dissimilarity metric learning in the belief function framework the evidential k nearest neighbor ek nn method provided a global treatment of imperfect knowledge regarding the class membership of training patterns it has outperformed traditional k nn rules in many applications but still shares some of their basic limitations e g 1 classification accuracy depends heavily on how to quantify the dissimilarity between different patterns and 2 no guarantee for satisfactory performance when training patterns contain unreliable imprecise and or uncertain input features in this paper we propose to address these issues by learning a suitable metric using a low dimensional transformation of the input space so as to maximize both the accuracy and efficiency of the ek nn classification to this end a novel loss function to learn the dissimilarity metric is constructed it consists of two terms the first one quantifies the imprecision regarding the class membership of each training pattern while by means of feature selection the second one controls the influence of unreliable input features on the output linear transformation the proposed method has been compared with some other metric learning methods on several synthetic and real datasets it consistently led to comparable performance with regard to testing accuracy and class structure visualization

adaptive fuzzy rule based classification systems this paper proposes an adaptive method to construct a fuzzy rule based classification system with high performance for pattern classification problems the proposed method consists of two procedures an error correction based learning procedure and an additional learning procedure the error correction based learning procedure adjusts the grade of certainty of each fuzzy rule by its classification performance that is when a pattern is misclassified by a particular fuzzy rule the grade of certainty of that rule is decreased on the contrary when a pattern is correctly classified the grade of certainty is increased because the error correction based learning procedure is not meaningful after all the given patterns are correctly classified we cannot adjust a classification boundary in such a case to acquire a more intuitively acceptable boundary we propose an additional learning procedure we also propose a method for selecting significant fuzzy rules by pruning unnecessary fuzzy rules which consists of the error correction based learning procedure and the concept of forgetting we can construct a compact fuzzy rule based classification system with high performance

on james and jordan von neumann constants of lorentz sequence spaces the nonsquare or james constant j x and the jordan von neumann constant cnj x are computed for two dimensional lorentz sequence spaces d 2 w q in the case where 2 q the jordan von neumann constant is also calculated in the case where 1 q 2

a new belief based k nearest neighbor classification method the k nearest neighbor k nn classification method originally developed in the probabilistic framework has serious difficulties to classify correctly the close data points objects originating from different classes to cope with such difficult problem and make the classification result more robust to misclassification errors we propose a new belief based k nearest neighbor bk nn method that allows each object to belong both to the specific classes and to the sets of classes with different masses of belief bk nn is able to provide a hyper credal classification on the specific classes the rejection classes and the meta classes as well thus the objects hard to classify correctly are automatically committed to a meta class or to a rejection class which can reduce the misclassification errors the basic belief assignment bba of each object is defined from the distance between the object and its neighbors and from the acceptance and rejection thresholds the bba s are combined using a new combination method specially developed for the bk nn several experiments based on simulated and real data sets have been carried out to evaluate the performances of the bk nn method with respect to several classical k nn approaches

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

interpretability of linguistic fuzzy rule based systems an overview of interpretability measures linguistic fuzzy modelling developed by linguistic fuzzy rule based systems allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings linguistic fuzzy modelling comes with two contradictory requirements interpretability and accuracy in recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown whereas the measures of accuracy are straightforward and well known interpretability measures are difficult to define since interpretability depends on several factors mainly the model structure the number of rules the number of features the number of linguistic terms the shape of the fuzzy sets etc moreover due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem in this paper we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule based systems to this end we will propose a taxonomy based on a double axis complexity versus semantic interpretability considering the two main kinds of measures and rule base versus fuzzy partitions considering the different components of the knowledge base to which both kinds of measures can be applied the main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works

credal classification rule for uncertain data based on belief functions abstract in this paper we present a new credal classification rule ccr based on belief functions to deal with the uncertain data ccr allows the objects to belong with different masses of belief not only to the specific classes but also to the sets of classes called meta classes which correspond to the disjunction of several specific classes each specific class is characterized by a class center i e prototype and consists of all the objects that are sufficiently close to the center the belief of the assignment of a given object to classify with a specific class is determined from the mahalanobis distance between the object and the center of the corresponding class the meta classes are used to capture the imprecision in the classification of the objects when they are difficult to correctly classify because of the poor quality of available attributes the selection of meta classes depends on the application and the context and a measure of the degree of indistinguishability between classes is introduced in this new ccr approach the objects assigned to a meta class should be close to the center of this meta class having similar distances to all the involved specific classes centers and the objects too far from the others will be considered as outliers noise ccr provides robust credal classification results with a relatively low computational burden several experiments using both artificial and real data sets are presented at the end of this paper to evaluate and compare the performances of this ccr method with respect to other classification methods

