restarted gmres preconditioned by deflation this paper presents a new preconditioning technique for the restarted gmres algorithm it is based on an invariant subspace approximation which is updated at each cycle numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm at a low cost of operations and memory

on approximate reasoning with graded rules this contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy if then rules formalized by approximating formulas of a special type two alternatives that follow from the dual character of approximating formulas are developed in parallel the link to the theory of fuzzy control systems is also explained

on the construction of deflation based preconditioners in this article we introduce new bounds on the effective condition number of deflated and preconditioned deflated symmetric positive definite linear systems for the case of a subdomain deflation such as that of nicolaides siam j numer anal 24 1987 pp 355 365 these theorems can provide direction in choosing a proper decomposition into subdomains if grid refinement is performed keeping the subdomain grid resolution fixed the condition number is insensitive to the grid size subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication numerical experiments for a steady state convection diffusion problem are included

on probability possibility transformations abstract a probability possibility transformation is a purely mechanical transformation of probabilistic support to possibilistic support and vice versa that is a conversion of the measure of support of one theory into that of the other that is independent of the problem domain in this paper it is shown that any bijective mapping between probability distributions and normalized possibility distributions that is symmetric and expansible cannot preserve the properties of projection noninteraction or conditionalization

possibilistic instance based learning a method of instance based learning is introduced which makes use of possibility theory and fuzzy sets particularly a possibilistic version of the similarity guided extrapolation principle underlying the instance based learning paradigm is proposed this version is compared to the commonly used probabilistic approach from a methodological point of view moreover aspects of knowledge representation such as the modeling of uncertainty are discussed taking the possibilistic extrapolation principle as a point of departure an instance based learning procedure is outlined which includes the handling of incomplete information methods for reducing storage requirements and the adaptation of the influence of stored cases according to their typicality first theoretical and experimental results showing the efficiency of possibilistic instance based learning are presented as well

what about fuzzy logic s linguistic soundness a fuzzy set does represent a concrete use of an imprecise predicate in a given context that is its meaning a la wittgenstein the diverse theories of fuzzy sets come from accepting that also the connectives are to be used in different ways that they do have meaning this paper aims to suggest the importance of finding counterparts of the theories of fuzzy sets in either natural or artificial languages as a first step towards a new experimental view of theoretical fuzzy logic

a survey of adaptive fuzzy controllers nonlinearities and classifications adaptive fuzzy controllers afcs i e fuzzy controllers using heuristic expert knowledge and equipped with adaption ability are widely used in the industry in the present days as they are capable of dealing with ill formulated nonlinear systems where exact mathematical formulas are not known or partially described nonlinear systems afcs have been widely deployed to control many nonlinear systems with different types of critical nonlinearities over the last two decades many different approaches toward adaptive fuzzy control have been reported in this paper we recapitulate the latest research works related to afcs and provide an up to date survey on the latest development made in adaptive fuzzy control theory and applications based on different types of fuzzy systems considered and nonlinearities associated with the nonlinear system to be controlled we categorize the different approaches based on different types of nonlinearities and consider basic nonlinear systems nonlinear systems with external disturbances nonlinear systems with some special type of nonlinearities nonaffine nonlinear systems and strict feedback nonlinear systems it turns out that many existing methods are subject to various limitations and none of them is capable of controlling a general nonlinear systems hence it is still an open problem to design and develop a universal afc for a general class of nonlinear systems

training ν support vector classifiers theory and algorithms the support vector machine svm for classification proposed by scholkopf smola williamson and bartlett 2000 has the advantage of using a parameter on controlling the number of support vectors in this article we investigate the relation between svm and c svm in detail we show that in general they are two different problems with the same optimal solution set hence we may expect that many numerical aspects of solving them are similar however compared to regular c svm the formulation of svm is more complicated so up to now there have been no effective methods for solving large scale svm we propose a decomposition method for svm that is competitive with existing methods for c svm we also discuss the behavior of svm by some numerical experiments

set theoretic methodology using fuzzy sets in rule extraction and validation consistency and coverage revisited abstract the use of set theoretic concepts of consistency and coverage and their fuzzified versions needs to be accompanied with an extension of the general non fuzzy methodology in the fuzzy context the same piece of data can provide partial evidence in favour of the existence of a given relationship and at the same time contribute to its disproof which constitutes a significant interpretation and methodological issue we point out the possible problems of the use of these measures fuzzified in direct analogy to their non fuzzy counterparts in the investigation of relationships and causality we propose two alternative fuzzifications of these measures to be used in the set theoretic framework and fuzzy set qualitative comparative analysis fsqca we also introduce novel degree of support and degree of disproof measures to handle simultaneous subset relations in this way simultaneous partial support for and disproof of a given rule by the given set of data can be analyzed in detail the suggested measures can enhance the insights needed in theory building and be widely used in research using fuzzified set theoretic methods for the assessment of the plausibility of the rules we compare the newly suggested consistency and coverage measures with standard ones and discuss their properties in practical examples

