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

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

multiobjective evolutionary optimization of type 2 fuzzy rule based systems for financial data classification classification techniques are becoming essential in the financial world for reducing risks and possible disasters managers are interested in not only high accuracy but in interpretability and transparency as well it is widely accepted now that the comprehension of how inputs and outputs are related to each other is crucial for taking operative and strategic decisions furthermore inputs are often affected by contextual factors and characterized by a high level of uncertainty in addition financial data are usually highly skewed toward the majority class with the aim of achieving high accuracies preserving the interpretability and managing uncertain and unbalanced data this paper presents a novel method to deal with financial data classification by adopting type 2 fuzzy rule based classifiers frbcs generated from data by a multiobjective evolutionary algorithm moea the classifiers employ an approach denoted as scaled dominance for defining rule weights in such a way to help minority classes to be correctly classified in particular we have extended paes rcs an moea based approach to learn concurrently the rule and data bases of frbcs for managing both interval type 2 fuzzy sets and unbalanced datasets to the best of our knowledge this is the first work that generates type 2 frbcs by concurrently maximizing accuracy and minimizing the number of rules and the rule length with the objective of producing interpretable models of real world skewed and incomplete financial datasets the rule bases are generated by exploiting a rule and condition selection rcs approach which selects a reduced number of rules from a heuristically generated rule base and a reduced number of conditions for each selected rule during the evolutionary process the weight associated with each rule is scaled by the scaled dominance approach on the fuzzy frequency of the output class in order to give a higher weight to the minority class as regards the data base learning the membership function parameters of the interval type 2 fuzzy sets used in the rules are learned concurrently to the application of rcs unbalanced datasets are managed by using in addition to complexity selectivity and specificity as objectives of the moea rather than only the classification rate we tested our approach named it2 paes rcs on 11 financial datasets and compared our results with the ones obtained by the original paes rcs with three objectives and with and without scaled dominance the frbcs fuzzy association rule based classification model for high dimensional dataset farc hd and fuzzy unordered rules induction algorithm furia the classical c4 5 decision tree algorithm and its cost sensitive version using nonparametric statistical tests we will show that it2 paes rcs generates frbcs with on average accuracy statistically comparable with and complexity lower than the ones generated by the two versions of the original paes rcs further the frbcs generated by farc hd and furia and the decision trees computed by c4 5 and its cost sensitive version despite the highest complexity result to be less accurate than the frbcs generated by it2 paes rcs finally we will highlight how these frbcs are easily interpretable by showing and discussing one of them

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

genetic training instance selection in multiobjective evolutionary fuzzy systems a coevolutionary approach when dealing with datasets that are characterized by a large number of instances multiobjective evolutionary learning moel of fuzzy rule based systems frbss suffers from high computational costs mainly because of the fitness evaluation the use of a reduced set of representative instances in place of the overall training set ts would considerably lessen the computational effort even though a large number of papers have proposed instance selection approaches mainly in classification problems how this selection should be performed especially in the context of regression is still an open issue in this paper we tackle the instance selection problem in the framework of moel of frbss through a coevolutionary approach in the execution of the moel periodically a single objective genetic algorithm soga evolves a population of reduced tss the soga aims to maximize a purposely defined index which measures how much the pareto fronts computed by using respectively the reduced ts and the overall ts are close to each other the closer the fronts the more the reduced ts is representative of the overall ts during the execution of the moel the rule base and the membership function parameters of the fuzzy sets are concurrently learned by maximizing the accuracy and minimizing the complexity we tested our approach on 12 large datasets we adopted reduced tss composed of 5 10 and 20 of the overall ts using nonparametric statistical tests we verified that with 10 and 20 of the overall ts the pareto front approximations that are generated by our coevolutionary approach are comparable with the ones generated by applying the moel with the overall ts although the coevolution allows us to save up to 86 36 of the execution time in addition the analysis of the behavior of three representative solutions on the test set highlights that the use of the reduced tss does not affect the generalization capabilities of the generated frbss

