asymmetric subsethood product fuzzy neural inference system asupfunis this work presents an asymmetric subsethood product fuzzy neural inference system asupfunis that directly extends the supfunis model by permitting signal and weight fuzzy sets to be modeled by asymmetric gaussian membership functions the asymmetric subsethood product network admits both numeric as well as linguistic inputs input nodes which act as tunable feature fuzzifiers fuzzify numeric inputs with asymmetric gaussian fuzzy sets and linguistic inputs are presented as is the antecedent and consequent labels of standard fuzzy if then rules are represented as asymmetric gaussian fuzzy connection weights of the network the model uses mutual subsethood based activation spread and a product aggregation operator that works in conjunction with volume defuzzification in a gradient descent learning framework despite the increase in the number of free parameters the proposed model performs better than supfunis on various benchmarking problems both in terms of the performance accuracy and architectural economy and compares excellently with other various existing models with a performance better than most of them

fuzzy rule selection by multi objective genetic local search algorithms and rule evaluation measures in data mining this paper shows how a small number of simple fuzzy if then rules can be selected for pattern classification problems with many continuous attributes our approach consists of two phases candidate rule generation by rule evaluation measures in data mining and rule selection by multi objective evolutionary algorithms in our approach first candidate fuzzy if then rules are generated from numerical data and prescreened using two rule evaluation measures i e confidence and support in data mining then a small number of fuzzy if then rules are selected from the prescreened candidate rules using multi objective evolutionary algorithms in rule selection we use three objectives maximization of the classification accuracy minimization of the number of selected rules and minimization of the total rule length thus the task of multi objective evolutionary algorithms is to find a number of non dominated rule sets with respect to these three objectives the main contribution of this paper is to propose an idea of utilizing the two rule evaluation measures as prescreening criteria of candidate rules for fuzzy rule selection an arbitrarily specified number of candidate rules can be generated from numerical data for high dimensional pattern classification problems through computer simulations we demonstrate that such a prescreening procedure improves the efficiency of our approach to fuzzy rule selection we also extend a multi objective genetic algorithm moga in our former studies to a multi objective genetic local search mogls algorithm where a local search procedure adjusts the selection i e inclusion or exclusion of each candidate rule furthermore a learning algorithm of rule weights i e certainty factors is combined with our mogls algorithm such extensions to our moga for fuzzy rule selection are another contribution of this paper

constructing a fuzzy controller from data abstract fuzzy control at the executive level can be interpreted as an approximation technique for a control function based on typical imprecisely specified input output tuples that are represented by fuzzy sets the imprecision is characterized by similarity relations that are induced by transformations of the canonical distance function between real numbers taking this interpretation of fuzzy controllers into account in order to derive a fuzzy controller from observed data typical input output tuples have to be identified in addition a concept of similarity based on a transformations of the canonical distance is needed in order to characterize the typical input output tuples by suitable fuzzy sets a variety of fuzzy clustering algorithms that are exactly working in this spirit exists they identify prototypes and assign fuzzy sets to the prototypes on the basis of a suitable transformed distance in this paper we discuss how such fuzzy clustering techniques can be applied to construct a fuzzy controller from data and introduce special clustering algorithms that are tailored for this problem

a new methodology of extraction optimization and application of crisp and fuzzy logical rules a new methodology of extraction optimization and application of sets of logical rules is described neural networks are used for initial rule extraction local or global minimization procedures for optimization and gaussian uncertainties of measurements are assumed during application of logical rules algorithms for extraction of logical rules from data with real valued features require determination of linguistic variables or membership functions contest dependent membership functions for crisp and fuzzy linguistic variables are introduced and methods of their determination described several neural and machine learning methods of logical rule extraction generating initial rules are described based on constrained multilayer perceptron networks with localized transfer functions or on separability criteria for determination of linguistic variables a tradeoff between accurary simplicity is explored at the rule extraction stage and between rejection error level at the optimization stage gaussian uncertainties of measurements are assumed during application of crisp logical rules leading to soft trapezoidal membership functions and allowing to optimize the linguistic variables using gradient procedures numerous applications of this methodology to benchmark and real life problems are reported and very simple crisp logical rules for many datasets provided

evolving fuzzy classifiers using different model architectures in this paper we present two novel approaches for on line evolving fuzzy classifiers called eclass and flexfis class both methods can be applied with different model architectures including single model sm with class labels as consequents classification hyper planes as consequents and multi model mm architecture additionally eclass can have a multi input multi output mimo architecture with multiple hyper planes as consequents of each fuzzy rule the difference between mm and mimo architectures is that the former one applies one separate and independent fuzzy rule based frb classifier for each class and is using an indicator labelling scheme while the latter one applies a single frb where the rules are mimo rather than miso both eclass and flexfis class methods are designed to work on a per sample basis and are thus one pass incremental additionally their structure frb is evolving rather than fixed it adapts their parameters in antecedent and consequent parts with any newly loaded sample a special emphasis is placed on advanced issues for improving accuracy and robustness including a thorough comparison between global and local learning of consequent functions a novel approach for detecting of and reacting on drifts in the data streams and an enhanced outlier treatment strategy the methods and their extensions according to the advanced issues are evaluated on one benchmark problem of handwritten images recognition as well as on a real life problem of image classification framework where images should be classified into good and bad ones during an on line and interactive production process

