed to generate aggregate crisp market share estimates from individual fuzzy set preference predictions the result is a substantial improvement in individual predictive validity using the fuzzy set model and more accurate aggregate market share prediction for use in product design demand forecasting and other management functions

type i and type ii fuzzy system modeling abstract two system modeling approaches are proposed with a unification of fuzzy methodologies type i and type ii knowledge representation are reviewed together with their proposed approximate reasoning schemes type i representation and inference are based on boolean normal forms type i models are myopic because in these models either the disjunctive or the conjunctive boolean normal forms are selected rather arbitrarily for knowledge representation and inference and then they are fuzzified whereas in type ii models representation and inference are based on fuzzy disjunctive and conjunctive normal forms which are derived from the fuzzy truth tables