a note on jordan von neumann constant and james constant abstract let x be a non trivial banach space l maligranda conjectured c nj x 1 j x 2 4 for james constant j x and von neumann jordan constant c nj x of x satit saejung gave a proof of it in 2006 in this note we show that the last step in satit saejung s proof is not valid using his proof the result should be c nj x 1 j x 2 4 j x 4 2 j x 2 2 4 on the other hand we give a new proof of c nj x 1 j x 2 4 as an application we give a relation between j x and j l p x

multi attribute search framework for optimizing extended belief rule based systems the advantages and applications of rule based systems have caused them to be widely recognized as one of the most popular systems in human decision making due to their accuracy and efficiency to improve the performance of rule based systems there are several issues proposed to be focused first it is unnecessary to take the entire rule base into consideration during each decision making process second there is no need to visit the entire rule base to search for the key rules last the key rules for each decision making process should be different this paper focuses on an advanced extended belief rule base ebrb system and proposes a multi attribute search framework masf to reconstruct the relationship between rules in the ebrb to form the masf based ebrb masfs can be divided into k dimensional tree kdt based masfs and burkhard keller bkt based masfs the former is targeted at decision making problems with small scale attribute datasets while the latter is for those with large scale attribute datasets based on the masf based ebrb the k neighbor search and the best activated rule set algorithms are further proposed to find both the unique and the desired rules for each decision making process without visiting the entire ebrb especially when handling classification problems with large attribute datasets two sets of experiments based on benchmark datasets with different numbers of attributes are performed to analyze the difference between kdt based masfs and bkt based masfs and to demonstrate how to use masfs to improve the accuracy and efficiency of ebrb systems masfs and their corresponding algorithms are also regarded as a general optimization framework that can be used with other rule based systems

segmentation of m fish images for improved classification of chromosomes with an adaptive fuzzy c means clustering algorithm an adaptive fuzzy c means algorithm was developed and applied to the segmentation and classification of multicolor fluorescence in situ hybridization m fish images which can be used to detect chromosomal abnormalities for cancer and genetic disease diagnosis the algorithm improves the classical fuzzy c means algorithm fcm by the use of a gain field which models and corrects intensity inhomogeneities caused by a microscope imaging system flairs of targets chromosomes and uneven hybridization of dna other than directly simulating the inhomogeneousely distributed intensities over the image the gain field regulates centers of each intensity cluster the algorithm has been tested on an m fish database that we have established which demonstrates improved performance in both segmentation and classification when compared with other fcm clustering based algorithms and a recently reported region based segmentation and classification algorithm our method gave the lowest segmentation and classification error which will contribute to improved diagnosis of genetic diseases and cancers

fuzzy modeling of high dimensional systems complexity reduction and interpretability improvement fuzzy modeling of high dimensional systems is a challenging topic this paper proposes an effective approach to data based fuzzy modeling of high dimensional systems an initial fuzzy rule system is generated based on the conclusion that optimal fuzzy rules cover extrema redundant rules are removed based on a fuzzy similarity measure then the structure and parameters of the fuzzy system are optimized using a genetic algorithm and the gradient method during optimization rules that have a very low firing strength are deleted finally interpretability of the fuzzy system is improved by fine training the fuzzy rules with regularization the resulting fuzzy system generated by this method has the following distinct features 1 the fuzzy system is quite simplified 2 the fuzzy system is interpretable and 3 the dependencies between the inputs and the output are clearly shown this method has successfully been applied to a system that has 11 inputs and one output with 20 000 training data and 80 000 test data

selecting fuzzy if then rules for classification problems using genetic algorithms this paper proposes a genetic algorithm based method for selecting a small number of significant fuzzy if then rules to construct a compact fuzzy classification system with high classification power the rule selection problem is formulated as a combinatorial optimization problem with two objectives to maximize the number of correctly classified patterns and to minimize the number of fuzzy if then rules genetic algorithms are applied to this problem a set of fuzzy if then rules is coded into a string and treated as an individual in genetic algorithms the fitness of each individual is specified by the two objectives in the combinatorial optimization problem the performance of the proposed method for training data and test data is examined by computer simulations on the iris data of fisher