cartesian closed categories of algebraic cpos abstract the results presented in this paper were inspired by the work of smyth who showed that there is a largest cartesian closed full subcategory inside the category of countably based algebraic cpos with least element namely the category of profinite domains removing the countability condition we show that there are exactly two maximal cartesian closed full subcategories inside the category of algebraic cpos with least element one is the natural extension of the class of profinite domains to the uncountable case the other is a new class of domains which are characterized by the property that every principal ideal is a complete lattice the name l domain is introduced for cpos with this property passing to the general situation where no least element is required anymore we find a pair of categories in place of each the profinite domains and the algebraic l domains thus there are four maximal cartesian closed categories of algebraic cpos in this case this complete overview over the possible classes of domains allows to prove general theorems about them this is illustrated by the result that a cpo has an algebraic function space if and only if its space of strict continuous functions is algebraic

quasi coherent states for unitary groups the author extends a recent construction of isotropic spin coherent states to unitary groups expectation values in such a quasi coherent state can be obtained from a generating functional which is given explicitly for the fundamental and the adjoint representations of the unitary groups su n and u n the close connection between the corresponding generating functional and the external field problem in qcd is pointed out free quantum fields in such a condensed colour singlet coherent state are formally related to two dimensional lattice gauge theories with in general a mixed action in the large n limit the author exhibits a phase transition for a free quantum field in a singlet quasi coherent state a path integral representation of the transition amplitude in terms of quasi coherent states is also given the corresponding effective action describes dynamics on a complex kahler manifold

on line incremental feature weighting in evolving fuzzy classifiers in this paper we present an approach for addressing the problem of dynamic dimension reduction during on line training evolution and updating of evolving fuzzy classifiers efc the basic idea of our approach is that instead of permanently changing the list of most important features with newly loaded data blocks we generalize the concept of incremental feature selection to an incremental feature weighting approach features are assigned weights in 0 1 according to their importance level these weights are permanently updated during on line mode and guarantee a smooth learning process in the evolving fuzzy classifiers as they change softly and continuously over time in some cases when the weights become approximately 0 an automatic switching off of some features and therefore a soft dimension reduction is achieved two novel incremental feature weighting strategies are proposed in this paper one based on a leave one feature out the other based on a feature wise separability criterion we will describe the integration concept of the feature weights in the evolving fuzzy classifiers using single and multi model architecture where flexfis class sm and flexfis class mm serve as training engines the whole approach of integrated incremental feature weighting in evolving fuzzy classifiers will be evaluated based on high dimensional on line real world classification scenarios and based on data from the internet the results will show that incremental feature weighting in efc in fact helps to reduce curse of dimensionality and therefore guides the evolving fuzzy classifiers to a higher on line predictive power

constructing measures for spin variable path integrals by exploiting the overcompleteness of the spin coherent states we derive expressions for spin kinematics path integrals specifically for spin 1 2 and spin 1 in terms of genuine wiener measures on continuous paths lying on the unit sphere and for certain dynamical systems which when projected onto the subspace spanned by the proper spin coherent state matrix elements yield the appropriate quantum mechanical propagator

generating an interpretable family of fuzzy partitions from data in this paper we propose a new method to construct fuzzy partitions from data the procedure generates a hierarchy including best partitions of all sizes from n to two fuzzy sets the maximum size n is determined according to the data distribution and corresponds to the finest resolution level we use an ascending method for which a merging criterion is needed this criterion is based on the definition of a special metric distance suitable for fuzzy partitioning and the merging is done under semantic constraints the distance we define does not handle the point coordinates but directly their membership degrees to the fuzzy sets of the partition this leads to the introduction of the notions of internal and external distances the hierarchical fuzzy partitioning is carried independently over each dimension and to demonstrate the partition potential they are used to build fuzzy inference system using a simple selection mechanism due to the merging technique all the fuzzy sets in the various partitions are interpretable as linguistic labels the tradeoff between accuracy and interpretability constitutes the most promising aspect in our approach well known data sets are investigated and the results are compared with those obtained by other authors using different techniques the method is also applied to real world agricultural data the results are analyzed and weighed against those achieved by other methods such as fuzzy clustering or discriminant analysis

learning fuzzy rules from iterative execution of games this paper discusses the linguistic knowledge extraction from the iterative execution of a multiplayer non cooperative repeated game linguistic knowledge is automatically extracted in the form of fuzzy if then rules our knowledge extraction is mainly based on the on line incremental learning of fuzzy rule based systems in this sense our linguistic knowledge extraction is the learning of fuzzy rules we first briefly describe a market selection game which is formulated as a non cooperative repeated game with many players and several alternative actions we also describe some simple strategies for our market selection game in our market selection game the payoff of each player depends on the actions of all players when a particular action is chosen by many players those players receive low payoff high payoff is obtained from actions chosen by only a small number of players this means that minority players with respect to their actions receive high payoff next we show how our market selection game can be handled as a pattern classification problem where a single training pattern is successively generated from every round of our game a fuzzy rule based classification system is used as a decision making system by each player for choosing an action in every round an on line incremental learning algorithm is proposed for adjusting the fuzzy rule based classification system then we show how our market selection game can be handled as a function approximation problem a fuzzy rule based approximation system is used as a value function for approximating the expected payoff from each action finally simulation results show that comprehensible linguistic knowledge is extracted by the learning of fuzzy rule based systems