a qualified description of extended fuzzy logic abstract the sense of approximate reasoning within fuzzy logic has given rise to speculation as to whether precise reasoning within classical logic can even be considered a feasible way of handling most of the world s problems now another question can be raised can fuzzy logic by itself justify handle the sort of uncertainty found in problems that belong to the class of open world problems this family of problems corresponds to the case of ambiguous mathematical theories mainly due to the fact that there is just imperfect incomplete information about them recently zadeh introduced an extension of fuzzy logic claimed to bridge the gaps in conventional approaches to the open world problems in this paper we present a basic background of extended fuzzy logic fle we address several main concepts such as f transformation and f validity enhance the definition of generalized constraints and propose that fle s rules have two layers one based on the implication model and another based on the conjunction model in addition we explore how fle can handle those conditions described in antonym pairs as well as those occurred without considering rationality in the framework both cases challenge most decision making theories we also suggest fle and second order probability are f complementary rather than f competitive the paper asserts that fle is highly capable of constructing mathematical solutions to problems including computational problems that are stated in natural language

on generalized modus ponens with multiple rules and a residuated implication we consider a multiple rule generalized modus ponens inference scheme with an interpretation based on compositional rule of inference cri and a residuated implication we show that such a system is equivalent as far as cri is concerned to a system that satisfies the basic requirement for fuzzy reasoning proposed by turksen and tian fuzzy sets and systems 58 1993 3 40 we establish an analogous conclusion for an alternative interpretation due to magrez and smets of generalized modus ponens which we call consequent dilation rule cdr this method is computationally faster than cri but it produces less specific inference results

revision of possibility distributions a bayesian inference pattern abstract in probability theory the bayes rule of inference plays a central role as corroborated by the ever increasing number of applications in various fields the rule allows to revise a prior probability distribution when new information becomes available the posterior probability distribution takes the form of a conditional distribution although several similarities between the possibilistic and probabilistic frameworks have already been reported very few studies in possibility theory have dealt with bayesian inference the objective of this work is to thoroughly study this type of inference and to develop the counterpart of the bayes rule in the possibilistic framework with the use of conditional possibility distributions application of possibility theory wherever bayes theory has already been applied can now be envisaged as a new perspective to uncertainty modeling and processing in the last part of this paper the suitability of the proposed framework for the problem of forecast processing is discussed and an example illustrates the application of various rules of inference corresponding to different aggregation operators

the triple rotation method for constructing t norms given an involutive negator n and a left continuous t norm t that either has no zero divisors or is rotation invariant we build a rotation invariant t norm from a rescaled version of t and its left right and front rotation depending on the involutive negator n and the set of zero divisors of t some reshaping of the rescaled version of t may occur during the rotation process the rescaled version of t itself can be understood as the b zoom of the newly constructed rotation invariant t norm with b the unique fixpoint of n starting with a rotation invariant t norm t there is however one important restriction the triple rotation method based on the involutive negator n will yield a t norm if and only if the companion q of t is commutative on 0 1 2 when q is not commutative on 0 1 2 there even does not exist a rotation invariant t norm with b zoom equal to t

practical inference with systems of gradual implicative rules a general approach to practical inference with gradual implicative rules and fuzzy inputs is presented gradual rules represent constraints restricting outputs of a fuzzy system for each input they are tailored for interpolative reasoning our approach to inference relies on the use of inferential independence it is based on fuzzy output computation under an interval valued input a double decomposition of fuzzy inputs is done in terms of alpha cuts and in terms of a partitioning of these cuts according to areas where only a few rules apply the case of 1 d and 2 d inputs is considered as well as higher dimensional cases an application to a cheese making process illustrates the approach

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

searching for basic properties obtaining robust implication operators in fuzzy control abstract this paper deals with the problem of searching basic properties for robust implication operators in fuzzy control we use the word robust in the sense of good average behavior in different applications and in combination with different defuzzification methods we study the behavior of the two main families of implication operators in the fuzzy control inference process these two families are composed by those operators that extend the boolean implication implication functions and those ones that extend the boolean conjunction t norms and force implications in order to develop the comparative study we will build different fuzzy controllers by means of these implication operators and will apply them to the fuzzy modeling of the real function y x and two three dimensional surfaces we analyze whether one of these two properties extension of the boolean implication and extension of the boolean conjunction is sufficient for obtaining a good implication operator or whether some complementary properties are necessary next we analyze whether we can get basic properties for good implication operators presenting three basic properties for the so called robust implication operators

continuity issues of the implicational interpretation of fuzzy rules the implicational interpretation of fuzzy rules has received little attention in real world applications so far this is largely due to the fact that ensuring continuity of the resulting function is not a straightforward task this paper targets this subject departing from consistent linguistic descriptions rule bases we introduce sufficient conditions for the continuity of the implicational interpretation with mean of maximum defuzzification we demonstrate that continuity can be achieved under practically feasible conditions regardless of the dimensionality of the input

checking the coherence and redundancy of fuzzy knowledge bases checking the coherence of a set of rules is an important step in knowledge base validation coherence is also needed in the field of fuzzy systems indeed rules are often used regardless of their semantics and it sometimes leads to sets of rules that make no sense avoiding redundancy is also of interest in real time systems for which the inference engine is time consuming a knowledge base is potentially inconsistent or incoherent if there exists a piece of input data that respects integrity constraints and that leads to logical inconsistency when added to the knowledge base we more particularly consider knowledge bases composed of parallel fuzzy rules then coherence means that the projection on the input variables of the conjunctive combination of the possibility distributions representing the fuzzy rules leaves these variables completely unrestricted i e any value for these variables is possible or at least not more restrictive than integrity constraints fuzzy rule representations can be implication based or conjunction based we show that only implication based models may lead to coherence problems however unlike conjunction based models they allow to design coherence checking processes some conditions that a set of parallel rules has to satisfy in order to avoid inconsistency problems are given for certainty or gradual rules the problem of redundancy which is also of interest for fuzzy knowledge bases validation is addressed for these two kinds of rules