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

a dynamically constrained multiobjective genetic fuzzy system for regression problems in this paper a multiobjective genetic fuzzy system gfs to learn the granularities of fuzzy partitions tuning the membership functions mfs and learning the fuzzy rules is presented it uses dynamic constraints which enable three parameter mf tuning to improve the accuracy while guaranteeing the transparency of fuzzy partitions the fuzzy models fms are initialized by a method that combines the benefits of wang mendel wm and decision tree algorithms thus the initial fms have less rules rule conditions and input variables than if wm initialization were to be used moreover the fuzzy partitions of initial fms are always transparent our approach is tested against recent multiobjective and monoobjective gfss on six benchmark problems it is concluded that the accuracy and interpretability of our fms are always comparable or better than those in the comparative studies furthermore on some benchmark problems our approach clearly outperforms some comparative approaches suitability of our approach for higher dimensional problems is shown by studying three benchmark problems that have up to 21 input variables

integration of an index to preserve the semantic interpretability in the multiobjective evolutionary rule selection and tuning of linguistic fuzzy systems in this paper we propose an index that helps preserve the semantic interpretability of linguistic fuzzy models while a tuning of the membership functions mfs is performed the proposed index is the aggregation of three metrics that preserve the original meanings of the mfs as much as possible while a tuning of their definition parameters is performed additionally rule selection mechanisms can be used to reduce the model complexity which involves another important interpretability aspect to this end we propose a postprocessing multiobjective evolutionary algorithm that performs rule selection and tuning of fuzzy rule based systems with three objectives accuracy semantic interpretability maximization and complexity minimization we tested our approach on nine real world regression datasets in order to analyze the interaction between the fuzzy rule selection approach and the tuning approach these are also individually proved in a multiobjective framework and compared with their respective single objective counterparts we compared the different approaches by applying nonparametric statistical tests for pairwise and multiple comparisons taking into consideration three representative points from the obtained pareto fronts in the case of the multiobjective based approaches results confirm the effectiveness of our approach and a wide range of solutions is obtained which are not only more interpretable but are also more accurate

uniqueness and value sharing of meromorphic functions in this paper we study the uniqueness of meromorphic functions and prove the following theorem let f z and g z be two non constant meromorphic functions n k two positive integers with n 3k 8 if f n z k and g n z k share 1 cm then either f z c 1e c z g z c 2e c z where c 1 c 2 and c are three constants satisfying 1 k c 1c 2 n nc 2 k 1 or f z tg z for a constant t such that t n 1 our results improves the results of fang m l fang uniqueness and value sharing of entire functions comput math appl 44 2002 823 831 7 fang and hong m l fang w hong a unicity theorem for entire functions concerning differential polynomials indian j pure appl math 32 9 2001 1343 1348 8 and lin and yi w c lin h x yi uniqueness theorems for meromorphic function indian j pure appl math 32 9 2004 121 132 9

the combination of evidence in the transferable belief model a description of the transferable belief model which is used to quantify degrees of belief based on belief functions is given the impact of open and closed world assumption on conditioning is discussed the nature of the frame of discernment on which a degree of belief will be established is discussed a set of axioms justifying dempster s rule for the combination of belief functions induced by two distinct evidences is presented

constructing fuzzy models with linguistic integrity from numerical data afreli algorithm this paper presents an algorithm to extract rules re lating input output data and including prior knowledge the rules are created in the environment of fuzzy systems the fuzzy sets describing the system are constructed within a framework of lin guistic integrity to guarantee its interpretability in the linguistic context two algorithms are presented in this paper the main al gorithm is the autonomous fuzzy rule extractor with linguistic in tegrity afreli this algorithm is complemented with the use of the fuzion algorithm created to merge consecutive member ship functions while guaranteeing the distinguish ability between fuzzy sets comparisons with other proposed methods show a good tradeoff between accuracy and interpretability

a cloned linguistic decision tree controller for real time path planning in hostile environments abstract the idea of a cloned controller to approximate optimised control algorithms in a real time environment is introduced a cloned controller is demonstrated using linguistic decision trees ldts to clone a model predictive controller mpc based on mixed integer linear programming milp for unmanned aerial vehicle uav path planning through a hostile environment modifications to the ldt algorithm are proposed to account for attributes with circular domains such as bearings and discontinuous output functions the cloned controller is shown to produce near optimal paths whilst significantly reducing the decision period further investigation shows that the cloned controller generalises to the multi obstacle case although this can lead to situations far outside of the training dataset and consequently result in decisions with a high level of uncertainty a modification to the algorithm to improve the performance in regions of high uncertainty is proposed and shown to further enhance generalisation the resulting controller combines the high performance of mpc milp with the rapid response of an ldt while providing a degree of transparency interpretability of the decision making

designing fuzzy inference systems from data an interpretability oriented review fuzzy inference systems fis are widely used for process simulation or control they can be designed either from expert knowledge or from data for complex systems fis based on expert knowledge only may suffer from a loss of accuracy this is the main incentive for using fuzzy rules inferred from data designing a fis from data can be decomposed into two main phases automatic rule generation and system optimization rule generation leads to a basic system with a given space partitioning and the corresponding set of rules system optimization can be done at various levels variable selection can be an overall selection or it can be managed rule by rule rule base optimization aims to select the most useful rules and to optimize rule conclusions space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters structure optimization is of a major importance selecting variables reducing the rule base and optimizing the number of fuzzy sets over the years many methods have become available for designing fis from data their efficiency is usually characterized by a numerical performance index however for human computer cooperation another criterion is needed the rule interpretability an implicit assumption states that fuzzy rules are by nature easy to be interpreted this could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts the paper analyzes the main methods for automatic rule generation and structure optimization they are grouped into several families and compared according to the rule interpretability criterion for this purpose three conditions for a set of rules to be interpretable are defined