deep tsk fuzzy classifier with stacked generalization and triplely concise interpretability guarantee for large data although takagi sugeno kang tsk fuzzy classifier has been applied to a wide range of practical scenarios how to enhance its classification accuracy and interpretability simultaneously is still a challenging task in this paper based on the powerful stacked generalization principle a deep tsk fuzzy classifier d tsk fc is proposed to achieve the enhanced classification accuracy and triplely concise interpretability for fuzzy rules d tsk fc consists of base building units just like the existing popular deep learning d tsk fc can be built in a layer by layer way in terms of the stacked generalization principle the training set plus random shifts obtained from random projections of prediction results of current base building unit are presented as the input of the next base building unit the hidden layer in each base building unit of d tsk fc is represented by triplely concise interpretable fuzzy rules in the sense of randomly selected features with the fixed five fuzzy partitions random rule combinations and the same input space kept in every base building unit of d tsk fc the output layer of each base building unit can be learnt quickly by least learning machine llm besides benefiting from llm d tsk fc s deep learning can be well scaled up for large datasets our extensive experimental results witness the power of the proposed deep tsk fuzzy classifier

data driven fuzzy modeling for takagi sugeno kang fuzzy system this paper presents a systematic approach to design first order tagaki sugeno kang tsk fuzzy systems this approach attempts to obtain the fuzzy rules without any assumption about the structure of the data the structure identification and parameter optimization steps in this approach are carried out automatically and are capable of finding the optimal number of the rules with an acceptable accuracy starting with an initial structure the system first tries to improve the structure and then as soon as an improved structure is found it fine tunes its rules parameters then it goes back to improve the structure again to find a better structure and re fine tune the rules parameters this loop continues until a satisfactory solution tsk model is found the proposed approach has successfully been applied to well known benchmark datasets and real world problems the obtained results are compared with those obtained with other methods from the literature experimental studies demonstrate that the predicted properties have a good agreement with the measured data by using the elicited fuzzy model with a small number of rules finally as a case study the proposed approach is applied to the desulfurization process of a real steel industry comparing the proposed approach with some other fuzzy systems and neural networks it is shown that the developed tsk fuzzy system exhibits better results with higher accuracy and smaller size of architecture

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

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

a study of the behaviour of linguistic fuzzy rule based classification systems in the framework of imbalanced data sets in the field of classification problems we often encounter classes with a very different percentage of patterns between them classes with a high pattern percentage and classes with a low pattern percentage these problems receive the name of classification problems with imbalanced data sets in this paper we study the behaviour of fuzzy rule based classification systems in the framework of imbalanced data sets focusing on the synergy with the preprocessing mechanisms of instances and the configuration of fuzzy rule based classification systems we will analyse the necessity of applying a preprocessing step to deal with the problem of imbalanced data sets regarding the components of the fuzzy rule base classification system we are interested in the granularity of the fuzzy partitions the use of distinct conjunction operators the application of some approaches to compute the rule weights and the use of different fuzzy reasoning methods

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

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

bayesian takagi sugeno kang fuzzy classifier in this paper the takagi sugeno kang tsk fuzzy classifier is casted into the bayesian inference framework and a new fuzzy classifier called bayesian tsk fuzzy classifier b tsk fc is proposed accordingly the proposed classifier can be constructed by learning both the antecedent and consequent parameters of the involved fuzzy rules simultaneously as a result of the introduction of bayesian inference the proposed b tsk fc can be distinguished as follows 1 unlike most existing tsk fuzzy classifiers where the antecedent and consequent parameters of fuzzy rules are learnt in a decoupled manner and the antecedent parameters are learnt only in the input space the antecedent parameters of the fuzzy rules in b tsk fc are learnt by developing a fuzzy clustering method in the input output space and the consequent parameters of fuzzy rules are learnt in accordance with the maximum margin of separation principle thereby resulting to form an intrinsic link between the input and output spaces to achieve improved classification performance and better interpretability 2 with a dirichlet prior assumption about fuzzy memberships in fuzzy clustering a markov chain monte carlo technique is employed to estimate the parameters of the proposed classifier from a sampling perspective 3 rather than being rivals fuzziness and probability in b tsk fc are collaboratively modeled to enhance the performance of tsk fuzzy classifier in terms of classification and interpretability our experimental results in synthetic datasets as well as several real world datasets confirm such merits of the proposed fuzzy classifier