multi label core vector machine with a zero label abstract multi label core vector machine rank cvm is an efficient and effective algorithm for multi label classification but there still exist two aspects to be improved reducing training and testing computational costs further and detecting relevant labels effectively in this paper we extend rank cvm via adding a zero label to construct its variant with a zero label i e rank cvmz which is formulated as the same quadratic programming form with a unit simplex constraint and non negative ones as rank cvm and then is solved by frank wolfe method efficiently attractively our rank cvmz has fewer variables to be solved than rank cvm which speeds up training procedure dramatically further the relevant labels are effectively detected by the zero label experimental results on 12 benchmark data sets demonstrate that our method achieves a competitive performance compared with six existing multi label algorithms according to six indicative instance based measures moreover on the average our rank cvmz runs 83 times faster and has slightly fewer support vectors than its origin rank cvm

logical structure of fuzzy if then rules abstract this paper provides a logical basis for manipulation with fuzzy if then rules our theory is wide enough and it encompasses not only finding a conclusion by means of the compositional rule of inference due to lotfi a zadeh but also other kinds of approximate reasoning methods e g perception based deduction provided that there exists a possibility to characterize them within a formal logical system in contrast with other approaches employing variants of multiple valued first order logic the approach presented here employs fuzzy type theory of v novak which has sufficient expressive power to present the essential concepts and results in a compact elegant and justifiable form within the effectively formalized representation developed here based on a complete logical system it is possible to reconstruct numerous well known properties of cri related fuzzy inference methods albeit not from the analytic point of view as usually presented but as formal derivations of the logical system employed the authors are confident that eventually all relevant knowledge about fuzzy inference methods based on fuzzy if then rule bases will be represented formalized and backed up by proof within the well founded logical representation presented here an immediate positive consequence of this approach is that suddenly all elements of a fuzzy inference method based on fuzzy if then rules are first class citizens of the representation there are clear logically founded definitions for fuzzy if then rule bases to be consistent complete or independent

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

measuring efficiency by a fuzzy statistical approach methods of measuring economic efficiency of input output systems by employing a fuzzy statistical approach are explored here in the context of the recent technique of data envelopment analysis which utilizes a nonparametric approach to measure efficiency three applied fuzzy measures are illustrated here e g fuzzy linear programs fuzzy linear regressions and finally the fuzzy entropy measures in the context of a two person game theory model

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

on equivalence of approximate reasoning results using different interpretations of fuzzy if then rules abstract initially an axiomatic approach to the definition of fuzzy implication has been recalled in this paper based on this definition several important fuzzy implications have been investigated with respect to the results obtained from inference system a specific type of equivalence of inference results under defuzzification using both logical implication interpretation and conjunctive interpretation of the fuzzy if then rules has been described the theoretical considerations are illustrated by means of numerical examples in the field of fuzzy modeling

an efficient preconditioned cg method for the solution of a class of layered problems with extreme contrasts in the coefficients knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs a mathematical model for the prediction of fluid pressures is given by a time dependent diffusion equation application of the finite element method leads to a system of linear equations a complication is that the underground consists of layers with very large differences in permeability this implies that the symmetric and positive definite coefficient matrix has a very large condition number bad convergence behavior of the cg method has been observed moreover a classical termination criterion is not valid in this problem after diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability for the ic preconditioner the same behavior is observed to annihilate the effect of the extreme eigenvalues a deflated cg method is used the convergence rate improves considerably and the termination criterion becomes again reliable finally a cheap approximation of the eigenvectors is proposed

fuzzy inferences and conditional possibility distributions abstract let x i i 1 2 be two variables taking values in finite universes of discourse u i and associated with possibility distributions f i u i 0 1 after an observation of x 1 we can deduce a characterization of x 2 by means of an inference process we define q x 2 x 1 for x i u i as some fuzzy implication and we introduce the possibility distribution p x 1 x 2 f 1 x 1 q x 2 x 1 for a t norm we exhibit some reasons to say that q and f may be respectively regarded as a conditional and a joint possibility distribution corresponding to marginal properties with regard to the use of a t conorm a property of non interactivity of the variables with respect to is bound with the definition of q and p in this way we suggest justifications of the choice of particular implications in inference processes

a fuzzy set based approach for modeling dependence among human errors the assessment of dependence among human errors is an important aspect of human reliability analysis when dependence between two tasks exists the probability of the operators failure on one task is higher if they have failed on the preceding task compared to when they have succeeded in current practice the task of assessing dependence among successive operator actions relies to a great extent to expert judgment often with lack of traceability and repeatability to overcome these limitations this work presents a systematic framework for the elicitation of expert knowledge on the factors influencing the dependence between two successive tasks the framework is based on a fuzzy expert system in which a set of transparent fuzzy logic rules is used to represent the relationship between the input factors and the conditional human error probability the proposed modeling approach is applied to two tasks required in response to an accident scenario at a nuclear power plant given the methodological scope of the work the fuzzy expert system is tested directly on a working model of dependence whereas no actual exercise of expert judgment elicitation is carried out