representing uncertainty on set valued variables using belief functions a formalism is proposed for representing uncertain information on set valued variables using the formalism of belief functions a set valued variable x on a domain w is a variable taking zero one or several values in w while defining mass functions on the frame 2 2 w is usually not feasible because of the double exponential complexity involved we propose an approach based on a definition of a restricted family of subsets of 2 w that is closed under intersection and has a lattice structure using recent results about belief functions on lattices we show that most notions from dempster shafer theory can be transposed to that particular lattice making it possible to express rich knowledge about x with only limited additional complexity as compared to the single valued case an application to multi label classification in which each learning instance can belong to several classes simultaneously is demonstrated

towards neuro linguistic modeling constraints for optimization of membership functions abstract in neuro fuzzy or fuzzy neural systems using unconstrained optimization schemes like backpropagation it is not possible to guarantee that the resulting membership functions represent human interpretable linguistic terms however one of the most interesting features of fuzzy systems is the insight provided on the linguistic relationship between their variables this work is devoted to the study of constraints which when used within an optimization scheme obviate the subjective task of interpreting membership functions to achieve this a comprehensive set of semantic properties that membership functions should have is postulated and discussed then a set of constraints is introduced and shown to be able to fulfil the properties implementation issues and one example illustrating the importance of the proposed constraints are included

a proposal for the genetic lateral tuning of linguistic fuzzy systems and its interaction with rule selection linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings however since the accuracy and the interpretability of the obtained model are contradictory properties the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled to solve this problem one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level in this paper a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability to do so we consider a new rule representation scheme base on the linguistic 2 tuples representation model which allows the lateral variation of the involved labels furthermore the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper presenting results on different real applications the obtained results show the good performance of the proposed approach in high dimensional problems and its ability to cooperate with methods to remove unnecessary rules

distributed representation of fuzzy rules and its application to pattern classification abstract this paper introduces the concept of distributed representation of fuzzy rules and applies it to classification problems distributed representation is implemented by superimposing many fuzzy rules corresponding to different fuzzy partitions of a pattern space this means that we simulatenously employ many fuzzy rule tables corresponding to different fuzzy partitions in fuzzy inference in order to apply distributed representation of fuzzy rules to pattern classification problems we first propose an algorithm to generate fuzzy rules from numerical data next we propose a fuzzy inference method using the generated fuzzy rules the classification power of distributed representation is compared with that of ordinary fuzzy rules which can be viewed as local representation

an extended rule based inference for general decision making problems complex decision making problems are usually modeled in terms of a number of decisive variables that are related hierarchically pieces of evidence are aggregated in a bottom up way to determine the final decision goal for dealing with such problems ishizuka et al 5 6 developed a rule based inference procedure by using fuzzy sets to represent imprecise data and employed the dempster shafer theory to aggregate evidence for structural damage assessment this paper highlights the necessity of applying the rules with all possible consequent clauses in the procedure with the considerations of information quality and quantity when applying the procedure to more general decision making problems the use of such rules can provide high quality information which contains much less ignorance in the resultant information also it can afford more information due to the decrease in total ignorance a simple example with two cases is used to compare the results based on the two different presentations of rules in the inference procedure for illustrating the importance of considering the rules proposed in this paper

on a new geometric constant related to the von neumann jordan constant abstract the von neumann jordan constant c nj x is computed for x being l 2 l 1 and l l 1 space by introducing a new geometric constant x t these partly give an answer to an open question posed by kato et al some basic properties of this new coefficient are investigated moreover we obtain a new class of banach spaces with uniform normal structure

uniqueness and value sharing of entire functions abstract in this paper we study the uniqueness of entire functions and prove the following theorem let f z and g z be two nonconstant entire functions n k two positive integers with n 2 k 4 if f n z k and g n z k share 1 with counting the multiplicity then either f z c 1 e cz g z c 2 e cz where c 1 c 2 and c are three constants satisfying 1 k c 1 c 2 n nc 2 k 1 or f z tg z for a constant t such that t n 1