design of interpretable fuzzy rule based classifiers using spectral analysis with structure and parameters optimization this paper presents a design method for fuzzy rule based systems that performs data modeling consistently according to the symbolic relations expressed by the rules the focus of the model is the interpretability of the rules and the model s accuracy such that it can be used as tool for data understanding the number of rules is defined by the eigenstructure analysis of the similarity matrix which is computed from data the rule induction algorithm runs a clustering algorithm on the dataset and associates one rule to each cluster each rule is selected among all possible combinations of one dimensional fuzzy sets as the one nearest to a cluster s center the rules are weighted in order to improve the classifier performance and the weights are computed by a bounded quadratic optimization problem the model complexity is minimized in a structure selection search performed by a genetic algorithm that selects simultaneously the most representative subset of variables and also the number of fuzzy sets in the fuzzy partition of the selected variables the resulting model is evaluated on a set of benchmark datasets for classification problems the results show that the proposed approach produces accurate and yet compact fuzzy classifiers the resulting model is also evaluated from an interpretability point of view showing how the rule weights provide additional information to help data understanding and model exploitation

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

a defuzzification free hierarchical fuzzy system df hfs rock mass rating prediction abstract when the use of a single fuzzy system becomes inapplicable due to the increase in the number of input parameters hierarchical fuzzy systems are commonly used for the solution this inapplicability arises from both computational cost and the challenging process of fuzzy rule creation the conventional application of hierarchical fuzzy systems performs the steps from fuzzification to defuzzification one by one in each subsystem and the provided crisp result is transferred to the higher layer the major drawback of this process is that the defuzzification steps performed in the inner layers degenerate the fuzziness level of information this drawback leads to two outcomes the output of the hierarchical system and single fuzzy system may be highly different from each other and the output of the hierarchical system can change according to its hierarchical structure as a result the preservation of fuzziness during the hierarchical inference flow should be considered to employ hierarchical approaches to the problems in this study the defuzzification free hierarchical fuzzy inference system df hfs is proposed in which the misleading defuzzification steps are eliminated from the hierarchical inference flow and the fuzziness is propagated up to the highest layer without being exposed to any degeneration to test the accuracy of data transmission experiments are performed on two different problems the modeling of the logical xor and rock mass rating the obtained experimental results indicate that the proposed hierarchical flow achieves more successful data transmission than its counterparts and that it provides the closest outputs to the corresponding single fuzzy system

c psa constrained pareto simulated annealing for constrained multi objective optimization in recent years evolutionary algorithms eas have been extensively developed and utilized to solve multi objective optimization problems however some previous studies have shown that for certain problems an approach which allows for non greedy or uphill moves unlike eas can be more beneficial one such approach is simulated annealing sa sa is a proven heuristic for solving numerical optimization problems but owing to its point to point nature of search limited efforts has been made to explore its potential for solving multi objective problems the focus of the presented work is to develop a simulated annealing algorithm for constrained multi objective problems the performance of the proposed algorithm is reported on a number of difficult constrained benchmark problems a comparison with other established multi objective optimization algorithms such as infeasibility driven evolutionary algorithm idea non dominated sorting genetic algorithm ii nsga ii and multi objective scatter search ii moss ii has been included to highlight the benefits of the proposed approach

modeling with discrete time recurrent fuzzy systems via mixed integer optimization in this paper we present a new approach for modeling of dynamical systems with discrete time recurrent fuzzy systems dtrfs the modeling problem is divided into two subproblems the first subproblem considers the identification of the rule base of the dtrfs which can be interpreted as structural modeling this problem is recast into integer and mixed integer optimization problems respectively dependent on the chosen cost function and system structure the second subproblem considers the optimization of the parameters of the system i e quantitative modeling as a result we obtain a linguistically interpretable model of the dynamical system which additionally can be interpreted as a linguistic automaton the applicability of the approach is shown by modeling a thermofluidic process

conservative numerical methods for x f x abstract two distinctly different numerical methods are developed for solving the conservative initial value problem x f x x o dot x o both methods conserve exactly the same total energy as does the given differential equation computer examples are described and

