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

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

the first sign change of fourier coefficients of cusp forms let f sum no a n q n a n real be an arbitrary nonzero cusp form of even integral weight k geq 2 on gamma 0 n square free n we obtain an effective bound that the first sign change of a n occurs

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

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

fourier coefficients of cusp forms of half integral weight let mathfrak f be a cusp form of half integral weight whose fourier coefficients mathfrak a mathfrak f n are all real we study the sign change problem of mathfrak a mathfrak f n when n runs over some specific sets of integers lower bounds of the best possible order of magnitude are established for the number of those coefficients that have the same signs these give an improvement on some recent results of bruinier and kohnen modular forms on schiermonnikoong cambridge university press cambridge 57 66 2008 and kohnen int j number theory 6 1255 1259 2010

acyclic automata and small expressions using multi tilde bar operators a regular expression with n occurrences of symbol can be converted into an equivalent automaton with n 1 states the so called glushkov automaton of the expression conversely it is possible to decide whether a given n 1 state automaton is a glushkov one and if so to convert it back to an equivalent regular expression of width n our goal is to extend the class of automata for which such a linear retranslation is possible we define new regular operators called multi tilde bars allowing us to simultaneously apply a multi tilde operator and a multi bar one to a list of expressions the main results are that a multi tilde bar expression of width n can be converted into an n 1 state position like automaton and that any acyclic n state automaton can be turned into an extended expression of width o n

applicability of the fuzzy operators in the design of fuzzy logic controllers a study of the different roles played by the fuzzy operators in fuzzy control is developed in this paper the behavior of a very large amount of fuzzy operators is analyzed and a comparison of the accuracy of many fuzzy logic controllers designed by means of these operators is carried out in order to do that a comparison methodology is defined and two fuzzy control applications are selected the inverted pendulum problem and the fuzzy modeling of the real curve y x

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

periodicity of signs of fourier coefficients of eta quotients we study the periodicity of signs of fourier coefficients of the function d f qd rd where is a positive integer f q n 1 1 qn and rd z as an application we will prove the periodicity of signs for the crank differences conjectured by g e andrews and r lewis and for the weighted counts of certain types of partitions

design of fuzzy rule based classifiers with semantic cointension in computing with words cww knowledge is linguistically represented and has an explicit semantics defined through fuzzy information granules the linguistic representation in turn naturally bears an implicit semantics that belongs to users reading the knowledge base hence a necessary condition for achieving interpretability requires that implicit and explicit semantics are cointensive interpretability is definitely stringent when knowledge must be acquired from data through inductive learning therefore in this paper we propose a methodology for designing interpretable fuzzy models through semantic cointension we focus our analysis on fuzzy rule based classifiers frbcs where we observe that rules resemble logical propositions thus semantic cointension can be partially regarded as the fulfillment of the logical view i e the set of basic logical laws that are required in any logical system the proposed approach is grounded on the employment of a couple of tools dcf which extracts interpretable classification rules from data and espresso that is capable of fast minimization of boolean propositions our research demonstrates that it is possible to design models that exhibit good classification accuracy combined with high interpretability in the sense of semantic cointension also structural parameters that quantify model complexity show that the derived models are also simple enough to be read and understood

a weighting function for improving fuzzy classification systems performance this paper considers the automatic design of fuzzy rule based classification systems from labeled data the classification accuracy and interpretability of generated rules are of major importance in fuzzy classification systems we propose a weighting function for compatibility grade of patterns that improves the performance of fuzzy classification system without degrading the interpretability of fuzzy rules our approach divides the covering subspace of each fuzzy rule into two subdivisions based on a threshold any pattern with compatibility grade above this threshold should be classified truly so the weighting function enhances their association degree for patterns below threshold their compatibility grades remain unchanged the splitting threshold for each rule i e the compatibility grade of a specific pattern is found using distribution of patterns in the covering subspace of that rule we also show that how the proposed approach is applicable when fuzzy rules have certainty grades experiments on some well known data sets are used to evaluate the performance of our approach

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

on the first sign change of hecke eigenvalues of newforms let f be a non zero cusp form of even integral weight k 2 and level n with real fourier coefficients a n n n using a classical theorem of landau together with the facts that the hecke l function of f is entire and the rankin selberg zeta function of f has a pole at its abscissa of convergence with residue essentially equal to the square of the petersson norm of f one easily shows that there are infinitely many n n such that a n 0 as well as infinitely many n with a n 1 siegel s argument does not apply and the result above is not known now suppose that n is squarefree and f is a newform which is a normalized hecke eigenform so a 1 1 and a n is the n th hecke eigenvalue of f we shall prove

adaptability interpretability and rule weights in fuzzy rule based systems this paper discusses interpretability in two main categories of fuzzy systems fuzzy rule based classifiers and interpolative fuzzy systems our goal is to show that the aspect of high level interpretability is more relevant to fuzzy classifiers whereas fuzzy systems employed in modeling and control benefit more from low level interpretability we also discuss the interpretability accuracy tradeoff and observe why various rule weighting schemes that have been brought into play to increase adaptability of fuzzy systems rather just increase computational overhead and seriously compromise interpretability of fuzzy systems