fuzzy inference systems preserving moser navara axioms abstract satisfiability of three naturally proposed axioms which are desirable for fuzzy inference system was discussed by moser and navara in 36 the fuzzy rule base models considered in this study were mainly the mamdani assilian ones and partly also logically motivated implicative fuzzy systems the investigation leads to the fact that often none of these fuzzy systems do satisfy the given axioms simultaneously therefore moser and navara proposed so called conditionally firing rules and proved that the simultaneous satisfaction of the three axioms is feasible under very mild and practically non restrictive conditions note that the inference mechanism tailored to the fuzzy rules in the investigation was the compositional rule of inference abb cri and similarly the concept of conditionally firing rules were stemming from the cri and the mamdani assilian model of a fuzzy rule base for the implicative interpretation of fuzzy rules the original axioms proposed by moser and navara were later on re defined by st pnicka and mandal with the aim to express the meaning of the original axioms initially proposed for the mamdani assilian rules furthermore conditionally firing implicative rules were proposed based on the so called bandler kohout subproduct abb bks inference mechanism and a study similar to the one by moser and navara was undertaken for them the study encompasses the satisfaction of the modified axioms for the use of implicative rules and the bks furthermore some additional results confirming the applicability of the conditionally firing rules satisfying the moser navara axioms are also provided including the universal approximation ability however the solvability criterions for fuzzy relational equations give a strong preference to the usage of the other combinations namely to the combination of cri with the implicative interpretation of fuzzy rules and to the combination of bks with the mamdani assilian interpretation of fuzzy rules therefore also these combinations of fuzzy inference mechanisms and interpretations of fuzzy rules should be investigated with respect to the satisfaction of moser navara axioms the main purpose of this article which extends the short conference contribution 47 by st pnicka is to provide such an investigation jointly with the review on conditionally firing rules and supplementary results

a deflation technique for linear systems of equations iterative methods for solving linear systems of equations can be very efficient if the structure of the coefficient matrix can be exploited to accelerate the convergence of the iterative process however for classes of problems for which suitable preconditioners cannot be found or for which the iteration scheme does not converge iterative techniques may be inappropriate this paper proposes a technique for deflating the eigenvalues and associated eigenvectors of the iteration matrix which either slow down convergence or cause divergence this process is completely general and works by approximating the eigenspace bbb p corresponding to the unstable or slowly converging modes and then applying a coupled iteration scheme on bbb p and its orthogonal complement bbb q

similarity based inverse approximate reasoning this paper investigates the inverse problem of ap proximate reasoning to conclude a from b and a b from different angles we develop a method to and a solution to the inverse problem by the use of the law of contrapositive symme try in the process generalized modus ponens generalized modus tollens and their generating functions are studied extensively an attempt is made here to establish that the concept of similarity plays an important role in inverse approximate reasoning we apply the concept of similarity to model inverse approximate reasoning and develop a new methodology that is named similarity based inverse approximate reasoning for a solution to the problem algorithms are presented and illustrated with simple yet concrete examples we then use zadeh s compositional rule of inference cri as a combi nation of cylindrical extension and projection and thereby model generalized modus tollens in a different way we thus develop a third way to find a solution to the inverse problem of approximate reasoning a comparison between the result of application of our technique and the real static characteristic of a dc shunt motor is presented

on the representation measurement and discovery of fuzzy associations the use of fuzzy sets to describe associations between data extends the types of relationships that may be represented facilitates the interpretation of rules in linguistic terms and avoids unnatural boundaries in the partitioning of the attribute domains in addition the partial membership values provide a method for incorporating the distribution of the data into the assessment of a rule this paper investigates techniques to identify and evaluate associations in a relational database that are expressible by fuzzy if then rules extensions of the classical confidence measure based on the spl alpha cut decompositions of the fuzzy sets are proposed to incorporate the distribution of the data into the assessment of a relationship and identify robustness in an association a rule learning strategy that discovers both the presence and the type of an association is presented

logical foundations of rule based systems the evolution of a theory of fuzzy if then rules and its contribution to the establishment of fuzzy logic is a focus of this paper i am advocating in favour of hajek s fuzzy logic regarding it as a right methodology for special logical theories a theory of fuzzy if then rules as a special theory in the above considered sense is proposed this theory is for all practitioners who want to create a system of fuzzy if then rules free of conflicts logically consistent and rich enough to be able to make non trivial conclusions or answer inquires

stabilization of unstable procedures the recursive projection method fixed point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes the recursive projection method rpm which stabilizes such procedures by computing a projection onto the unstable subspace is presented on this subspace a newton or special newton iteration is performed and the fixed point iteration is used on the complement as continuation in the parameter proceeds the projection is efficiently updated possibly increasing or decreasing the dimension of the unstable subspace the method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system convergence proofs are given and pseudo arclength continuation on the unstable subspace is introduced to allow continuation past folds examples are presented for an important application of the rpm in which a black box time integration scheme is stabilized enabling it to compute unstable steady states the rpm can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