including a simplicity criterion in the selection of the best rule in a genetic fuzzy learning algorithm learning algorithms can obtain very useful descriptions of several problems many different alternative descriptions can be generated in many cases a simple description is preferable since it has a higher possibility of being valid in unseen cases and also it is usually easier to understand by a human expert thus the main idea of this paper is to propose simplicity criteria and to include them in a learning algorithm in this case the learning algorithm will reward the simplest descriptions we study simplicity criteria in the selection of fuzzy rules in the genetic fuzzy learning algorithm called slave

reduced interval type 2 neural fuzzy system using weighted bound set boundary operation for computation speedup and chip implementation this paper proposes a reduced interval type 2 neural fuzzy system using weighted bound set boundaries rit2nfs wb for the simplification of type reduction operations the objective of this simplification is to reduce the system training time in software implementation and chip size in hardware implementation especially when the number of rules is large the antecedent part in the rit2nfs wb uses interval type 2 fuzzy sets it2fss and the consequent part can be of the takagi sugeno kang tsk or mamdani type the rit2nfs wb is built through an online structure and parameter learning to improve model accuracy in addition the interpretability of the rit2nfs wb is improved by considering distributions of the it2fss in input variables a distinguishability oriented cost function is used in parameter learning to generate distinguishable it2fss and improve semantics based interpretability for highly overlapped it2fss they are merged to reduce the number of it2fss and improve complexity based interpretability the software implemented tsk type rit2nfs wb is hardware implemented on a field programmable gate array chip to accelerate the chip execution speed the chip utilizes not only the parallel execution properties of fuzzy rules and bound set boundaries but the pipeline technique as well in particular the flexibility of the chip is considered so that no redesign of the circuits is required when the rit2nfs wb is applied to different problems the characteristics of the software and hardware implemented rit2nfs wb are verified through various examples and comparisons with various type 1 and interval type 2 fuzzy models

ga fuzzy modeling and classification complexity and performance the use of genetic algorithms gas and other evolutionary optimization methods to design fuzzy rules for systems modeling and data classification have received much attention in recent literature authors have focused on various aspects of these randomized techniques and a whole scale of algorithms have been proposed we comment on some recent work and describe a new and efficient two step approach that leads to good results for function approximation dynamic systems modeling and data classification problems first fuzzy clustering is applied to obtain a compact initial rule based model then this model is optimized by a real coded ga subjected to constraints that maintain the semantic properties of the rules we consider four examples from the literature a synthetic nonlinear dynamic systems model the iris data classification problem the wine data classification problem and the dynamic modeling of a diesel engine turbocharger the obtained results are compared to other recently proposed methods

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

a self stabilized algorithm for enforcing constraints in multibody systems a new algorithm is developed for the enforcement of constraints within the framework of nonlinear flexible multibody system modeled with the finite element approach the proposed algorithm exactly satisfies the constraints at the displacement and velocity levels and furthermore it achieves nonlinear unconditional stability by imposing the vanishing of the work done by the constraint forces when combined with specific discretizations of the inertial and elastic forces identical convergence rates are observed for the displacements velocities and lagrange multipliers the proposed algorithm is closely related to the stabilized index 2 or ggl method although no additional multipliers are introduced in the present approach these desirable characteristics are obtained without resorting to numerically dissipative algorithms if high frequencies are present in the system i e the system is physically stiff dissipative schemes become necessary the proposed algorithm is extended to deal with this situation

gp coach genetic programming based learning of compact and accurate fuzzy rule based classification systems for high dimensional problems in this paper we propose gp coach a genetic programming based method for the learning of compact and accurate fuzzy rule based classification systems for high dimensional problems gp coach learns disjunctive normal form rules generated by means of a context free grammar coded as one rule per tree the population constitutes the rule base so it is a genetic cooperative competitive learning approach gp coach uses a token competition mechanism to maintain the diversity of the population and this obliges the rules to compete and cooperate among themselves and allows the obtaining of a compact set of fuzzy rules the results obtained have been validated by the use of non parametric statistical tests showing a good performance in terms of accuracy and interpretability