self adaptive neuro fuzzy inference systems for classification applications this paper presents a self adaptive neuro fuzzy inference system sanfis that is capable of self adapting and self organizing its internal structure to acquire a parsimonious rule base for interpreting the embedded knowledge of a system from the given training data set a connectionist topology of fuzzy basis functions with their universal approximation capability is served as a fundamental sanfis architecture that provides an elasticity to be extended to all existing fuzzy models whose consequent could be fuzzy term sets fuzzy singletons or functions of linear combination of input variables without a priori knowledge of the distribution of the training data set a novel mapping constrained agglomerative clustering algorithm is devised to reveal the true cluster configuration in a single pass for an initial sanfis construction estimating the location and variance of each cluster subsequently a fast recursive linear nonlinear least squares algorithm is performed to further accelerate the learning convergence and improve the system performance good generalization capability fast learning convergence and compact comprehensible knowledge representation summarize the strength of sanfis computer simulations for the iris wisconsin breast cancer and wine classifications show that sanfis achieves significant improvements in terms of learning convergence higher accuracy in recognition and a parsimonious architecture

constructing measures for path integrals the overcompleteness of the coherent states for the heisenberg weyl group implies that many different integral kernels can be used to represent the same operator within such an equivalence class we construct an integral kernel to represent the quantum mechanical evolution operator for certain dynamical systems in the form of a path integral that involves genuine wiener measures on continuous phase space paths to achieve this goal it is necessary to employ an expression for the classical action different from the usual one

flexfis a robust incremental learning approach for evolving takagi sugeno fuzzy models in this paper we introduce a new algorithm for incremental learning of a specific form of takagi sugeno fuzzy systems proposed by wang and mendel in 1992 the new data driven online learning approach includes not only the adaptation of linear parameters appearing in the rule consequents but also the incremental learning of premise parameters appearing in the membership functions fuzzy sets together with a rule learning strategy in sample mode a modified version of vector quantization is exploited for rule evolution and an incremental learning of the rules premise parts the modifications include an automatic generation of new clusters based on the nature distribution and quality of new data and an alternative strategy for selecting the winning cluster rule in each incremental learning step antecedent and consequent learning are connected in a stable manner meaning that a convergence toward the optimal parameter set in the least squares sense can be achieved an evaluation and a comparison to conventional batch methods based on static and dynamic process models are presented for high dimensional data recorded at engine test benches and at rolling mills for the latter the obtained data driven fuzzy models are even compared with an analytical physical model furthermore a comparison with other evolving fuzzy systems approaches is carried out based on nonlinear dynamic system identification tasks and a three input nonlinear function approximation example

effect of rule weights in fuzzy rule based classification systems this paper examines the effect of rule weights in fuzzy rule based classification systems each fuzzy if then rule in our classification system has antecedent linguistic values and a single consequent class we use a fuzzy reasoning method based on a single winner rule in the classification phase the winner rule for a new pattern is the fuzzy if then rule that has the maximum compatibility grade with the new pattern when we use fuzzy if then rules with certainty grades the winner is determined as the rule with the maximum product of the compatibility grade and the certainty grade in this paper the effect of rule weights is illustrated by drawing classification boundaries using fuzzy if then rules with without certainty grades it is also shown that certainty grades play an important role when a fuzzy rule based classification system is a mixture of general rules and specific rules through computer simulations we show that comprehensible fuzzy rule based systems with high classification performance can be designed without modifying the membership functions of antecedent linguistic values when we use fuzzy if then rules with certainty grades

rule acquisition and adjustment based on set valued mapping based on the theory of set valued mapping a new approach on how to acquire fuzzy rule sets from the corresponding relations between the group of data on the two universes of discourse is presented moreover some theories and methods on how to adjust fuzzy rule sets with the change of the group of data are established finally a recurrence theorem of the group of data is also given

on an open problem of amadio and curien the finite antichain condition more than a dozen years ago amadio bifinite domains stable case in lecture notes in computer science vol 530 1991 pp 16 33 see amadio and curien domains and lambda calculi cambridge tracts in theoretical computer science vol 46 cambridge university press 1998 as well raised the question of whether the category of stable bifinite domains of amadio droste r m amadio bifinite domains stable case in lecture notes in computer science vol 530 1991 pp 16 33 m droste on stable domains theor comput sci 111 1993 89 101 m droste cartesian closed categories of stable domains for polymorphism preprint universitat ghs essen is the largest cartesian closed full sub category of the category of w algebraic meet cpos with stable functions an affirmative solution to this problem has two major steps 1 show that for any w algebraic meet cpo d if all higher order stable function spaces built from d are w algebraic then d is finitary i e it satisfies the so called axiom i 2 show that for any w algebraic meet cpo d if d violates mi then d d violates either m or i we solve the first part of the problem in this paper i e for any w algebraic meet cpo d if the stable function space d d satisfies m then d is finitary our notion of mub meet closed set which is introduced for step 1 will also be used for treating some example cases in step 2