survey of utilisation of fuzzy technology in medicine and healthcare the complexity of biological systems unlike physical science applications makes the development of computerised systems for medicine not a straightforward algorithmic solution because of the inherent uncertainty which arises as a natural occurrence in these types of applications human minds work from approximate data extract meaningful information from massive data and produce crisp solutions fuzzy logic provides a suitable basis for the ability to summarise and extract from masses of data impinging upon the human brain those facts that are related to the performance of the task at hand in practice a precise model may not exist for biological systems or it may be too difficult to model in these cases fuzzy logic is considered as an appropriate tool for modelling and control since our knowledge and experience are directly contained and presented in control strategies without explicit mathematical models this paper surveys the utilisation of fuzzy logic in medical sciences with an analysis of its possible future penetration

multilabel classifiers with a probabilistic thresholding strategy in multilabel classification tasks the aim is to find hypotheses able to predict for each instance a set of classes or labels rather than a single one some state of the art multilabel learners use a thresholding strategy which consists in computing a score for each label and then predicting the set of labels whose score is higher than a given threshold when this score is the estimated posterior probability the selected threshold is typically 0 5 in this paper we introduce a family of thresholding strategies which take into account the posterior probability of all possible labels to determine a different threshold for each instance thus we exploit some kind of interdependence among labels to compute this threshold which is optimal regarding a given expected loss function we found experimentally that these strategies outperform other thresholding options for multilabel classification they provide an efficient method to implement a learner which considers the interdependence among labels in the sense that the overall performance of the prediction of a set of labels prevails over that of each single label

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

maxima and minima in fuzzified linear orderings the paper studies the notions of fuzzy maxima minima suprema and infima in the setting of fuzzified linear orderings the main result shows that in fuzzified linear orders satisfying certain reasonable assumptions the fuzzy maxima and minima of all non empty fuzzy sets are non empty

application of prediction models using fuzzy sets a fuzzy inference framework based on fuzzy relations is developed adapted and applied to temperature and humidity measurements from a specific coffee crop site in brazil this framework consists of fuzzy relations over possibility distributions resulting in a model analogous to a bayesian inference process the application of this fuzzy model to a data set of experimental measurements and its correspondent forecasts of temperature and humidity resulted in a set of revised forecasts that incorporate information from a historical record of the problem each set of revised forecasts was compared with the correspondent set of experimental data using two different statistical measures mape mean absolute percentage error and willmott s d this comparison showed that the sets of forecasts revised by the fuzzy model exhibited better results than the original forecasts on both statistical measures for more than two thirds of the evaluated cases

adaptive fuzzy interpolation fuzzy interpolative reasoning strengthens the power of fuzzy inference by the enhancement of the robustness of fuzzy systems and the reduction of the systems complexity however after a series of interpolations it is possible that multiple object values for a common variable are inferred leading to inconsistency in interpolated results such inconsistencies may result from defective interpolated rules or incorrect interpolative transformations this paper presents a novel approach for identification and correction of defective rules in interpolative transformations thereby removing the inconsistencies in particular an assumption based truth maintenance system atms is used to record dependences between interpolations and the underlying technique that the classical general diagnostic engine gde employs for fault localization is adapted to isolate possible faulty interpolated rules and their associated interpolative transformations from this an algorithm is introduced to allow for the modification of the original linear interpolation to become first order piecewise linear the approach is applied to a realistic problem which predicates the diarrheal disease rates in remote villages to demonstrate the potential of this study

fuzzy rule base ensemble generated from data by linguistic associations mining as there are many various methods for time series prediction developed but none of them generally outperforms all the others there always exists a danger of choosing a method that is inappropriate for a given time series to overcome such a problem distinct ensemble techniques that combine several individual forecasts are being proposed in this contribution we employ the so called fuzzy rule based ensemble this method is constructed as a linear combination of a small number of forecasting methods where the weights of the combination are determined by fuzzy rule bases based on time series features such as trend seasonality or stationarity for identification of fuzzy rule bases we use the linguistic association mining a huge experimental justification is provided

fuzzy decision support system for demand forecasting with a learning mechanism in this paper a new decision support system for demand forecasting dss df is presented a demand forecast is generated in dss df by combining four forecasts values two of them are obtained independently one by a customer and the other by a market expert they represent subjective judgments on future demand given as linguistic values such as demand is around a certain value or demand is not lower than a certain value etc two additional forecasts are crisp values obtained using conventional statistical methods one using time series analysis based on decomposition tsad and the other using an auto regressive integrated moving average arma model the combination of these four forecast values into one improved forecast is made by applying fuzzy if then rules a modified mamdani style inference is used which enables reasoning with fuzzy inputs a new learning mechanism is developed and incorporated into the dss df to adapt the rule bases that combine the individual forecasted values the rule bases are adapted taking into consideration the performance of each of the forecast methods recorded in the past the application of dss df is demonstrated by an illustrative example the forecasts obtained by dss df are compared with results procured by applying the conventional tsad and arma methods separately the results obtained are encouraging and indicate that combining forecasts obtained by different methods may be beneficial