comparison of some single step methods for the numerical solution of the structural dynamic equation this paper compares the performance of the ss22 and ss23 references 1 and 2 single step algorithms for the numerical solution of the second order structural dynamic equation and a related new algorithm ss32b applied to the equivalent first order system with sine and step forcing functions various aspects of stability relevant to these equations are discussed

cooperative evolutionary learning of linguistic fuzzy rules and parametric aggregation connectors for mamdani fuzzy systems there are two tasks in the design of linguistic fuzzy models for a concrete application the derivation of the linguistic rule base and the setup of the inference system and the defuzzification method traditionally the derivation of the linguistic rule base has been considered the most important task but the use of the appropriate aggregation connectors in the inference system and the defuzzification interface can improve the fuzzy system behavior in this paper we take in consideration this idea we propose an evolutionary learning method to learn a linguistic rule base and the parametric aggregation connectors of the inference and defuzzification in a single step the aim of this methodology is to make possible a high level of positive synergy between the linguistic rule base and the aggregation connectors improving the accuracy of the linguistic mamdani fuzzy systems our proposal has shown good results solving three different applications we introduce a statistical analysis of results for validating the model behavior on the applications used in the experimental study we must remark that we present an experimental study with a double intention a to compare the behavior of the new approach in comparison with those ones that first learn the rule base and then adapt the connectors and b to analyze the rule bases obtained with fixed aggregation connectors and with the adaptive ones for showing the changes on the consequent rules changes on labels that produce a better behavior of the linguistic model than the classic ones

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

a fast and efficient multi objective evolutionary learning scheme for fuzzy rule based classifiers during the last years multi objective evolutionary algorithms moeas have been extensively used to generate fuzzy rule based systems characterized by different trade offs between accuracy and complexity in this paper we propose an moea based approach to learn concurrently the rule and data bases of fuzzy rule based classifiers frbcs in particular the rule bases are generated by exploiting a rule and condition selection rcs strategy which selects a reduced number of rules from a heuristically generated set of candidate rules and a reduced number of conditions for each selected rule during the evolutionary process rcs can be considered as a rule learning in a constrained search space as regards the data base learning the membership function parameters of each linguistic term used in the rules are learned concurrently to the application of rcs we tested our approach on twenty four classification benchmarks and compared our results with the ones obtained by two similar state of the art moea based approaches and by two well known non evolutionary classification algorithms namely furia and c4 5 using non parametric statistical tests we show that our approach generates frbcs with accuracy and complexity statistically comparable to and sometimes better than the ones generated by the two moea based approaches exploiting however only the 5 of the number of fitness evaluations used by these approaches further the classifiers generated by our approach result to be more interpretable than the ones generated by the furia and c4 5 algorithms while achieving the same accuracy level

genetic local search for multi objective combinatorial optimization abstract the paper presents a new genetic local search gls algorithm for multi objective combinatorial optimization moco the goal of the algorithm is to generate in a short time a set of approximately efficient solutions that will allow the decision maker to choose a good compromise solution in each iteration the algorithm draws at random a utility function and constructs a temporary population composed of a number of best solutions among the prior generated solutions then a pair of solutions selected at random from the temporary population is recombined local search procedure is applied to each offspring results of the presented experiment indicate that the algorithm outperforms other multi objective methods based on gls and a pareto ranking based multi objective genetic algorithm ga on travelling salesperson problem tsp

non linear dynamics of three dimensional rods exact energy and momentum conserving algorithms the long term dynamic response of non linear geometrically exact rods under going finite extension shear and bending accompanied by large overall motions is addressed in detail the central objective is the design of unconditionally stable time stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum the total angular momentum and for the hamiltonian case the total energy this objective is accomplished in two steps first a class of algorithms is introduced which conserves linear and angular momentum this result holds independently of the definition of the algorithmic stress resultants second an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved conventional schemes exhibiting no numerical dissipation symplectic algorithms in particular are shown to lead to unstable solutions when the high frequencies are not resolved compared to conventional schemes there is little if any additional computational cost involved in the proposed class of energy momentum methods the excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations

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

comparison and design of interpretable linguistic vs scatter frbss gm3m generalization and new rule meaning index for global assessment and local pseudo linguistic representation this work is devoted to defining more general interpretability indexes to be applied to any scatter or linguistic model implemented by any type of membership functions they are based on metrics that should take into account the semantic and inference issues the semantic issue in order to preserve the meaning of the linguistic labels and the inference issue since this can influence the behavior of the rules on the other hand these metrics have been designed to be intuitive in order to support the analysis or selection of a final model and to favor a low computational cost within an optimization process in order to check their usefulness a multi objective evolutionary algorithm simultaneously performing a rule selection and an adjustment of the fuzzy partitions is guided by the proposed indexes on several benchmark data sets to obtain models with different degrees of accuracy and interpretability in addition using these metrics a local analysis can be carried out between models of a different nature this local analysis through the model components gives support to the user to make the best choice from among the models

a new method for design and reduction of neuro fuzzy classification systems in this paper we propose a new class of neuro fuzzy systems moreover we develop a novel method for reduction of such systems without the deterioration of their accuracy the reduction algorithm gradually eliminates inputs rules antecedents and the number of discretization points of integrals in the center of area defuzzification method it then automatically detects and merges similar input and output fuzzy sets through computer simulations it is shown that accuracy of the system after reduction and merging has not deteriorated despite the fact that in some cases up to 54 of various parameters and 74 of inputs were eliminated the reduction algorithm has been tested using well known classification benchmarks

genetic generation of fuzzy systems with rule extraction using formal concept analysis fuzzy classification systems have been widely researched with many approaches proposed in the literature several methods are available for the automatic definition of fuzzy classification systems which basically comprehend two tasks i the definition of the attributes in terms of fuzzy sets and ii the generation of a rule set containing the domain knowledge named fuzzy rule base genetic fuzzy systems are used to learn or tune in fuzzy classification systems some genetic approaches for learning the fuzzy rule base require the previous extraction of a set of rules to be used as the genetic search space in this paper we present the fca based method a proposal for the automatic generation of fuzzy rule bases which extracts a set of rules using the formal concept analysis theory directly from data after extracting the rules forming the genetic search space fca based uses a genetic algorithm to select the final rule base the last step of the fca based method is a rule pruning step in order to improve the interpretability of the fuzzy rule bases the extraction of rules proposed for the fca based algorithm presents polynomial complexity and does not require the predefinition of the number of rules to be extracted as it extracts rules directly from data the proposed method avoids the random extraction of rules it also presents the advantage of automatically extracting rules with variable number of conditions in their antecedents a feature subset selection method specifically designed for fuzzy classification systems is integrated into the fca based method in order to reduce the search space of solutions the fca based method is detailed and compared to eight different rule based fuzzy systems experimental results using 27 benchmark datasets and a 10 fold cross validation strategy show that fca based presents higher accuracy and statistically significant difference with seven of the eight compared methods

improving generalization of fuzzy if then rules by maximizing fuzzy entropy when fuzzy if then rules initially extracted from data have not a satisfying performance we consider that the rules require refinement distinct from most existing rule refinement approaches that are based on the further reduction of training error this paper proposes a new rule refinement scheme that is based on the maximization of fuzzy entropy on the training set the new scheme which is realized by solving a quadratic programming problem is expected to have the advantages of improving the generalization capability of initial fuzzy if then rules and simultaneously overcoming the overfitting of refinement experimental results on a number of selected databases demonstrate the expected improvement of generalization capability and the prevention of overfitting by a comparison of both training and testing accuracy before and after the refinement

a review of the application of multiobjective evolutionary fuzzy systems current status and further directions over the past few decades fuzzy systems have been widely used in several application fields thanks to their ability to model complex systems the design of fuzzy systems has been successfully performed by applying evolutionary and in particular genetic algorithms and recently this approach has been extended by using multiobjective evolutionary algorithms which can consider multiple conflicting objectives instead of a single one the hybridization between multiobjective evolutionary algorithms and fuzzy systems is currently known as multiobjective evolutionary fuzzy systems this paper presents an overview of multiobjective evolutionary fuzzy systems describing the main contributions on this field and providing a two level taxonomy of the existing proposals in order to outline a well established framework that could help researchers who work on significant further developments finally some considerations of recent trends and potential research directions are presented