non deterministic information systems and their domains abstract in the theory of denotational semantics of programming languages dedekind complete algebraic partial orders domains frequently have been considered since scott s and strachey s fundamental work in 1971 stoy 1977 as scott 1982 showed these domains can be represented canonically by deterministic information systems however recently more complicated constructions such as power domains have led to more general domains plotkin 1976 smyth and plotkin 1977 smyth 1983 we introduce non deterministic information systems and establish the representation theorem similar to scott 1982 for these more general domains this result will be the basis for solving recursive domain equations

designing of classifiers based on immune principles and fuzzy rules this paper proposed an algorithm to design a fuzzy classification system based on immune principles the proposed algorithm evolves a population of antibodies based on the clonal selection and hypermutation principles the membership function parameters and the fuzzy rule set including the number of rules inside it are evolved at the same time each antibody candidate solution corresponds to a fuzzy classification rule set we compared our algorithm with other classification schemes on some benchmark datasets the results demonstrated the effectiveness of the proposed immune algorithm

rule weight specification in fuzzy rule based classification systems this paper shows how the rule weight of each fuzzy rule can be specified in fuzzy rule based classification systems first we propose two heuristic methods for rule weight specification next the proposed methods are compared with existing ones through computer simulations on artificial numerical examples and real world pattern classification problems simulation results show that the proposed methods outperform the existing ones in the case of multiclass pattern classification problems with many classes

weighting fuzzy classification rules using receiver operating characteristics roc analysis in fuzzy rule based classification systems rule weight has often been used to improve the classification accuracy in past research a number of heuristic methods for rule weight specification have been proposed in this paper a method of fuzzy rule weight specification using receiver operating characteristic roc analysis is proposed in order to specify the weight of a fuzzy rule using 2 class roc analysis the threshold that the rule achieves its maximum accuracy is found this threshold is used as the weight of the rule the proposed method is compared with existing ones through computer simulations on some well known classification problems with continuous attributes simulation results show that the proposed method performs better than existing methods of fuzzy rule weight specification

measures for path integrals by exploitation of the overcompleteness of coherent states expressions are presented for path integrals in terms of genuine wiener path space measures for driven harmonic oscillators which when projected onto the subspace spanned by coherent state matrix elements yield the appropriate quantum mechanical propagator

a hybrid coevolutionary algorithm for designing fuzzy classifiers rule learning is one of the most common tasks in knowledge discovery in this paper we investigate the induction of fuzzy classification rules for data mining purposes and propose a hybrid genetic algorithm for learning approximate fuzzy rules a novel niching method is employed to promote coevolution within the population which enables the algorithm to discover multiple rules by means of a coevolutionary scheme in a single run in order to improve the quality of the learned rules a local search method was devised to perform fine tuning on the offspring generated by genetic operators in each generation after the ga terminates a fuzzy classifier is built by extracting a rule set from the final population the proposed algorithm was tested on datasets from the uci repository and the experimental results verify its validity in learning rule sets and comparative advantage over conventional methods

a learning algorithm of fuzzy neural networks with triangular fuzzy weights abstract in this paper first we propose an architecture of fuzzy neural networks with triangular fuzzy weights the proposed fuzzy neural network can handle fuzzy input vectors as well as real input vectors in both cases outputs from the fuzzy neural network are fuzzy vectors the input output relation of each unit of the fuzzy neural network is defined by the extension principle of zadeh next we define a cost function for the level sets i e cuts of fuzzy outputs and fuzzy targets then we derive a learning algorithm from the cost function for adjusting three parameters of each triangular fuzzy weight finally we illustrate our approach by computer simulations on numerical examples

a characterisation of the least fixed point operator by dinaturality simpson a k a characterisation of the least fixed point operator by dinaturality theoretical computer science 118 1993 301 314 the paper addresses the question of when the least fixed point operator in a cartesian closed category of domains is characterised as the unique dinatural transformation from the exponentiation bifunctor to the identity functor we give a sufficient condition on a cartesian closed full subcategory of the category of algebraic cpos for the characterisation to hold the condition is quite mild and the least fixed point operator is so characterised in many of the most commonly used categories of domains by using retractions the characterisation extends to the associated cartesianclosed categories of continuous cpos however dinaturality does not always characterise the leastfixed point operator we show that in cartesian closed full subcategories of the category of continuous lattices the characterisation fails

stone duality and representation of stable domain abstract in this paper the author studies the stone duality and representation of l domain w r t stable functions two basic notions are introduced one is d semilattice and the other is semitopological system the author first gives the representation of stable d lattice then establishes the representation theorem of l domain and gives the stone duality of l domains w r t stable functions in the scheme of lattice the author introduces the theory of semitopological systems which is a generalization of vickers topological systems by using it the author gives the stone duality of l domains w r t stable functions and that of scott domains w r t functions preserving the directed sups and meets of nonempty subsets in the scheme of topology

the largest cartesian closed category of domains the main result of this paper provides a necessary and sufficient condition that the function space d d of a domain algebraic cpo d be again a domain the condition confirming a conjecture of plotkin is that d be strongly algebraic sfp the importance of this is that it enables us to conclude that sfp is the largest category of domains that is closed under the constructions of interest in semantics in the final section we briefly consider how our results may need to be extended if our admittedly rather restrictive notion of category of domains is relaxed in certain ways