the superlinear convergence behaviour of gmres abstract gmres is a rather popular iterative method for the solution of nonsingular nonsymmetric linear systems it is well known that gmres often has a so called superlinear convergence behaviour i e the rate of convergence seems to improve as the iteration proceeds for the conjugate gradients method this phenomenon has been related to a modest degree of convergence of the ritz values it has been observed in experiments that for gmres too changes in the convergence behaviour seem to be related to the convergence of ritz values in this paper we prove that as soon as eigenvalues of the original operator are sufficiently well approximated by ritz values gmres from then on converges at least as fast as for a related system in which these eigenvalues and their eigenvector components are missing

threshold optimisation for multi label classifiers many multi label classifiers provide a real valued score for each class a well known design approach consists of tuning the corresponding decision thresholds by optimising the performance measure of interest we address two open issues related to the optimisation of the widely used f measure and precision recall p r curve with respect to the class related decision thresholds on a given data set i we derive properties of the micro averaged f which allow its global maximum to be found by an optimisation strategy with a low computational cost so far only a suboptimal threshold selection rule and a greedy algorithm with no optimality guarantee were known ii we rigorously define the macro and micro p r curves analyse a previously suggested strategy for computing them based on maximising f and develop two possible implementations which can be also exploited for optimising related performance measures we evaluate our algorithms on five data sets related to three different application domains

what are fuzzy rules and how to use them abstract fuzzy rules have been advocated as a key tool for expressing pieces of knowledge in fuzzy logic however there does not exist a unique kind of fuzzy rules nor is there only one type of fuzzy logic this diversity has caused many a misunderstanding in the literature of fuzzy control the paper is a survey of different possible semantics for a fuzzy rule and shows how they can be captured in the framework of fuzzy set and possibility theory it is pointed out that the interpretation of fuzzy rules dictates the way the fuzzy rules should be combined the various kinds of fuzzy rules considered in the paper gradual rules certainty rules possibility rules and others have different inference behaviors and correspond to various intended uses and applications the representation of fuzzy unless rules is briefly investigated on the basis of their intended meaning the problem of defining and checking the coherence of a block of parallel fuzzy rules is also briefly addressed this issue has been neglected in the fuzzy control literature although it looks important for validation purposes

implication based models of monotone fuzzy rule bases abstract in many modelling problems there is some inherent monotone relationship between one or more of the input variables and the output variable we consider the prototypical case of an increasing relationship between each of the input variables and the output variable when using fuzzy rule based models this desired monotonicity is reflected in the rule base given an appropriate ordering on the fuzzy sets involved in the respective input and output domains more specifically the larger the antecedent fuzzy sets the larger the consequent fuzzy set however fuzzy rule based modelling involves a final defuzzification step possibly resulting in a function that is no longer monotone in the context of mamdani assilian conjunctive fuzzy models ample attention has been paid to this problem both for the centre of gravity defuzzification and mean of maxima defuzzification methods in this paper we show that for implicative fuzzy models the non monotonicity problem can be circumvented by making explicit the semantics of the fuzzy rules by subjecting the antecedent and consequent fuzzy sets to the at least and or at most modifiers

normal forms for fuzzy logic functions and their approximation ability abstract aiming at developing theoretical backgrounds of fuzzy logic we consider an mv algebra of fuzzy logic functions two kinds of formulas regarded as disjunctive and conjunctive normal forms are introduced as a formal translation of if then rules their ability for an approximate representation of uniformly continuous functions are investigated the problem of transformation of any formula of propositional fuzzy logic into either of the normal forms is discussed

on the representation of fuzzy rules in terms of crisp rules there are two classical ways of viewing a fuzzy set in terms of a combination of crisp sets either as a weighted max aggregation of its level cuts or as a convex combination of the latter these views are applied to the particular case of fuzzy relations which are induced by fuzzy rules where the rules are modeled by implication connectives defined on a finite subset of the unit interval the paper studies more closely the representation of such fuzzy rules in terms of a convex combination of gradual rules a special type of implication based fuzzy rule inducing a crisp relation this representation which might be interpreted in a probabilistic way is shown to be unique on the assumption that the implication operator used for modeling the fuzzy rule does not satisfy a special kind of strict monotonicity condition in this case the crisp relations induced by the involved gradual rules correspond to level cuts of the fuzzy relation associated with the fuzzy rule however other representations might exist if the aforementioned property is not satisfied under a slightly stronger strict monotonicity condition the existence of further non consonant representations is even guaranteed then the crisp relations induced by gradual rules do not necessarily correspond to level cuts of the underlying fuzzy relation in this sense the decomposition of a fuzzy rule into a class of crisp implication based rules is more general than the classical decomposition of fuzzy sets

a short note on fuzzy relational inference systems this paper is a short note contribution to the topic of fuzzy relational inference systems and the preservation of their desirable properties it addresses the two main fuzzy relational inferences compositional rule of inference cri and the bandler kohout subproduct bk subproduct and their combination with two fundamental fuzzy relational models of fuzzy rule bases namely the mamdani assilian and the implicative models the goal of this short note article is twofold firstly we show that the robustness related to the combination of bk subproduct and implicative fuzzy rule base model was not proven correctly in 24 however we will show that the result itself is still valid and a valid proof will be provided secondly we shortly discuss the preservation of desirable properties of fuzzy inference systems and conclude that neither the above mentioned robustness nor any other computational advantages should automatically lead to a preference of the combinations of cri with mamdani assilian models or of the bk subproduct with the implicative models