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

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

language operations with regular expressions of polynomial size this work deals with questions regarding to what extent regularity preserving language operations affect the descriptional complexity of regular expressions some language operations are identified which are feasible for regular expressions in the sense that the result of the operation can be represented as a regular expression of size polynomial in that of the operands we prove that taking language quotients in particular the prefix and suffix closures of a regular set can incur at most a quadratic blow up on the required expression size the circular shift operation can cause only a cubic increase in size and at least a quadratic bloat can be necessary in the worst case

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

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

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

fuzzy min max neural networks i classification a supervised learning neural network classifier that utilizes fuzzy sets as pattern classes is described each fuzzy set is an aggregate union of fuzzy set hyperboxes a fuzzy set hyperbox is an n dimensional box defined by a min point and a max point with a corresponding membership function the min max points are determined using the fuzzy min max learning algorithm an expansion contraction process that can learn nonlinear class boundaries in a single pass through the data and provides the ability to incorporate new and refine existing classes without retraining the use of a fuzzy set approach to pattern classification inherently provides a degree of membership information that is extremely useful in higher level decision making the relationship between fuzzy sets and pattern classification is described the fuzzy min max classifier neural network implementation is explained the learning and recall algorithms are outlined and several examples of operation demonstrate the strong qualities of this new neural network classifier

interpolative reasoning with insufficient evidence in sparse fuzzy rule bases abstract rule based fuzzy approximate reasoning uses various techniques of modified modus ponens the observation is in most cases not identical with any of the antecedents in the rules however a conclusion still can be computed by using some combination of all consequents where an overlapping of observation and antecedent is present if the rule base is sparse i e it contains insufficient information on the total state space it might occur that an observation has absolutely no overlapping with any of the antecedents and so not even a single rule is fired i e no conclusion can be computed on the basis of modus ponens in such a case interpolative reasoning in the strict sense can be applied some kind of weighted average of the flanking rules is calculated this technique can be extended to a form of extrapolation when the observation is not flanked from both sides linear interpolation and extrapolation is presented and then the idea is extended to arbitrary approximation

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

a proposed method for learning rule weights in fuzzy rule based classification systems in fuzzy rule based classification systems frbcss rule weighting has often been used as a simple mechanism to tune the classifier in past research a number of heuristic rule weight specification methods have been proposed for this purpose a learning algorithm based on reward and punishment has also been proposed to adjust the weights of each fuzzy rule in the rule base in this paper a new method of learning rule weight in frbcss is proposed the method can be used when single winner or weighted vote methods of reasoning is used compared with reward and punishment scheme the proposed method is much faster and more effective another advantage of the proposed method is that during the learning process redundant rules are removed i e by setting their weights to zero the final rule base usually contains much fewer rules than the initial one this feature is very useful since a compact rule base is usually desired from the point of efficiency and interpretability a number of uci data sets are used to assess the performance of the proposed method in comparison with reward and punishment scheme

oscillations of fourier coefficients of modular forms a p 2p cos 0 p since we know the truth of the ramanujan petersson conjecture it follows that the 0 p s are real inspired by the sato tate conjecture for elliptic curves serre 14 conjectured that the 0 p s are uniformly distributed in the interval 0 rc with respect to the 1 measure sin2odo following serre we shall refer to this as the sato tate r conjecture there being no room for confusion it has been known for a long time that the truth of this conjecture implies much about the oscillatory behaviour of the fourier coefficients in particular the following is implied by the sato tate conjecture theorem 1 for any normalized hecke eigenform

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

identification of transparent compact accurate and reliable linguistic fuzzy models transparency accuracy compactness and reliability all appear to be vital even though somewhat contradictory requirements when it comes down to linguistic fuzzy modeling this paper presents a methodology for simultaneous optimization of these criteria by chaining previously published various algorithms a heuristic fully automated identification algorithm that is able to extract sufficiently accurate yet reliable and transparent models from data and two algorithms for subsequent simplification of the model that are able to reduce the number of output parameters as well as the number of fuzzy rules with only a marginal negative effect to the accuracy of the model

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

regular expression for a language without empty word in 1 a bruggemann klein asks the following question is there a linear time algorithm transforming a regular expression e into an expression e with oliver in this paper we give a recursive definition of e which enables us to provide such an algorithm furthermore we show that e e 12 log e 1 e 12 where e is the size of e and e is the size of e

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