support vector learning for fuzzy rule based classification systems to design a fuzzy rule based classification system fuzzy classifier with good generalization ability in a high dimensional feature space has been an active research topic for a long time as a powerful machine learning approach for pattern recognition problems the support vector machine svm is known to have good generalization ability more importantly an svm can work very well on a high or even infinite dimensional feature space this paper investigates the connection between fuzzy classifiers and kernel machines establishes a link between fuzzy rules and kernels and proposes a learning algorithm for fuzzy classifiers we first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function fuzzy inference on the if part of a fuzzy rule can be viewed as evaluating the kernel function the kernel function is then proven to be a mercer kernel if the reference functions meet a certain spectral requirement the corresponding fuzzy classifier is named positive definite fuzzy classifier pdfc a pdfc can be built from the given training samples based on a support vector learning approach with the if part fuzzy rules given by the support vectors since the learning process minimizes an upper bound on the expected risk expected prediction error instead of the empirical risk training error the resulting pdfc usually has good generalization moreover because of the sparsity properties of the svms the number of fuzzy rules is irrelevant to the dimension of input space in this sense we avoid the curse of dimensionality finally pdfcs with different reference functions are constructed using the support vector learning approach the performance of the pdfcs is illustrated by extensive experimental results comparisons with other methods are also provided

on james and von neumann jordan constants and sufficient conditions for the fixed point property abstract in this paper we prove that a banach space x and its dual space x have uniform normal structure if c nj x 1 3 2 the garcia falset coefficient r x is estimated by the c nj x constant and the weak orthogonality coefficient introduced by b sims finally we present an affirmative answer to a conjecture by l maligranda concerning the relation between the james and c nj x constants for a banach space

interpretability issues in data based learning of fuzzy systems this paper presents a method for an automatic and complete design of fuzzy systems from data the main objective is to build fuzzy systems with a user controllable trade off between accuracy and interpretability whereas criteria for accuracy mostly follow straightforwardly from the application definition of interpretability and its criteria are subject to controversial discussion for this reason a set of interpretability criteria is given which guide the design process consequently interpretability is maintained by structural choices regarding the type of membership functions rules and inference mechanism on the one hand and by including interpretability criteria in the rule rule base evaluation on the other hand an application in instrumented gait analysis to characterize a certain group of patients in comparison to healthy subjects illustrates the proposed algorithm

ecmdd evidential c medoids clustering with multiple prototypes in this work a new prototype based clustering method named evidential c medoids ecmdd which belongs to the family of medoid based clustering for proximity data is proposed as an extension of fuzzy c medoids fcmdd on the theoretical framework of belief functions in the application of fcmdd and original ecmdd a single medoid prototype which is supposed to belong to the object set is utilized to represent one class for the sake of clarity this kind of ecmdd using a single medoid is denoted by secmdd in real clustering applications using only one pattern to capture or interpret a class may not adequately model different types of group structure and hence limits the clustering performance in order to address this problem a variation of ecmdd using multiple weighted medoids denoted by wecmdd is presented unlike secmdd in wecmdd objects in each cluster carry various weights describing their degree of representativeness for that class this mechanism enables each class to be represented by more than one object experimental results in synthetic and real data sets clearly demonstrate the superiority of secmdd and wecmdd moreover the clustering results by wecmdd can provide richer information for the inner structure of the detected classes with the help of prototype weights

adaptive imputation of missing values for incomplete pattern classification in classification of incomplete pattern the missing values can either play a crucial role in the class determination or have only little influence or eventually none on the classification results according to the context we propose a credal classification method for incomplete pattern with adaptive imputation of missing values based on belief function theory at first we try to classify the object incomplete pattern based only on the available attribute values as underlying principle we assume that the missing information is not crucial for the classification if a specific class for the object can be found using only the available information in this case the object is committed to this particular class however if the object cannot be classified without ambiguity it means that the missing values play a main role for achieving an accurate classification in this case the missing values will be imputed based on the k nearest neighbor k nn and self organizing map som techniques and the edited pattern with the imputation is then classified the original or edited pattern is classified according to each training class and the classification results represented by basic belief assignments are fused with proper combination rules for making the credal classification the object is allowed to belong with different masses of belief to the specific classes and meta classes which are particular disjunctions of several single classes the credal classification captures well the uncertainty and imprecision of classification and reduces effectively the rate of misclassifications thanks to the introduction of meta classes the effectiveness of the proposed method with respect to other classical methods is demonstrated based on several experiments using artificial and real data sets highlightsmissing values are adaptively imputed in classification according to context som and k nn are used for the imputation with admissible computation burden ensemble classifier is introduced for credal classification the imprecision of classification can be well captured using belief functions the proposed method has been tested by artificial and real data sets

uniqueness of meromorphic functions sharing one value in this paper we study the uniqueness of meromorphic resp entire functions sharing one value and obtain some results which improve and generalize the related results due to fang bhoosnurmath dyavanal yang hua fang hua bai han etc

uniqueness of meromorphic functions sharing a small function and its applications in this paper we shall study the uniqueness problems of meromorphic functions sharing a small function our results improve or generalize many previous results on value sharing of meromorphic functions