repairs to glvq a new family of competitive learning schemes first we identify an algorithmic defect of the generalized learning vector quantization glvq scheme that causes it to behave erratically for a certain scaling of the input data we show that glvq can behave incorrectly because its learning rates are reciprocally dependent on the sum of squares of distances from an input vector to the node weight vectors finally we propose a new family of models the glvq f family that remedies the problem we derive competitive learning algorithms for each member of the glvq f model and prove that they are invariant to all scalings of the data we show that glvq f offers a wide range of learning models since it reduces to lvq as its weighting exponent a parameter of the algorithm approaches one from above as this parameter increases glvq f then transitions to a model in which either all nodes may be excited according to their inverse distances from an input or in which the winner is excited while losers are penalized and as this parameter increases without limit glvq f updates all nodes equally we illustrate the failure of glvq and success of glvq f with the iris data

multipolar aggregation operators in reasoning methods for fuzzy rule based classification systems the reasoning methods in fuzzy rule based classification systems are studied and their relation to multipolar aggregation operators is discussed we describe the change in the output of the classification system based on the change of the fuzzy reasoning method we propose a new fuzzy reasoning method based on multipolar fusion owa operator which can control the transition between the maximum rule method and the maximum vote method by a weighting vector this new method uses principles of owa operators and assigns the rule weight according to the strength of activation of the rule we compare different fuzzy reasoning methods in case studies

adaptive asymmetric fuzzy neural network controller design via network structuring adaptation this paper proposes a self structuring fuzzy neural network sfnn using asymmetric gaussian membership functions in the structure and parameter learning phases an adaptive self structuring asymmetric fuzzy neural network control asafnc system which consists of an sfnn controller and a robust controller is proposed the sfnn controller uses an sfnn with structure and parameter learning phases to online mimic an ideal controller simultaneously the structure learning phase consists of the growing and pruning algorithms of fuzzy rules to achieve an optimal network structure and the parameter learning phase adjusts the interconnection weights of neural network to achieve favorable approximation performance the robust controller is designed to compensate for the modeling error between the sfnn controller and the ideal controller an online training methodology is developed in the lyapunov sense and thus the stability of the closed loop control system can be guaranteed finally the proposed asafnc system is applied to a second order chaotic dynamics system the simulation results show that the proposed asafnc can achieve favorable tracking performance

quantum mechanical path integrals with wiener measure for all polynomial hamiltonians ii the coherent state representation of quantum mechanical propagators as well defined phase space path integrals involving wiener measure on continuous phase space paths in the limit that the diffusion constant diverges is formulated and proved this construction covers a wide class of self adjoint hamiltonians including all those which are polynomials in the heisenberg operators in fact this method also applies to maximal symmetric hamiltonians that do not possess a self adjoint extension this construction also leads to a natural covariance of the path integral under canonical transformations an entirely parallel discussion for spin variables leads to the representation of the propagator for an arbitrary spin operator hamiltonian as well defined path integrals involving wiener measure on the unit sphere again in the limit that the diffusion constant diverges

on rule pruning using fuzzy neural networks shann and fu sf proposed a fuzzy neural network fnn for rule pruning in a fuzzy controller in this paper we first analyze the fnn of sf and discuss some of its limitations sf attempted to eliminate redundant rules interpreting some of the connection weights as certainty factors of rules in their strategy the connection weights are unrestricted in sign and hence their interpretation as certainty factors introduces some inconsistencies into the scheme we propose a modification of this fnn which eliminates these inconsistencies moreover we also propose a pruning scheme which unlike the scheme of sf always produces a compatible rule set superiority of the modified fnn is established using the inverted pendulum problem

an extension to fuzzy cognitive maps for classification and prediction fuzzy cognitive maps fcms as an illustrative causative representation of modeling and manipulation of complex systems can be used to model the dynamic behavior of the investigated systems however due to defects in expression and architecture the traditional fcms and most of their relevant extensions are not applicable to classification problems to solve this problem this paper presents an approach that directly extends the model by translating the reasoning mechanism of traditional fcms to a set of fuzzy if then rules moreover the proposed approach fully considers the contribution of the inputs to the activation of the fuzzy rules and quantifies the causalities using mutual subsethood which works in conjunction with volume defuzziflcation in a gradient descent learning framework in this manner our approach enhances the capability of the conventional fcms to automatically identify membership functions and quantify causalities despite the increase in the number of tunable parameters experimental results show that the proposed approach efficiently extends the application of the traditional fcms into classification problems while keeping the ability for prediction and approximation

nonsingleton fuzzy logic systems theory and application in this paper we present a formal derivation of general nonsingleton fuzzy logic systems nsflss and show how they can be efficiently computed we give examples for special cases of membership functions and inference and we show how an nsfls can be expressed as a nonsingleton fuzzy basis function expansion and present an analytical comparison of the nonsingleton and singleton fuzzy logic systems formulations we prove that an nsfls can uniformly approximate any given continuous function on a compact set and show that our nsfls does a much better job of predicting a noisy chaotic time series than does a singleton fuzzy logic system fls