gradual inference rules in approximate reasoning abstract gradual inference rules of the form the more x is f the more y is g or of similar forms with the less instead of the more which express a progressive change of the degree to which the entity y satisfies the gradual property g when the degree to which the entity x satisfies the gradual property f is modified are often encountered in commonsense reasoning a representation of such rules by means of fuzzy sets is proposed and discussed this representation turns out to be based on a special implication function already considered in multiple valued logic patterns of reasoning involving gradual inference rules are formalized their links with interpolation mechanisms are pointed out

fuzzy class theory the paper introduces a simple yet powerful axiomatization of zadeh s notion of fuzzy set based on formal fuzzy logic the presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics its essence is elementary fuzzy set theory cast as two sorted first order theory over fuzzy logic which is generalized to simple type theory we show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism in this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed however we also demonstrate how to add structure to the universe of discourse within our framework

on the suitability of the bandler kohout subproduct as an inference mechanism fuzzy relational inference fri systems form an important part of approximate reasoning schemes using fuzzy sets the compositional rule of inference cri which was introduced by zadeh has attracted the most attention so far in this paper we show that the fri scheme that is based on the bandler kohout bk subproduct along with a suitable realization of the fuzzy rules possesses all the important properties that are cited in favor of using cri viz equivalent and reasonable conditions for their solvability their interpolative properties and the preservation of the indistinguishability that may be inherent in the input fuzzy sets moreover we show that under certain conditions the equivalence of first infer then aggregate fita and first aggregate then infer fati inference strategies can be shown for the bk subproduct much like in the case of cri finally by addressing the computational complexity that may exist in the bk subproduct we suggest a hierarchical inferencing scheme thus this paper shows that the bk subproduct based fri is as effective and efficient as the cri itself

monotonicity of siso fuzzy relational inference with an implicative rule base a fuzzy relational inference fri mechanism is appraised based on the different desirable properties it possesses among these properties monotonicity of an fri has not received much attention in this work we investigate the monotonicity of a single input single output fri with an implicative form of the rule base in all the previous works that deal with monotonicity of an fri with the implicative form of rule base the employed fuzzy implications come from a residuated lattice it can be noticed that this rich underlying structure plays a major role in proving the results further we also modify the given monotone rule base this work differs from the previous works in that 1 the fuzzy implications employed in it do not come from any known residuated structure on text 0 1 and 2 the original rule base is employed without any alteration we determine conditions under which monotonicity of an fri where the rule base is modeled by a strict fuzzy implication can be ensured without transforming the original rule base thus the results in this work further augment the case for considering fuzzy implications other than those from the residuated setting to be used in applications

fuzzy set based methods in instance based reasoning a formal framework of instance based prediction is presented in which the generalization beyond experience is founded on the concepts of similarity and possibility the underlying extrapolation principle is formalized within the framework of fuzzy rules thus instance based reasoning can be realized as fuzzy set based approximate reasoning more precisely our model makes use of so called possibility rules these rules establish a relation between the concepts of similarity and possibility which takes the uncertain character of similarity based inference into account inductive inference is possibilistic in the sense that predictions take the form of possibility distributions on the set of outcomes rather than precise deterministic estimations the basic model is extended by means of fuzzy set based modeling techniques this extension provides the basis for incorporating domain specific expert knowledge thus our approach favors a view of instance based reasoning according to which the user interacts closely with the system

conditional measures and their applications to fuzzy sets abstract given a decomposable measure with respect to a continuous t conorm as introduced by the author in an earlier paper see section 1 we can construct conditional measures as implications these fulfil a generalized product law replacing the product in the classical law by any other strict t norm and turn out to be decomposable with respect to an operation v depending on and the condition set v section 2 more general conditional measures are introduced axiomatically and are shown to be conditional measures with respect to some decomposable measure section 3 bayesian like models are given which are alternatives to that presented by the author in a recent paper section 4 finally some applications to fuzzy sets are sketched section 5

radial fuzzy systems the class of radial fuzzy systems is introduced the fuzzy systems in this class use radial functions to implement membership functions of fuzzy sets and exhibit a shape preservation property in antecedents of their rules the property is called the radial property it enables the radial fuzzy systems to have their computational model mathematically tractable under both conjunctive and implicative representations of their rule bases coherence of radial implicative fuzzy systems is discussed and a sufficient condition for coherence is stated

comparison of fuzzy reasoning methods l a zadeh e h mamdani and m mizumoto et al have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition if x is a then y is b with a and b being fuzzy concepts mizumoto et al have investigated the properties of their methods in the case of generalized modus ponens this paper deals with the properties of their methods in the case of generalized modus tollens and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems finally the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

fuzzy reasoning method by optimizing the similarity of truth tables this paper presents a new fuzzy reasoning method by optimizing the similarity of truth tables os method for short its basic idea is to find a fuzzy set such that the truth tables generated by the antecedent rule and the consequent rule are as similar as possible based on this idea the principle of os method and the fuzzy reasoning with os method are given and discussed and then the os methods with certain similarity measure and several fuzzy implications are investigated finally numerical examples are analyzed to compare the proposed method with compositional rule of inference cri method