fuzzy modeling using genetic algorithms with fuzzy entropy as conciseness measure abstract in this paper a fuzzy modeling method using genetic algorithms gas with a conciseness measure is presented this paper introduces de luca and termini s fuzzy entropy to evaluate the shape of a membership function and proposes another measure to evaluate the deviation of a membership function from symmetry a combined measure is then derived from these two measures and a new conciseness measure is defined for evaluation of the shape and allocation of the membership functions of a fuzzy model numerical results show that the new conciseness measure is effective for fuzzy modeling formulated as a multi objective optimization problem

genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction tuning fuzzy rule based systems for linguistic fuzzy modeling is an interesting and widely developed task it involves adjusting some of the components of the knowledge base without completely redefining it this contribution introduces a genetic tuning process for jointly fitting the fuzzy rule symbolic representations and the meaning of the involved membership functions to adjust the former component we propose the use of linguistic hedges to perform slight modifications keeping a good interpretability to alter the latter component two different approaches changing their basic parameters and using nonlinear scaling factors are proposed as the accomplished experimental study shows the good performance of our proposal mainly lies in the consideration of this tuning approach performed at two different levels of significance the paper also analyzes the interaction of the proposed tuning method with a fuzzy rule set reduction process a good interpretability accuracy tradeoff is obtained combining both processes with a sequential scheme first reducing the rule set and subsequently tuning the model

distinguishability quantification of fuzzy sets distinguishability is a semantic property of fuzzy sets that has a great relevance in the design of interpretable fuzzy models distinguishability has been mathematically defined through different measures which are addressed in this paper special emphasis is given to similarity which exhibits sound theoretical properties but its calculation is usually computationally intensive and possibility whose calculation can be very efficient but it does not exhibit the same properties of similarity it is shown that under mild conditions usually met in interpretable fuzzy modeling possibility can be used as a valid measure for assessing distinguishability thus overcoming the computational inefficiencies of similarity measures moreover procedures that minimize possibility also minimize similarity and consequently improve distinguishability in this sense the use of possibility is fully justified in interpretable fuzzy modeling

conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence dempster s rule plays a central role in the theory of belief functions however it assumes the combined bodies of evidence to be distinct an assumption which is not always verified in practice in this paper a new operator the cautious rule of combination is introduced this operator is commutative associative and idempotent this latter property makes it suitable to combine belief functions induced by reliable but possibly overlapping bodies of evidence a dual operator the bold disjunctive rule is also introduced this operator is also commutative associative and idempotent and can be used to combine belief functions issues from possibly overlapping and unreliable sources finally the cautious and bold rules are shown to be particular members of infinite families of conjunctive and disjunctive combination rules based on triangular norms and conorms

a novel generic hebbian ordering based fuzzy rule base reduction approach to mamdani neuro fuzzy system there are two important issues in neuro fuzzy modeling 1 interpretability the ability to describe the behavior of the system in an interpretable way and 2 accuracy the ability to approximate the outcome of the system accurately as these two objectives usually exert contradictory requirements on the neuro fuzzy model certain compromise has to be undertaken this letter proposes a novel rule reduction algorithm namely hebb rule reduction and an iterative tuning process to balance interpretability and accuracy the hebb rule reduction algorithm uses hebbian ordering which represents the degree of coverage of the samples by the rule as an importance measure of each rule to merge the membership functions and hence reduces the number of the rules similar membership functions mfs are merged by a specified similarity measure in an order of hebbian importance and the resultant equivalent rules are deleted from the rule base the rule with a higher hebbian importance will be retained among a set of rules the mfs are tuned through the least mean square lms algorithm to reduce the modeling error the tuning of the mfs and the reduction of the rules proceed iteratively to achieve a balance between interpretability and accuracy three published data sets by nakanishi nakanishi turksen sugeno 1993 the pat synthetic data set pal mitra mitra 2003 and the traffic flow density prediction data set are used as benchmarks to demonstrate the effectiveness of the proposed method good interpretability as well as high modeling accuracy are derivable simultaneously and are suitably benchmarked against other well established neuro fuzzy models

fuzzy support vector machines for multilabel classification the problem of one against all support vector machines svms for multilabel classification is that a data sample may be classified into a multilabel class that is not defined or it may not be classified into any class to solve this problem in this paper we propose fuzzy svms fsvms for multilabel classification in which for each multilabel class a region with the associated membership function is defined and a data point is classified into a multilabel class whose membership function is the largest by computer experiments we show that the accuracy is improved by the fsvm over the conventional one against all svm highlightswe develop a fuzzy support vector machine fsvm for multilabel classification we resolve unclassifiable regions by membership functions we also resolve undefined multilabels by membership functions effectiveness of the proposed fsvm is demonstrated by computer experiments