improving classification performance using fuzzy mlp and two level selective partitioning of the feature space a fuzzy mlp model developed by one of the authors is used for obtaining selective two level partitioning of the feature space in order to improve its classification performance the model can handle uncertainty and or impreciseness in the input as well as the output the input to the network is modelled in terms of linguistic pi sets whose centres and radii along the feature axes in each partition are generated automatically from the distribution of the training data the performance of the model at the end of the first stage is used as a criterion for guiding the selection of the appropriate partition to be subdivided at the second stage in order to improve the effectiveness of the model a comparative study of the performance of the two level technique with other methods viz the conventional mlp linear discriminant analysis and the k nearest neighbours algorithms is also provided to demonstrate its superiority

path integral for system with spin a path integral representation for the system with spin is considered explicit path integral representations are given in classical phase space for spin realized as the complex kahler manifold and as the grassmann algebra for the spin case

approximative fuzzy rules approaches for classification with hybrid ga techniques abstract in this paper the use of different methods from the fuzzy modeling field for classification tasks is evaluated and the potential of their integration in producing better classification results is investigated the methods considered approximative in their nature consider different integrations of techniques with an initial rule generation step and a following rule tuning approach using different evolutionary algorithms we analyse the adaptation of existing techniques in the fuzzy modeling context for the classification problem and the integration of these techniques in order to improve the classifiers performance finally a genetic algorithm ga for translation from approximative rules to similar descriptive ones trying to preserve the accuracy of the approximative classifier is presented the classical iris and cancer data set are used throughout the evaluation process to form a common ground for comparison and performance analysis

coherent state path integral calculation of the wigner function we consider a set of operators 1 n with diagonal representatives p n in the space of generalized coherent states n d n p n nn we regularize the coherent state path integral as a limit of a sequence of averages l over polygonal paths with l vertices n1 l the distribution of the path centroid 1 l l 1lp nl tends to the wigner function w x the joint distribution for the operators w x lim l n x l this result is proved in the case where the hamiltonian commutes with the wigner function is non positive if the dominant paths with path centroid in a certain region have berry phases close to odd multiples of for finite l the path centroid distribution is a wigner function convolved with a gaussian of variance inversely proportional to l the results are illustrated by numerical calculations of the spin wigner function from su 2 coherent states the relevance to the quantum monte carlo sign problem is also discussed

fuzzy classification using pattern discovery rule based classifiers allow rationalization of classifications made this in turn improves understanding which is essential for effective decision support as a rule based classifier the pattern discovery pd algorithm functions well in discrete nominal and continuous data domains a drawback when using pd as a classifier for decision support is that it has an unbounded decision space that confounds the understanding of the degree of support for a decision incorporating pd into a fuzzy inference system fis allows the the degree of support for a decision to be expressed with intuitively understandable terms in addition using discrete algorithms in continuous domains can result in reduced accuracy due to quantization fuzzification reduces this ldquocost of quantizationrdquo and improves classification performance in this work the pd algorithm was used as a source of rules for a series of fiss implemented using different rule weighting and defuzzification schemes each providing a linguistic basis for rule description and a bounded space for expression of decision support the output of each fis consists of a suggested outcome a strong confidence metric describing suggestions within this space and a linguistic expression of the rules this constitutes a stronger basis for decision making than that provided by pd alone a variety of synthetic continuous class distributions with varying degrees of separation was used to evaluate the performance of fuzzy pd back propagation and bayesian classifiers overall the accuracy of the fuzzy system was found to be similar but slightly below that of the inherently continuous valued classifiers and was somewhat improved with respect to the pd classifiers for the difficult spiral class distributions studied the fuzzy classifiers were able to make more classifications than the pd classifiers the correct classification rates for the fuzzy classifiers were similar across the various rule weighting and defuzzification schemes demonstrating the strength of the statistical method for rule generation analysis of several real world data sets shows that a pd based fis has comparable performance to a neuro fuzzy system the use of a pd based fis however provides insight into the structure of the data analyzed not available through the other approaches

a logical approach to stable domains building on earlier work by guo qiang zhang on disjunctive information systems and by thomas ehrhard pasquale malacaria and the first author on stable stone duality we develop a framework of disjunctive propositional logic in which theories correspond to algebraic l domains disjunctions in the logic can be indexed by arbitrary sets as in geometric logic but must be provably disjoint this raises several technical issues which have to be addressed before clean notions of axiom system and theory can be defined we show soundness and completeness of the proof system with respect to distributive disjunctive semilattices and prove that every such semilattice arises as the lindenbaum algebra of a disjunctive theory via stable stone duality we show how to use disjunctive propositional logic for a logical description of algebraic l domains

genetic feature selection in a fuzzy rule based classification system learning process for high dimensional problems abstract the inductive learning of a fuzzy rule based classification system frbcs is made difficult by the presence of a large number of features that increases the dimensionality of the problem being solved the difficulty comes from the exponential growth of the fuzzy rule search space with the increase in the number of features considered in the learning process in this work we present a genetic feature selection process that can be integrated in a multistage genetic learning method to obtain in a more efficient way frbcss composed of a set of comprehensible fuzzy rules with high classification ability the proposed process fixes a priori the number of selected features and therefore the size of the search space of candidate fuzzy rules the experimentation carried out using sonar example base shows a significant improvement on simplicity precision and efficiency achieved by adding the proposed feature selection processes to the multistage genetic learning method or to other learning methods