annotated fuzzy logic programs several fuzzy logic programming systems that deal with fuzzy sets as data in programs have been developed but they lack the fundamentals of a theorem prover whence the soundness and the completeness cannot be proved annotated logic programs have been developed as an extension of classical logic programs and proved to be an essential formalism for lattice based reasoning it appears to be a suitable framework but is still not sufficient for development from the point of view of a theorem prover of fuzzy logic programs computing with fuzzy sets this paper proposes annotated fuzzy logic programs aflps as a formal logical basis for fuzzy logic programming systems involving fuzzy sets as soft data i e data pervaded with uncertainty the syntax of aflps is introduced and their declarative semantics is studied the aflp sld style proof procedure is then defined and proved to be sound and complete with respect to the declarative semantics of aflps c 2000 elsevier science b v all rights reserved

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

reasoning about mathematical fuzzy logic and its future this paper is devoted to reasoning about fuzzy logic which is based on various personal observations of the author our goal is to think of the state of the art in mathematical fuzzy logic mfl and to outline some of the tasks on which in the author s opinion mfl should focus in the future in our discussion we will mention not only the basic theory but also its extension called fuzzy logic in broader sense flb the paradigm of the latter is to be the logic of natural human reasoning whose most essential characteristic is the use of natural language besides brief description of flb we will also mention some of its applications on the basis of that we will ponder on other possible directions for research namely the possibility of using fln as a metatheory of fuzzy mathematics as a proper tool for modeling of the main manifestations of the phenomenon of vagueness and as a reasonable tool for developing models of linguistic semantics

arithmetic fuzzy models it is well known that a fuzzy rule base can be interpreted in different ways from a logical point of view the conjunctive interpretation is preferred while from a practical point of view the disjunctive interpretation has been dominantly present each of these interpretations results in a specific fuzzy relation that models the fuzzy rule base basic interpolation requirements naturally suggest a corresponding inference mechanism the direct image for the conjunctive interpretation and the subdirect image for the disjunctive interpretation interpolation then corresponds to solvability of some system of fuzzy relational equations in this paper we show that other types of fuzzy relations which are closely related to takagi sugeno t s models are of major interest as well these fuzzy relations are based on addition and multiplication only from which we get the name arithmetic fuzzy models under some mild requirements these fuzzy relations turn out to be solutions of the same systems of fuzzy relational equations the impact of these results is both theoretical and practical there exist simple solutions to systems of fuzzy relational equations other than the extremal solutions that have received all the attention so far which are moreover easy to implement

a probabilistic framework for interval type 2 fuzzy linguistic summarization many studies on linguistic summarization have addressed the use of ordinary fuzzy set type 1 fuzzy set t1fs for modeling words however few of them have exploited interval type 2 fuzzy set it2fs although it2fs is better able to deal with the uncertainty associated with words the existing studies on linguistic summarization using it2fs have focused on scalar cardinality based degree of truth in this paper for the first time we propose a probabilistic framework based on the idea of interval mass assignment to evaluate interval type 2 fuzzy linguistic summaries based on the type ii quantified sentences and the semi fuzzy quantifiers as an alternative method to the scalar cardinality based methods we implement a real case study on linguistic summarization of time series data of europe brent spot price as well as a comparison of the results obtained with our approach and those of the existing approaches

penalized partial least square discriminant analysis with l 1 norm for multi label data multi label data are prevalent in real world due to its great potential applications multi label learning has now been receiving more and more attention from many fields however how to effectively exploit the correlations of variables and labels and tackle the high dimensional problems of data are two major challenging issues for multi label learning in this paper we make an attempt to cope with these two problems by proposing an effective multi label learning algorithm specifically we make use of the technique of partial least square discriminant analysis to identify a common latent space between the variable space and the label space of multi label data moreover considering the label space of the multi label data is sparse a l1 norm penalty is further performed to constrain the y loadings of the optimization problem of partial least squares making them sparse the merit of our method is that it can capture the correlations and perform dimension reduction at the same time the experimental results conducted on eleven public data sets show that our method is promising and superior to the state of the art multi label classifiers in most cases highlightsa new and effective learning method for the multi label data is proposed the method exploits the sparse property of the label space fully the method captures the correlations of the variables by using partial least squares discriminant analysis dimension reduction has also applied to the high dimensional data the sparse purpose is achieved by performing the l1 norm penalty

core vector machines fast svm training on very large data sets standard svm training has o m3 time and o m2 space complexities where m is the training set size it is thus computationally infeasible on very large data sets by observing that practical svm implementations only approximate the optimal solution by an iterative strategy we scale up kernel methods by exploiting such approximateness in this paper we first show that many kernel methods can be equivalently formulated as minimum enclosing ball meb problems in computational geometry then by adopting an efficient approximate meb algorithm we obtain provably approximately optimal solutions with the idea of core sets our proposed core vector machine cvm algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m experiments on large toy and real world data sets demonstrate that the cvm is as accurate as existing svm implementations but is much faster and can handle much larger data sets than existing scale up methods for example cvm with the gaussian kernel produces superior results on the kddcup 99 intrusion detection data which has about five million training patterns in only 1 4 seconds on a 3 2ghz pentium 4 pc