the transferable belief model abstract we describe the transferable belief model a model for representing quantified beliefs based on belief functions beliefs can be held at two levels 1 a credal level where beliefs are entertained and quantified by belief functions 2 a pignistic level where beliefs can be used to make decisions and are quantified by probability functions the relation between the belief function and the probability function when decisions must be made is derived and justified four paradigms are analyzed in order to compare bayesian upper and lower probability and the transferable belief approaches

a multiobjective evolutionary approach to concurrently learn rule and data bases of linguistic fuzzy rule based systems in this paper we propose the use of a multiobjective evolutionary approach to generate a set of linguistic fuzzy rule based systems with different tradeoffs between accuracy and interpretability in regression problems accuracy and interpretability are measured in terms of approximation error and rule base rb complexity respectively the proposed approach is based on concurrently learning rbs and parameters of the membership functions of the associated linguistic labels to manage the size of the search space we have integrated the linguistic two tuple representation model which allows the symbolic translation of a label by only considering one parameter with an efficient modification of the well known 2 2 pareto archived evolution strategy paes we tested our approach on nine real world datasets of different sizes and with different numbers of variables besides the 2 2 paes we have also used the well known nondominated sorting genetic algorithm nsga ii and an accuracy driven single objective evolutionary algorithm ea we employed these optimization techniques both to concurrently learn rules and parameters and to learn only rules we compared the different approaches by applying a nonparametric statistical test for pairwise comparisons thus taking into consideration three representative points from the obtained pareto fronts in the case of the multiobjective eas finally a data complexity measure which is typically used in pattern recognition to evaluate the data density in terms of average number of patterns per variable has been introduced to characterize regression problems results confirm the effectiveness of our approach particularly for possibly high dimensional datasets with high values of the complexity metric

ecm an evidential version of the fuzzy c means algorithm a new clustering method for object data called ecm evidential c means is introduced in the theoretical framework of belief functions it is based on the concept of credal partition extending those of hard fuzzy and possibilistic ones to derive such a structure a suitable objective function is minimized using an fcm like algorithm a validity index allowing the determination of the proper number of clusters is also proposed experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics

approximate reasoning on a basis of z number valued if then rules approximate reasoning is about reasoning with imperfect information nowadays a large diversity of approaches to approximate reasoning with fuzzy information and fuzzy type 2 information exists it should be stressed however that real world imperfect information is characterized by combination of fuzzy and probabilistic uncertainties which is referred to as bimodal information in view of this zadeh introduced the concept of a z number regarded as an ordered pair z a b of fuzzy numbers a and b where a is a linguistic value of a variable of interest and b is a linguistic value of probability measure of a playing a role of its reliability unfortunately up to day there is no research on approximate reasoning realized on the basis of if then rules with z number valued antecedents and consequents briefly z rules zadeh addressed this problem as related to an uncharted territory in this paper a new approach is developed to study approximate reasoning with z rules on a basis of linear interpolation we provide an application of the approach to job satisfaction evaluation and to students educational achievement evaluation problems related to psychological and perceptual issues naturally characterized by imperfect information the obtained results show applicability and validity of the proposed approach

editorial special issue on interpretable fuzzy systems interpretability is acknowledged as one of the most appreciated advantages of fuzzy systems in many applications especially in those with high human interaction where it actually becomes a strong requirement however it is important to remark that there is a somehow misleading but widely extended belief even in part of the fuzzy community regarding fuzzy systems as interpretable no matter how they were designed of course we are aware the use of fuzzy logic favors the interpretability of designed models thanks to their semantic expressivity close to natural language fuzzy variables and rules can be used to formalize linguistic propositions which are likely to be easily understandood by human beings obviously this fact makes easier the knowledge extraction and representation tasks carried out when modeling real world complex systems notwithstanding fuzzy logic is not enough by itself to guarantee the interpretability of the final model as it is thoroughly illustrated in this special issue achieving interpretable fuzzy systems is a matter of careful design because fuzzy systems cannot be deemed as interpretable per se thus several constraints have to be imposed along the whole design process with the aim of producing really interpretable fuzzy systems in the sense that every element of the whole system may be checked and understood by a human being otherwise fuzzy systems may even become black boxes