combining boosting and evolutionary algorithms for learning of fuzzy classification rules this paper presents a novel boosting algorithm for genetic learning of fuzzy classification rules the method is based on the iterative rule learning approach to fuzzy rule base system design the fuzzy rule base is generated in an incremental fashion in that the evolutionary algorithm optimizes one fuzzy classifier rule at a time the boosting mechanism reduces the weight of those training instances that are classified correctly by the new rule therefore the next rule generation cycle focuses on fuzzy rules that account for the currently uncovered or misclassified instances the weight of a fuzzy rule reflects the relative strength the boosting algorithm assigns to the rule class when it aggregates the casted votes the approach is compared with other classification algorithms for a number problem sets from the uci repository

fuzzy self organizing map abstract kohonen s self organizing map is one of the best known neural network models in this paper we introduce a fuzzy version of the model called fuzzy self organizing map we replace the neurons of the original model by fuzzy rules which are composed of fuzzy sets the fuzzy sets define an area in the input space where each fuzzy rule fires the output of each rule is a singleton the outputs are combined together by a weighted average where the firing strengths of the fuzzy rules act as the weights the weighted average gives a continuous valued output for the system thus the fuzzy self organizing map performs a mapping from a n dimensional input space to one dimensional output space the learning capability of the fuzzy self organizing map enables it to model a continuous valued function to an arbitrary accuracy the learning is done by first self organizing the centers of the fuzzy sets according to kohonen s self organizing map learning laws after that the fuzzy sets and the outputs of the fuzzy rules are initialized finally in the last phase of the new learning method the fuzzy sets are tuned by an algorithm similar to kohonen s learning vector quantization 2 1 simulation results of a two dimensional sinc function show good accuracy and fast convergence

low level interpretability and high level interpretability a unified view of data driven interpretable fuzzy system modelling this paper aims at providing an in depth overview of designing interpretable fuzzy inference models from data within a unified framework the objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real world systems whose first principle models are unknown because system behaviour can be described naturally as a series of linguistic rules data driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose however fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability model accuracy and interpretability are two conflicting objectives so interpretation preservation during adaptation in data driven fuzzy system modelling is a challenging task which has received much attention in fuzzy system modelling community in order to clearly discriminate the different roles of fuzzy sets input variables and other components in achieving an interpretable fuzzy model a taxonomy of fuzzy model interpretability is first proposed in terms of low level interpretability and high level interpretability in this paper the low level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level while the high level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage completeness and consistency of the rules in terms of the criteria on fuzzy rule level some criteria for low level interpretability and high level interpretability are identified respectively different data driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low level interpretability and high level interpretability furthermore some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested

neuro fuzzy rule generation survey in soft computing framework the present article is a novel attempt in providing an exhaustive survey of neuro fuzzy rule generation algorithms rule generation from artificial neural networks is gaining in popularity in recent times due to its capability of providing some insight to the user about the symbolic knowledge embedded within the network fuzzy sets are an aid in providing this information in a more human comprehensible or natural form and can handle uncertainties at various levels the neuro fuzzy approach symbiotically combining the merits of connectionist and fuzzy approaches constitutes a key component of soft computing at this stage to date there has been no detailed and integrated categorization of the various neuro fuzzy models used for rule generation we propose to bring these together under a unified soft computing framework moreover we include both rule extraction and rule refinement in the broader perspective of rule generation rules learned and generated for fuzzy reasoning and fuzzy control are also considered from this wider viewpoint models are grouped on the basis of their level of neuro fuzzy synthesis use of other soft computing tools like genetic algorithms and rough sets are emphasized rule generation from fuzzy knowledge based networks which initially encode some crude domain knowledge are found to result in more refined rules finally real life application to medical diagnosis is provided

the largest cartesian closed category of stable domains abstract berry s category of di domains with stable functions is a relatively intricate yet elegant framework for semantics of programming languages despite over 15 years of work in the area the exact reasons for distributivity axiom d and finitariness axiom i have not been fully explicated this paper shows that axiom d and axiom i are important when one works within the realm of scott domains in particular it has been shown that i if d s d has a countable basis then d must be finitary for any scott domain d ii if d s d is bounded complete then d must be distributive for any finitary scott domain d therefore the category of di domains is the largest cartesian closed category within omega algebraic bounded complete domains with the exponential being the stable function space

a neuro fuzzy method to learn fuzzy classification rules from data abstract neuro fuzzy systems have recently gained a lot of interest in research and application neuro fuzzy models as we understand them are fuzzy systems that use local learning strategies to learn fuzzy sets and fuzzy rules neuro fuzzy techniques have been developed to support the development of e g fuzzy controllers and fuzzy classifiers in this paper we discuss a learning method for fuzzy classification rules the learning algorithm is a simple heuristics that is able to derive fuzzy rules from a set of training data very quickly and tunes them by modifying parameters of membership functions our approach is based on nefclass a neuro fuzzy model for pattern classification we also discuss some results obtained by our software implementation of nefclass which is freely available on the internet