iterative fuzzy clustering algorithm with supervision to construct probabilistic fuzzy rule base from numerical data to deal with data patterns with linguistic ambiguity and with probabilistic uncertainty in a single framework we construct an interpretable probabilistic fuzzy rule based system that requires less human intervention and less prior knowledge than other state of the art methods specifically we present a new iterative fuzzy clustering algorithm that incorporates a supervisory scheme into an unsupervised fuzzy clustering process the learning process starts in a fully unsupervised manner using fuzzy c means fcm clustering algorithm and a cluster validity criterion and then gradually constructs meaningful fuzzy partitions over the input space the corresponding fuzzy rules with probabilities are obtained through an iterative learning process of selecting clusters with supervisory guidance based on the notions of cluster pureness and class separability the proposed algorithm is tested first with synthetic data sets and benchmark data sets from the uci repository of machine learning database and then with real facial expression data and tv viewing data

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

using bounds on the data in linear programming the tolerance approach to sensitivity analysis in contrast to traditional sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent variations in a number of parameters a primary focus of this approach is to determine a maximum tolerance percentage for selected right hand side terms in which the same basis is optimal as long as each term is accurate to within that percentage of its estimated value similarly the approach yields a maximum tolerance percentage for selected objective function coefficients this paper shows how the tolerance approach can exploit information on the range of possible values over which terms and coefficients can vary to yield larger maximum tolerance percentages

an explicit finite difference scheme with exact conservation properties abstract a finite difference scheme for the numerical study of the korteweg de vries equation is constructed it is explicit and yet conserves exactly the energy of the computed solutions the underlying idea can also be applied to more general equations or systems numerical experiments are included

how to render second order accurate time stepping algorithms fourth order accurate while retaining the stability and conservation properties abstract for a fairly general class of second order accurate algorithms a sub stepping procedure is described which achieves fourth order accuracy while retaining the stability conservation properties and implementation of the underlying second order method two features render the proposed approach especially attractive from a computational perspective over conventional fourth order methods first the improved fourth order accuracy is attained merely by two additional residual evaluations with no changes in the implementation of the underlying second order method in particular no additional storage or extra computations of high order gradients are required second for hamiltonian systems the stability properties and possible conservation properties of the underlying second order method such as energy momentum and symplectic character are preserved the new technique is applied to a number of standard algorithms for elastodynamics structural and rigid body dynamics in particular when applied to the recently proposed second order accurate energy momentum method this approach results in a fourth order accurate energy linear and angular momentum conserving algorithm for general hamiltonian systems

sgerd a steady state genetic algorithm for extracting fuzzy classification rules from data this paper considers the automatic design of fuzzy rule based classification systems from labeled data the performance of classifiers and the interpretability of generated rules are of major importance in these systems in past research some genetic based algorithms have been used for the rule learning process these genetic fuzzy systems have utilized different approaches to encode rules in this paper we have proposed a novel steady state genetic algorithm to extract a compact set of good fuzzy rules from numerical data sgerd the selection mechanism of this algorithm is nonrandom and only the best individuals can survive our approach is very simple and fast and can be applied to high dimensional problems with numerical attributes to select the rules having high generalization capabilities our algorithm makes use of some rule and data dependent parameters we have also proposed an enhancing function that modifies the rule evaluation measures in order to assess the candidate rules more effectively before their selection experiments on some well known data sets are performed to show the performance of sgerd

a geometrically exact rod model incorporating shear and torsion warping deformation abstract a fully nonlinear three dimensional rod model is developed that incorporates transverse shear and torsion warping deformation the geometric setting is that of a constrained body model with configuration space modeled on r3 so 3 r a differentiable manifold the proposed model incorporates the classical notion of bi moment and bi shear in a fully nonlinear geometrically exact context explicit properly invariant constitutive equations that generalize those of the linearized theory are developed the underlying variational formulation of the model is discussed and computational procedures employing a galerkin projection are addressed numerical examples are presented that illustrate the performance of the formulation