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

fingrams visual representations of fuzzy rule based inference for expert analysis of comprehensibility since zadeh s proposal and mamdani s seminal ideas interpretability is acknowledged as one of the most appreciated and valuable characteristics of fuzzy system identification methodologies it represents the ability of fuzzy systems to formalize the behavior of a real system in a human understandable way by means of a set of linguistic variables and rules with a high semantic expressivity close to natural language interpretability analysis involves two main points of view readability of the knowledge base description regarding complexity of fuzzy partitions and rules and comprehensibility of the fuzzy system regarding implicit and explicit semantics embedded in fuzzy partitions and rules as well as the fuzzy reasoning method readability has been thoroughly treated by many authors who have proposed several criteria and metrics unfortunately comprehensibility has usually been neglected because it involves some cognitive aspects related to human reasoning which are very hard to formalize and to deal with this paper proposes the creation of a new paradigm for fuzzy system comprehensibility analysis based on fuzzy systems inference maps so called fuzzy inference grams fingrams by analogy with scientograms used for visualizing the structure of science fingrams show graphically the interaction between rules at the inference level in terms of co fired rules i e rules fired at the same time by a given input the analysis of fingrams offers many possibilities measuring the comprehensibility of fuzzy systems detecting redundancies and or inconsistencies among fuzzy rules identifying the most significant rules etc some of these capabilities are explored in this study for the case of fuzzy models and classifiers

an evidential classifier based on feature selection and two step classification strategy in this paper we investigate ways to learn efficiently from uncertain data using belief functions in order to extract more knowledge from imperfect and insufficient information and to improve classification accuracy we propose a supervised learning method composed of a feature selection procedure and a two step classification strategy using training information the proposed feature selection procedure automatically determines the most informative feature subset by minimizing an objective function the proposed two step classification strategy further improves the decision making accuracy by using complementary information obtained during the classification process the performance of the proposed method was evaluated on various synthetic and real datasets a comparison with other classification methods is also presented highlightsa classifier is based on belief functions to tackle uncertain data the classifier composed by feature selection and a two step classification a new combination rule to better represent data uncertainty a new feature selection is based on minimizing uncertainty with sparse constraint two step classification improving accuracy of decision making

three objective genetics based machine learning for linguisitc rule extraction abstract this paper shows how a small number of linguistically interpretable fuzzy rules can be extracted from numerical data for high dimensional pattern classification problems one difficulty in the handling of high dimensional problems by fuzzy rule based systems is the exponential increase in the number of fuzzy rules with the number of input variables another difficulty is the deterioration in the comprehensibility of fuzzy rules when they involve many antecedent conditions our task is to design comprehensible fuzzy rule based systems with high classification ability this task is formulated as a combinatorial optimization problem with three objectives to maximize the number of correctly classified training patterns to minimize the number of fuzzy rules and to minimize the total number of antecedent conditions we show two genetic algorithm based approaches one is rule selection where a small number of linguistically interpretable fuzzy rules are selected from a large number of prespecified candidate rules the other is fuzzy genetics based machine learning where rule sets are evolved by genetic operations these two approaches search for non dominated rule sets with respect to the three objectives

a genetic learning process for the scaling factors granularity and contexts of the fuzzy rule based system data base abstract in this contribution we propose a new method to automatically learn the knowledge base of a fuzzy rule based system by finding an appropriate data base using a genetic algorithm and considering a simple generation method to derive the rule base our genetic process learns all the components of the data base number of labels working ranges and membership function shapes for each linguistic variable using a non linear scaling function to adapt the fuzzy partition contexts

belief rule based classification system among the computational intelligence techniques employed to solve classification problems the fuzzy rule based classification system frbcs is a popular tool capable of building a linguistic model interpretable to users however it may face lack of accuracy in some complex applications by the fact that the inflexibility of the concept of the linguistic variable imposes hard restrictions on the fuzzy rule structure in this paper we extend the fuzzy rule in frbcs with a belief rule structure and develop a belief rule based classification system brbcs to address imprecise or incomplete information in complex classification problems the two components of the proposed brbcs i e the belief rule base brb and the belief reasoning method brm are designed specifically by taking into account the pattern noise that existes in many real world data sets four experiments based on benchmark data sets are carried out to evaluate the classification accuracy robustness interpretability and time complexity of the proposed method

canonical decomposition of belief functions based on teugels representation of the multivariate bernoulli distribution abstract a canonical decomposition of belief functions is a unique decomposition of belief functions into elementary pieces of evidence smets found an equivalent representation of belief functions which he interpreted as a canonical decomposition however his proposal is not entirely satisfactory as it involves elementary pieces of evidence corresponding to a generalisation of belief function axioms whose semantics lacks formal justifications in this paper a new canonical decomposition relying only on well defined concepts is proposed in particular it is based on a means to induce belief functions from the multivariate bernoulli distribution and on teugels representation of this distribution which consists of the means and the central moments of the underlying bernoulli random variables according to our decomposition a belief function results from as many crisp pieces of information as there are elements in its domain and from simple probabilistic knowledge concerning their marginal reliability and the dependencies between their reliability in addition we show that instead of interpreting with some difficulty smets representation of belief functions as a canonical decomposition it is possible to give it a different and well defined semantics in terms of measures of information associated with the reliability of the pieces of information in our decomposition