two categories of effective continuous cpos this paper presents two categories of effective continuous complete partial orders cpos we define a new criterion on the basis of a cpo so as to make the resulting category of consistently complete continuous cpos cartesian closed we also generalise to continuous cpos the definition of a complete set which was used as a definition of effective bifinite domains in hamrin and stoltenberg hansen cartesian closed categories of effective domains in h schwichtenberg r steinbruggen eds proof and system reliability kluwer academic publishers dordrecht 2002 pp 1 20 and investigate the closure results that can be obtained

neuro fuzzy systems for function approximation abstract we present a neuro fuzzy architecture for function approximation based on supervised learning the learning algorithm is able to determine the structure and the parameters of a fuzzy system the approach is an extension to our already published nefcon and nefclass models which are used for control or classification purposes the proposed extended model which we call nefprox is more general and can be used for any application based on function approximation

a hybrid learning algorithm for a class of interval type 2 fuzzy neural networks in real life information about the world is uncertain and imprecise the cause of this uncertainty is due to deficiencies on given information the fuzzy nature of our perception of events and objects and on the limitations of the models we use to explain the world the development of new methods for dealing with information with uncertainty is crucial for solving real life problems in this paper three interval type 2 fuzzy neural network it2fnn architectures are proposed with hybrid learning algorithm techniques gradient descent backpropagation and gradient descent with adaptive learning rate backpropagation at the antecedents layer a interval type 2 fuzzy neuron it2fn model is used and in case of the consequents layer an interval type 1 fuzzy neuron model it1fn in order to fuzzify the rule s antecedents and consequents of an interval type 2 takagi sugeno kang fuzzy inference system it2 tsk fis it2 tsk fis is integrated in an adaptive neural network in order to take advantage the best of both models this provides a high order intuitive mechanism for representing imperfect information by means of use of fuzzy if then rules in addition to handling uncertainty and imprecision on the other hand neural networks are highly adaptable with learning and generalization capabilities experimental results are divided in two kinds in the first one a non linear identification problem for control systems is simulated here a comparative analysis of learning architectures it2fnn and anfis is done for the second kind a non linear mackey glass chaotic time series prediction problem with uncertainty sources is studied finally it2fnn proved to be more efficient mechanism for modeling real world problems

fuzzyfeaturerank bringing order into fuzzy classifiers through fuzzy expressions abstract this work presents fuzzyfeaturerank a new feature reduction method inspired on pagerank to reduce the dimensionality of the feature space in supervised classification problems more precisely as it relies on a weighted directed graph it is ultimately inspired on textrank a pagerank based method that adds weights to the edges to express the strength of the connections between nodes the method is based on dividing each original feature used to describe the data into a set of fuzzy predicates and then ranking all of them by their ability to differentiate among classes in the light of the training set in order to do that both the information gained by each predicate and their redundancy with other already selected predicates are taken into account the fuzzy predicates with the best scores can then be used as a reduced input to construct fuzzy classifiers that consider only the preselected predicates to build the antecedents of the fuzzy rules the novelty of the proposal relies on being an approach halfway between feature selection and feature extraction approaches being able to improve the discrimination ability of the original features but preserving the interpretability of the new features in the sense that they are fuzzy expressions the experimental results support the suitability of the proposal

fuzzy neural networks a survey abstract in this paper a fuzzy neural network will be a layered feedforward neural net that has fuzzy signals and or fuzzy weights we survey recent results on learning algorithms and applications for fuzzy neural networks

a note on the limiting procedures for path integrals limiting procedures for the feynman type path integral are considered the evolution operator is approximated with operators corresponding to the exponential of the hamiltonian s symbol the proof of convergence uses chernoff s theory and its extension on the class of stable operators the approach is especially adequate for the generalized coherent states path integral where decomposition of a hamiltonian into the sums used in the approach based on trotter s formula may be unnatural

structure and learning algorithms of a nonsingleton input fuzzy neural network based on the approximate analogical reasoning schema abstract a novel fuzzy neural network the pseudo outer product based fuzzy neural network using the nonsingleton fuzzifier together with the approximate analogical reasoning schema nonsingleton fuzzifier popfnn aars is proposed in this paper the nonsingleton fuzzifier popfnn aars is developed as an improvement over the singleton fuzzifier popfnn aars developed by quek and zhou popfnn aars a pseudo outer product based fuzzy neural network ieee trans systems man cybernetics 29 6 1999 859 870 the employment of the approximate analogical reasoning schema aars as the fuzzy inference model enables the integration of a nonsingleton fuzzifier in the nonsingleton fuzzifier popfnn aars compared against the commonly used singleton fuzzifier the nonsingleton fuzzifier is more powerful in handling imprecise data different similarity measures and modification functions proposed by zwick et al measures of similarity between fuzzy concepts a comparative analysis internat j approx reason 1 1987 221 242 for aars in the nonsingleton fuzzifier aars are investigated the structure and learning algorithms of the proposed nonsingleton fuzzifier popfnn aars are presented several sets of real life data are used to test the performance of the nonsingleton fuzzifier popfnn aars and their experimental results are presented for detailed discussion