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

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

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

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

competitive interaction reasoning in designing fuzzy rule based classification systems frbcss complex fuzzy rule extraction techniques and tuning membership functions are frequently used to enhance classification accuracy however these approaches not only decrease system s transparency which is the hallmark of fizzy design but also are oftentimes computationally expensive and require multiple parameters to be optimized using a large amount of training data in this paper inspired by the so called competitive behavior of mini columns in the brain neuronal circuitry we proposed a new reasoning method for fuzzy classifiers referred to as competitive interaction reasoning cir that employs the cumulative information provided by all fuzzy rules and adjusts the decision boundaries as if the membership functions are directly modified this mechanism is mathematically implemented by a linear transformation and resembles the competitive interaction observed in brain neuronal columns cross rule competition weights are optimized using hebbian reinforcement learning using a large number of simulations on benchmark data sets we show that the proposed cir significantly improves classification accuracy without compromising interpretability of the fuzzy classifier in addition cir can further facilitate formation of the fuzzy rules and incorporation of the expert knowledge by confining the destructive effects of noisy rules or expert inconsistencies experiments on 23 well known benchmark data sets confirm high performance of cir in comparison with a number of popular classifiers

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

a multiobjective evolutionary programming framework for graph based data mining subgraph mining is the process of identifying concepts describing interesting and repetitive subgraphs within graph based data the exponential number of possible subgraphs makes the problem very challenging existing methods apply a single objective subgraph search with the view that interesting subgraphs are those capable of not merely compressing the data but also enhancing the interpretation of the data considerably usually the methods operate by posing simple constraints or user defined thresholds such as returning all subgraphs whose frequency is above a specified threshold such search approach may often return either a large number of solutions in the case of a weakly defined objective or very few in the case of a very strictly defined objective in this paper we propose a framework based on multiobjective evolutionary programming to mine subgraphs by jointly maximizing two objectives support and size of the extracted subgraphs the proposed methodology is able to discover a nondominated set of interesting subgraphs subject to tradeoff between the two objectives which otherwise would not be achieved by the single objective search besides it can use different specific multiobjective evolutionary programming methods experimental results obtained by three of the latter methods on synthetically generated as well as real life graph based datasets validate the utility of the proposed methodology when benchmarked against classical single objective methods and their previous nonevolutionary multiobjective extensions

on rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3d geometrically exact beam model abstract in this work we discuss the finite element implementation of the internal constraints in a three dimensional 3d geometrically exact beam model which can be formulated as holonomic constraint relationships model problems chosen for a more detailed consideration include a general joint constraint between beams and the beam connected to a rigid component appropriate modifications of the standard form of the geometrically exact beam element arrays are carried out in order to impose explicitly these kinds of constraints into time integration schemes for nonlinear dynamics consequently only the minimum number of unknowns is retained for the global set of nonlinear equations to be solved thus avoiding the use of the extra variables the lagrange multipliers and the pertinent difficulties in integrating the system of differential algebraic equations a number of numerical simulations considering dynamic analysis of multibody systems with rigid flexible components and joint constraints are presented in order to illustrate versatility of the proposed procedure

data driven elastic fuzzy logic system modeling constructing a concise system with human like inference mechanism the construction of fuzzy logic systems flss using data driven techniques has become the most popular modeling approach however this approach still faces critical challenges including the difficulty in obtaining concise models for high dimensional data and generating accurate fuzzy rules to simulate human inference mechanism to tackle these issues a new fls modeling framework called data driven elastic fls dd efls is proposed in this paper the dd efls has two key characteristics first the fuzzy rules in the rule base can use different feature subspaces that are extracted from the original high dimensional space to yield simple and accurate rules in feature spaces of lower dimensionality second fuzzy inferences from various views are implemented by embedding different rules in the corresponding subspaces to imitate human inference mechanism based on the dd efls framework an elastic takagi sugeno kang tsk fls modeling method etsk fls is proposed to train the elastic tsk fls using the concise rules and a more human like inference mechanism for modeling tasks based on high dimensional datasets the characteristics and advantages of the proposed framework and the etsk fls method are validated experimentally using both synthetic and real world datasets

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells