a vectorization optimization method based type 2 fuzzy neural network for noisy data classification this paper proposes a vectorization optimization method vom based type 2 fuzzy neural network vom2fnn for noisy data classification in handling problems with uncertainties such as noisy data type 2 fuzzy systems usually outperform their type 1 counterparts hence type 2 fuzzy sets are adopted in the antecedent parts to model the uncertainty to consider the classification problems the discriminative capability is crucial to determine the performance therefore a vom is proposed in the consequent parts to increase the discriminability and reduce the parameters compared with other existing fuzzy neural networks the novelty of the proposed vom2fnn is its consideration of both uncertainty and discriminability the effectiveness of the proposed vom2fnn is demonstrated by three classification problems experimental results and theoretical analysis indicate that the proposed vom2fnn performs better than the other fuzzy neural networks

centroid of triangular and gaussian type 2 fuzzy sets abstract in this paper we present exact computation procedures for the centroid of type 2 fuzzy sets with either triangular or gaussian secondary membership functions we assume the standard formulation of the centroid based on the sup min extension principle the centroid of a triangular type 2 fuzzy set can be considered as a subsequent analytical extension of the km type reduction procedure dedicated to interval type 2 sets the centroid of a gaussian type 2 fuzzy set is recursively expressed by an independent procedure

granular neural networks with evolutionary interval learning to deal with different membership functions of the same linguistic term a new interval reasoning method using new granular sets is proposed based on yin yang methodology to make interval valued granular reasoning efficiently and optimize interval membership functions based on training data effectively a granular neural network gnn with a new high speed evolutionary interval learning is designed simulation results in nonlinear function approximation and bioinformatics have shown that the gnn with the evolutionary interval learning is able to extract interval valued granular rules effectively and efficiently from training data by using the new evolutionary interval learning algorithm

extremes of locally stationary gaussian random fields the main result of this contribution is the derivation of the exact asymptotic behavior of the supremum of a class of a t locally stationary gaussian random fields we present two applications of our result the first one deals with the extremes of aggregate multifractional brownian motions whereas the second one establishes the exact asymptotics of the supremum of the x process generated by multifractional brownian motions

a generalized type 2 fuzzy logic approach for dynamic parameter adaptation in bee colony optimization applied to fuzzy controller design abstract in this work a generalized type 2 fuzzy logic system gt2fls approach for dynamic parameter adaptation in metaheuristics and for optimal fuzzy controller design is presented in these two cases the efficiency of the gt2fls approach is verified with simulation results in the first case the gt2fls provides an approach to dynamically find the optimal values of the heuristic parameters that are a critical part of the bee colony optimization bco algorithm performance in the second case the gt2fls approach provides the basis for building a generalized type 2 fuzzy logic controller gt2flc which can be optimized with the traditional bco specifically to find the optimal design of the membership functions mfs in the fuzzy controller in both cases the gt2fls approach shows advantages in the optimization of the solutions to the problems for both cases we can considered them as hybrid systems combining gt2fls and bco although the combination is made in a different way and it can be noted that a gt2fls presents better stability in the minimization of the errors when applied to benchmark control problems simulation results illustrate that the implementation of the generalized type 2 fuzzy logic controller gt2flc approach improves its performance when using the bco algorithm and the stability of the fuzzy controller is better when compared with respect to a type 1 fuzzy logic controller t1flc and an interval type 2 fuzzy logic controller it2flc

fuzzy sets and type 2 under algebraic product and algebraic sum abstract the concept of fuzzy sets of type 2 has been proposed by l a zadeh as an extension of ordinary fuzzy sets a fuzzy set of type 2 can be defined by a fuzzy membership function the grade or fuzzy grade of which is taken to be a fuzzy set in the unit interval 0 1 rather than a point in 0 1 this paper investigates the algebraic properties of fuzzy grades that is fuzzy sets of type 2 under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring moreover the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join meet and algebraic product

theory of extended fuzzy discrete event systems for handling ranges of knowledge uncertainties and subjectivity in 2001 we originated a theory of fuzzy discrete event systems fdess that generalized the conventional crisp discrete event systems dess vagueness and imprecision concerning states and event transitions of dess were represented by membership grades and computed via fuzzy logic our application of the fdes theory to computerized human immunodeficiency virus acquired immune deficiency syndrome treatment regimen selection although preliminarily successful suggests that a more comprehensive fdes theory is needed to address two general issues critically important not only to biomedical applications but also to real world problems in other industries first domain experts should have means other than point estimates and type 1 fuzzy sets mandated in the current framework to describe uncertainties subjectivity and imprecision in their complex knowledge and experience second when a group of experts with distinct opinions is involved they should not be forced to reach consensus for the sake of system development this is because collective consensus may not be achievable which is often the case in medicine where individual expertspsila opinions should be equally respected since the underlying ground truth is unknown most of the time the theory of extended fdes presented in this paper addresses both the problems and contains the fdes theory as a special case experts are now allowed to use interval numbers and type 1 and type 2 fuzzy sets to intuitively and quantitatively express their diverse knowledge and experience which will then be processed by the new theory to form fuzzy state vectors and fuzzy event transition matrices accordingly we have established mathematical operations that cover the computations of fuzzy states fuzzy event transitions and parallel composition numerical examples are provided

clifford algebra approach to twistors local particle interpretation or equivalently an enlargement of a structure group to the poincare group at each point of a riemannian space time manifold naturally results in a complexification of the clifford algebra for the tangent minkowski space following crumeyrolle twistor space is identified with an appropriate one sided ideal of this algebra every antiautomorphism of the latter provides a unique projection from the complexified clifford algebra onto the affine complex minkowski space this projection commutes with the action of the poincare group using the above approach three projections the cases of symmetric antisymmetric and hermitian tensors are derived the projection in terms of the antisymmetric decomposable tensors is shown to give the penrose projection

measuring the range of an additive levy process the primary goal of this paper is to study the range of the random field x t nj 1xj tj where x1 xn vspace 1pt are independent levy processes in rd to cite a typical result of this paper let us suppose that i denotes the levy exponent of xi for each i 1 n then under certain mild conditions we show that a necessary and sufficient condition for x rn to have positive d dimensional lebesgue measure is the integrability of the function rd nj 1r 1 j 1 this extends a celebrated result of kesten and of bretagnolle in the one parameter setting furthermore we show that the existence of square integrable local times is yet another equivalent condition for the mentioned integrability criterion this extends a theorem of hawkes to the present random fields setting and completes the analysis of local times for additive levy processes initiated in a companion by paper khoshnevisan xiao and zhong

multiobjective programming for type 2 hierarchical fuzzy inference trees this paper proposes a design of hierarchical fuzzy inference tree hfit an hfit produces an optimum tree like structure i e a natural hierarchical structure that accommodates simplicity by combining several low dimensional fuzzy inference systems fiss such a natural hierarchical structure provides a high degree of approximation accuracy the construction of the hfit takes place in two phases first a nondominated sorting based multiobjective genetic programming mogp is applied to obtain a simple tree structure a low complexity model with a high accuracy second the differential evolution algorithm is applied to optimize the obtained tree s parameters in the derived tree each node acquires a different input s combination where the evolutionary process governs the input s combination hence hfit nodes are heterogeneous in nature which leads to a high diversity among the rules generated by the hfit additionally the hfit provides an automatic feature selection because it uses mogp for the tree s structural optimization that accepts inputs only relevant to the knowledge contained in data the hfit was studied in the context of both type 1 and type 2 fiss and its performance was evaluated through six application problems moreover the proposed multiobjective hfit was compared both theoretically and empirically with recently proposed fiss methods from the literature such as mcit2fis tscit2fnn sit2fnn rit2fns wb et2fis mrit2nfs it2fnn svr etc from the obtained results it was found that the hfit provided less complex and highly accurate models compared to the models produced by the most of other methods hence the proposed hfit is an efficient and competitive alternative to the other fiss for function approximation and feature selection

a self evolving interval type 2 fuzzy neural network with online structure and parameter learning this paper proposes a self evolving interval type 2 fuzzy neural network seit2fnn with online structure and parameter learning the antecedent parts in each fuzzy rule of the seit2fnn are interval type 2 fuzzy sets and the fuzzy rules are of the takagi sugeno kang tsk type the initial rule base in the seit2fnn is empty and the online clustering method is proposed to generate fuzzy rules that flexibly partition the input space to avoid generating highly overlapping fuzzy sets in each input variable an efficient fuzzy set reduction method is also proposed this method independently determines whether a corresponding fuzzy set should be generated in each input variable when a new fuzzy rule is generated for parameter learning the consequent part parameters are tuned by the rule ordered kalman filter algorithm for high accuracy learning performance detailed learning equations on applying the rule ordered kalman filter algorithm to the seit2fnn consequent part learning with rules being generated online are derived the antecedent part parameters are learned by gradient descent algorithms the seit2fnn is applied to simulations on nonlinear plant modeling adaptive noise cancellation and chaotic signal prediction comparisons with other type 1 and type 2 fuzzy systems in these examples verify the performance of the seit2fnn

join and meet operations for type 2 fuzzy sets with nonconvex secondary memberships in this paper we will present two theorems for the join and meet operations for general type 2 fuzzy sets with arbitrary secondary memberships which can be nonconvex and or nonnormal type 1 fuzzy sets these results will be used to derive the join and meet operations of the more general descriptions of interval type 2 fuzzy sets presented in a paper by bustince sola et al interval type 2 fuzzy sets are generalization of interval valued fuzzy sets towards a wider view on their relationship ieee trans fuzzy syst vol 23 pp 1876 1882 2015 where the secondary grades can be nonconvex hence this study will help to explore the potential of type 2 fuzzy logic systems which use the general forms of interval type 2 fuzzy sets which are not equivalent to interval valued fuzzy sets several examples for both general type 2 and the more general forms of interval type 2 fuzzy sets are presented

second order sliding fuzzy interval type 2 control for an uncertain system with real application a new second order sliding mode type 2 fuzzy controller for nonlinear uncertain perturbed systems is developed in this paper to overcome the constraint on the knowledge of the system model we have used local models that are related to some operating points to synthesize a type 2 nominal fuzzy global model the control is based on the super twisting algorithm which is among second order sliding mode controls smcs moreover two adaptive fuzzy type 2 systems have been introduced to generate the two super twisting signals to avoid both the chattering and the constraint on the knowledge of disturbances and uncertainties upper bounds these adaptive fuzzy type 2 systems are adjusted online by two adaptation laws that are deduced from the stability analysis in the lyapunov sense it has only one input i e the sliding surface and one output i e the optimal values of the gains control which are hard to compute with the original algorithm many results of the one link manipulator are obtained first by the simulation in order to compare the performances of the proposed method with that given by levant and then in a real time application in order to confirm the efficiency of the proposed approach the experimentation and simulation are done for the tracking control problem

reduction methods of type 2 uncertain variables and their applications to solid transportation problem uncertainty theory is a branch of mathematics for dealing with realistic uncertainties arising out of complexity changeability and non decidability of practical environments an uncertain variable is defined as a function from the uncertainty space to the set of real numbers and is characterized by an uncertainty distribution this paper proposes the definition of type 2 uncertain variables within the framework of uncertainty theory through introduction of generalized uncertain measures and focuses on more complex twofold uncertainties some uncertainty reduction methods associated with type 2 uncertain variables are also proposed for convenience of applicability including reduction of optimistic value pessimistic value and expected value moreover four classes of type 2 uncertain variables are reduced to type 1 uncertain variables with specific uncertainty distributions type 2 uncertain optimization methods are applied to solving the fixed charge solid transportation problem with the type 2 uncertain parameters where the solution methods are also provided for the proposed models finally numerical experiments are implemented to demonstrate application and sensitivity analysis of the proposed approaches

interval type 2 mutual subsethood fuzzy neural inference system it2msfunis this paper presents an interval type 2 mutual subsethood fuzzy neural inference system it2msfunis a mutual subsethood measure between two interval type 2 fuzzy sets it2 fs has been derived and has been used in determining the similarity between the it2 fs inputs and it2 fs antecedents the consequent weights are taken to be interval sets the inputs to the system are fuzzified into it2 fss with gaussian primary membership function having fixed center and uncertain variance aggregation of type 2 mutual subsethood based activation spreads is performed using product operator the output is obtained using simplified type reduction followed by defuzzification the system learns using memetic procedure involving differential evolution for global search and gradient descent for local exploitation in solution space the mathematical modeling and empirical studies of it2msfunis bring forth its efficacy in problems pertaining to function approximation time series prediction control and classification comparisons with other type 1 and type 2 neuro fuzzy systems verify that it2msfunis compares excellently with other models with a performance better than most of them both in terms of total number of trainable parameters and result accuracy empirical studies indicate the intelligent decision making capability of the proposed model the main contribution of this paper lies in the identification of mutual subsethood to find out the correlation between it2 fss and to find out its applicability in diverse application domains the improved performance of the proposed method can be attributed to the better contrast handling capacity of mutual subsethood method and uncertainty handling capacity of it2 fss the integration of mutual subsethood with interval type 2 fuzzy logic puts forth a novel model with various merits as demonstrated amply with the help of well known problems reported in the literature

an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators ii extension to three valued and interval valued fuzzy sets abstract previously in part i of the present paper and its supplement an approach was made to an interpretation of membership functions as probability and a proposal of the probabilistic operators with dependence relation of the sets this paper is an extension to three valued and interval valued fuzzy sets by taking into consideration uncertainty and contradiction in the judgment whether or not an element of universe of discourse belongs to a fuzzy subset upper lower and intermediate grades of membership are defined this definition leads to a notion of three valued fuzzy set and when only uncertainty is considered interval valued fuzzy set results under the proposed set operation rules the probabilistic operators with dependency various basic set operations and their properties are studied of the properties satisfied by the crisp set operators many of them are found to be valid some properties e g excluded middle laws are found to be invalid due to the uncertainty and contradiction involved also discussed are a representation of set operations of type 2 fuzzy sets by the three valued fuzzy set system and the connection of the proposed operators with other typical set operators

on the existence of paths between points in high level excursion sets of gaussian random fields research supported in part by us israel binational science foundation 2008262 by aro grant w911nf 07 1 0078 nsf grant dms 1005903 and nsa grant h98230 11 1 0154 at cornell university by israel science foundation 853 10 by afosr fa8655 11 1 3039 by office for absorption of new scientists and erc grant rcmt152

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

data based system modeling using a type 2 fuzzy neural network with a hybrid learning algorithm we propose a novel approach for building a type 2 neural fuzzy system from a given set of input output training data a self constructing fuzzy clustering method is used to partition the training dataset into clusters through input similarity and output similarity tests the membership function associated with each cluster is defined with the mean and deviation of the data points included in the cluster then a type 2 fuzzy takagi sugeno kang if then rule is derived from each cluster to form a fuzzy rule base a fuzzy neural network is constructed accordingly and the associated parameters are refined by a hybrid learning algorithm which incorporates particle swarm optimization and a least squares estimation for a new input a corresponding crisp output of the system is obtained by combining the inferred results of all the rules into a type 2 fuzzy set which is then defuzzified by applying a refined type reduction algorithm experimental results are presented to demonstrate the effectiveness of our proposed approach

similarity measures for general type 2 fuzzy sets based on the α plane representation abstract similarity measures are very important concepts in fuzzy sets fss theory there are many different definitions of similarity measures for both type 1 t1 fss and interval type 2 it2 fss in this paper one similarity measure for it2 fss which is extended from the jaccard similarity measure for t1 fss is reviewed then based on the plane representation for a general type 2 gt2 fs this similarity measure is generalized to such t2 fss some examples that demonstrate how to compute the similarity measures for different t2 fss are given

an evolving interval type 2 neurofuzzy inference system and its metacognitive sequential learning algorithm in this paper we propose an evolving interval type 2 neurofuzzy inference system it2fis and its fully sequential learning algorithm it2fis employs interval type 2 fuzzy sets in the antecedent part of each rule and the consequent realizes takagi sugeno kang fuzzy inference mechanism in order to render the inference fast and accurate we propose a data driven interval reduction approach to convert interval type 1 fuzzy set in antecedent to type 1 fuzzy number in the consequent during learning the sequential algorithm learns a sample one by one and only once the it2fis structure evolves automatically and adapts its network parameters using metacognitive learning mechanism concurrently the metacognitive learning regulates the learning process by appropriate selection of learning strategies and helps the proposed it2fis to approximate the input output relationship efficiently an evolving it2fis employing a metacognitive learning algorithm is referred to as mcti2fis performance of metacognitive interval type 2 neurofuzzy inference system mcit2fis is evaluated using a set of benchmark time series problems and is compared with existing type 2 and type 1 fuzzy inference systems finally the performance of the proposed mcit2fis has been evaluated using a practical stock price tracking problem the results clearly highlight that mcit2fis performs better than other existing results in the literature

geometric type 1 and type 2 fuzzy logic systems this paper presents a novel approach to the representation of type 1 and type 2 fuzzy sets utilising computational geometry to achieve this our approach borrows ideas from the field of computational geometry and applies these techniques in the novel setting of fuzzy logic we provide new algorithms for various operations on type 1 and type 2 fuzzy sets and for defuzzification results of experiments indicate that this approach reduces the execution speed of these operations

on the classification of clifford algebras and their relation to spinors in n dimensions a classification of all the clifford algebras is given in terms of kronecker products of the quaternion and dihedral groups the relationship to spinors in n dimensions is explicitly determined we show that the real clifford algebra in minkowski spacetime is distinct from both the algebra of dirac matrices and the algebra of majorana matrices and cannot be realized by the spinor framework the matrix representations of clifford algebras are discussed and are utilized to give a classification of the real forms of lie algebras we are thus able to relate clifford lie and spinor algebras in an intrinsic geometrical setting

footprint of uncertainty for type 2 fuzzy sets in the paper a new and comprehensible definition is proposed for type 2 fuzzy sets t2 fss and the primary and secondary memberships function are defined respectively by using multi valued mapping a new definition and formula for the footprint of uncertainty fou is presented and based on the new definitions the relation between fou and the original definition of t2 fs is discussed finally the partition method of fou is provided to represent the primary membership grade the fou and the interval type 2 fuzzy sets it2 fs when the upper and lower membership functions umf lmf of fou are given

type 2 fuzzy rough sets based on extended t norms a type 2 fuzzy rough set based on extended t norms is investigated from constructive and axiomatic approaches an approximation algorithm of type 2 fuzzy rough sets is listed based on cut sets weak type 2 fuzzy topologies are proposed to investigate the topological properties of type 2 fuzzy rough sets an interval valued type 2 fuzzy rough set is proposed and investigated this paper is devoted to investigate type 2 fuzzy rough sets based on extended t norms with respect to type 2 fuzzy relations with convex normal fuzzy truth values type 2 fuzzy rough sets are proposed on two finite universes of discourse and characterized from both constructive and axiomatic approaches in the constructive approach different classes of type 2 fuzzy rough sets are investigated an approximate algorithm of type 2 fuzzy rough sets based on cut sets is given with numerical examples in the axiomatic approach the axiomatic characterizations of type 2 fuzzy rough approximation operators are studied moreover the topological properties of type 2 fuzzy rough sets are discussed in the end this paper introduces interval valued type 2 fuzzy rough sets and discusses their properties

the case for an interval based representation of linguistic truth abstract this paper develops an interval based approach to the concept of linguistic truth a special purpose interval logic is defined and it is argued that for many applications this logic provides a potentially useful alternative to the conventional fuzzy logic the key idea is to interpret the numerical truth value v p of a proposition p as a degree of belief in the logical certainty of p in which case p is regarded as true for example if v p falls within a certain range say the interval 0 7 1 this leads to a logic which although being only a special case of fuzzy logic appears to be no less linguistically correct and at the same time offers definite advantages in terms of mathematical simplicity and computational speed it is also shown that this same interval logic can be generalized to a lattice based logic having the capacity to accommodate propositions p which employ fuzzy predicates of type 2

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

equalization of nonlinear time varying channels using type 2 fuzzy adaptive filters presents a kind of adaptive filter type 2 fuzzy adaptive filter faf one that is realized using an unnormalized type 2 takagi sugeno kang tsk fuzzy logic system fls we apply this filter to equalization of a nonlinear time varying channel and demonstrate that it can implement the bayesian equalizer for such a channel has a simple structure and provides fast inference a clustering method is used to adaptively design the parameters of the faf two structures are used for the equalizer transversal equalizer te and decision feedback equalizer dfe a decision tree structure is used to implement the decision feedback equalizer in which each leaf of the tree is a type 2 faf this dfe vastly reduces computational complexity as compared to a te simulation results show that equalizers based on type 2 fafs perform much better than nearest neighbor classifiers nnc or equalizers based on type 1 fafs

an enhanced type reduction algorithm for type 2 fuzzy sets karnik and mendel proposed an algorithm to compute the centroid of an interval type 2 fuzzy set efficiently based on this algorithm liu developed a centroid type reduction strategy to carry out type reduction for type 2 fuzzy sets a type 2 fuzzy set is decomposed into a collection of interval type 2 fuzzy sets by cuts then the karnik mendel algorithm is called for each interval type 2 fuzzy set iteratively however the initialization of the switch point in each application of the karnik mendel algorithm is not a good one in this paper we present an improvement to liu s algorithm we employ the previously obtained result to construct the starting values in the current application of the karnik mendel algorithm convergence in each iteration except the first one can then speed up and type reduction for type 2 fuzzy sets can be carried out faster the efficiency of the improved algorithm is analyzed mathematically and demonstrated by experimental results

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

state based decentralized diagnosis of bi fuzzy discrete event systems recently we have introduced bi fuzzy discrete event systems bfdess and established their supervisory control theory under full observations and partial observations respectively compared with the models of discrete event systems and fuzzy discrete event systems bfdess are more satisfactory models to characterize some high uncertainty systems in this paper we study the fault diagnosis problems of bfdess first the framework of state based decentralized diagnosis is introduced then the notion of state based codiagnosability is defined and a necessary and sufficient condition for state based codiagnosability is provided in particular a polynomial time algorithm for state based codiagnosability verification and an online decentralized diagnosis method are presented in addition a compact state based decentralized diagnosis framework is introduced and the notion of state based jointly diagnosability and its verification algorithm are proposed finally the implication relationships between the two diagnosabilities are investigated

multiobjective optimization and comparison of nonsingleton type 1 and singleton interval type 2 fuzzy logic systems singleton interval type 2 fuzzy logic systems flss have been widely applied in several real world applications where it was shown that the singleton interval type 2 flss outperform their singleton type 1 counterparts in applications with high uncertainty levels however one of the main criticisms of singleton interval type 2 flss is the fact that they outperform singleton type 1 flss solely based on their use of extra degrees of freedom extra parameters and that type 1 flss with a sufficiently large number of parameters may provide the same performance as interval type 2 flss in addition most works on type 2 flss only compare their results with singleton type 1 flss but fail to consider nonsingleton type 1 systems in this paper we aim to directly address and investigate this criticism in order to do so we will perform a comparative study between optimized singleton type 1 nonsingleton type 1 and singleton interval type 2 flss under the presence of noise we will also present a multiobjective evolutionary algorithm moea for the optimization of singleton type 1 nonsingleton type 1 and singleton interval type 2 fuzzy systems for function approximation problems the moea will aim to satisfy two objectives to maximize the accuracy of the fls and minimize the number of rules in the fls thus improving its interpretability furthermore we will present a methodology to obtain optimal consequents for the flss hence this paper has two main contributions first it provides a common methodology to learn the three types of flss i e singleton type 1 nonsingleton type 1 and singleton interval type 2 flss from data samples the second contribution is the creation of a common framework for the comparison of type 1 and type 2 flss that allows us to address the aforementioned criticism we provide details of a series of experiments and include statistical analysis showing that the type 2 fls is able to handle higher levels of noise than its nonsingleton and singleton type 1 counterparts

application of fuzzy sets of type 2 to the solution of fuzzy equation systems abstract the present paper continues the authors investigation on fuzzy equation systems with tolerances we consider the situation that the outputs of an object are modelled by fuzzy sets of type 2 for special membership functions this problem can be attributed to a problem with tolerances moreover this concept enables us to define solutions for fuzzy outputs of type 1 when classical solution concepts may fail

an interval type 2 neural fuzzy system for online system identification and feature elimination we propose an integrated mechanism for discarding derogatory features and extraction of fuzzy rules based on an interval type 2 neural fuzzy system nfs in fact it is a more general scheme that can discard bad features irrelevant antecedent clauses and even irrelevant rules high dimensional input variable and a large number of rules not only enhance the computational complexity of nfss but also reduce their interpretability therefore a mechanism for simultaneous extraction of fuzzy rules and reducing the impact of or eliminating the inferior features is necessary the proposed approach namely an interval type 2 neural fuzzy system for online system identification and feature elimination it2nfs sife uses type 2 fuzzy sets to model uncertainties associated with information and data in designing the knowledge base the consequent part of the it2nfs sife is of takagi sugeno kang type with interval weights the it2nfs sife possesses a self evolving property that can automatically generate fuzzy rules the poor features can be discarded through the concept of a membership modulator the antecedent and modulator weights are learned using a gradient descent algorithm the consequent part weights are tuned via the rule ordered kalman filter algorithm to enhance learning effectiveness simulation results show that it2nfs sife not only simplifies the system architecture by eliminating derogatory irrelevant antecedent clauses rules and features but also maintains excellent performance

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

design of a new adaptive neuro fuzzy inference system based on a solution for clustering in a data potential field in this study we propose a new method for building adaptive neuro fuzzy inference systems anfis via datasets in order to improve the performance of conventional anfis to handle noisy data we focus on ameliorating the cluster data space established from a given dataset to achieve this we propose a weighted clustering process in the joint input output data space thus during the clustering process the cluster with the smallest potential distance which is a combination of the euclidean distance and the size of the clusters has priority when obtaining the surveyed sample based on this principle we formulate a new algorithm for synthesizing an anfis via the proposed data potential field called anfis pf which has the following features it establishes a data potential field that covers the entire initial data space a cluster data space is built based on the generated data potential field and the anfis is synthesized using this cluster data space finally we performed experiments using datasets with and without noise to demonstrate the effectiveness of the proposed method in several applications including dynamic response noisy datasets obtained from a magnetorheological damper

on clarifying some definitions and notations used for type 2 fuzzy sets as well as some recommended changes this paper is about some important changes in type 2 fuzzy set t2 fs definitions and notations that have occurred during the past 16 years it summarizes the evolution of how the primary membership jx has been used in both the mathematical descriptions of a t2 fs and its footprint of uncertainty fou discusses notational problems associated with the secondary membership function explains why and when one should distinguish between the fou and the domain of uncertainty dou explains why no errors have been introduced into t2 fs computations because of notation and it provides recommendations notational changes that can be used by all authors

monotone centroid flow algorithm for type reduction of general type 2 fuzzy sets recently type 2 fuzzy logic systems t2 flss have received increased research attention due to their potential to model and cope with the dynamic uncertainties ubiquitous in many engineering applications however because of the complex nature and the computational intensity of the inference process only the constrained version of t2 flss i e the interval t2 flss was typically used fortunately the very recently introduced concepts of planes and zslices allow for efficient representation as well as a computationally fast inference process with general t2 gt2 flss this paper addresses the type reduction phase in gt2 flss using gt2 fuzzy sets fss represented in the plane framework the monotone property of centroids of a set of planes is derived and leveraged toward developing a simple to implement but fast algorithm for type reduction of gt2 fss i e the monotone centroid flow mcf algorithm when compared with the centroid flow cf algorithm which was previously developed by zhai and mendel the mcf algorithm features the following advantages 1 the mcf algorithm computes numerically identical centroid as the karnik mendel km iterative algorithms unlike the approximated centroid which is obtained with the cf algorithm 2 the mcf algorithm is faster than the cf algorithm as well as the independent application of the km algorithms 3 the mcf algorithm is easy to implement unlike the cf algorithm which requires computation of the derivatives of the centroid and 4 the mcf algorithm completely eliminates the need to apply the km iterative procedure to any planes of the gt2 fs the performance of the algorithm is presented on benchmark problems and compared with other type reduction techniques that are available in the literature

t2fela type 2 fuzzy extreme learning algorithm for fast training of interval type 2 tsk fuzzy logic system a challenge in modeling type 2 fuzzy logic systems is the development of efficient learning algorithms to cope with the ever increasing size of real world data sets in this paper the extreme learning strategy is introduced to develop a fast training algorithm for interval type 2 takagi sugeno kang fuzzy logic systems the proposed algorithm called type 2 fuzzy extreme learning algorithm t2fela has two distinctive characteristics first the parameters of the antecedents are randomly generated and parameters of the consequents are obtained by a fast learning method according to the extreme learning mechanism in addition because the obtained parameters are optimal in the sense of minimizing the norm the resulting fuzzy systems exhibit better generalization performance the experimental results clearly demonstrate that the training speed of the proposed t2fela algorithm is superior to that of the existing state of the art algorithms the proposed algorithm also shows competitive performance in generalization abilities

long term learning for type 2 neural fuzzy systems abstract the development of a new long term learning framework for interval valued neural fuzzy systems is presented for the first time in this article the need for such a framework is twofold to address continuous batch learning of data sets and to take advantage the extra degree of freedom that type 2 fuzzy logic systems offer for better model predictive ability the presented long term learning framework uses principles of granular computing grc to capture information knowledge from raw data in the form of interval valued sets in order to build a computational mechanism that has the ability to adapt to new information in an additive and long term learning fashion the latter is to accommodate new input output mappings and new classes of data without significantly disturbing existing input output mappings therefore maintaining existing performance while creating and integrating new knowledge rules this is achieved via an iterative algorithmic process which involves a two step operation iterative rule base growth capturing new knowledge and iterative rule base pruning removing redundant knowledge for type 2 rules the two step operation helps create a growing but sustainable model structure the performance of the proposed system is demonstrated using a number of well known non linear benchmark functions as well as a highly nonlinear multivariate real industrial case study simulation results show that the performance of the original model structure is maintained and it is comparable to the updated model s performance following the incremental learning routine the study is concluded by evaluating the performance of the proposed framework in frequent and consecutive model updates where the balance between model accuracy and complexity is further assessed

on type 2 fuzzy relations and interval valued type 2 fuzzy sets this paper introduces new operations on the algebra of fuzzy truth values extended supremum and extended infimum which are generalizations of the extended operations of maximum and minimum between fuzzy truth values for type 2 fuzzy sets respectively using these new operations the properties of type 2 fuzzy relations are discussed especially the compositions of type 2 fuzzy relations on this basis this paper introduces interval valued type 2 fuzzy sets and interval valued type 2 fuzzy relations and discusses their properties

a recurrent self evolving interval type 2 fuzzy neural network for dynamic system processing this paper proposes a recurrent self evolving interval type 2 fuzzy neural network rseit2fnn for dynamic system processing an rseit2fnn incorporates type 2 fuzzy sets in a recurrent neural fuzzy system in order to increase the noise resistance of a system the antecedent parts in each recurrent fuzzy rule in the rseit2fnn are interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang tsk type with interval weights the antecedent part of rseit2fnn forms a local internal feedback loop by feeding the rule firing strength of each rule back to itself the tsk type consequent part is a linear model of exogenous inputs the rseit2fnn initially contains no rules all rules are learned online via structure and parameter learning the structure learning uses online type 2 fuzzy clustering for the parameter learning the consequent part parameters are tuned by a rule ordered kalman filter algorithm to improve learning performance the antecedent type 2 fuzzy sets and internal feedback loop weights are learned by a gradient descent algorithm the rseit2fnn is applied to simulations of dynamic system identifications and chaotic signal prediction under both noise free and noisy conditions comparisons with type 1 recurrent fuzzy neural networks validate the performance of the rseit2fnn

modified ahp for gene selection and cancer classification using type 2 fuzzy logic this paper proposes a modification to the analytic hierarchy process ahp to select the most informative genes that serve as inputs to an interval type 2 fuzzy logic system it2fls for cancer classification unlike the conventional ahp the modified ahp allows us to process quantitative factors that are ranking outcomes of individual gene selection methods including t test entropy receiver operating characteristic curve wilcoxon test and signal to noise ratio the it2fls is introduced for the classification task due to its great ability for handling nonlinear noisy and outlier data which are common problems in cancer microarray gene expression profiles an unsupervised learning strategy using the fuzzy c means clustering is employed to initialize parameters of the it2fls other classifiers such as multilayer perceptron network support vector machine and fuzzy artmap are also implemented for comparisons experiments are carried out on three well known microarray datasets diffuse large b cell lymphoma leukemia cancer and prostate rather than the traditional cross validation leave one out cross validation strategy is applied for the experiments results demonstrate the performance dominance of the it2fls against the competing classifiers more noticeably the modified ahp improves the classification performance not only of the it2fls but of all other classifiers as well accordingly the proposed combination between the modified ahp and it2fls is a powerful tool for cancer classification and can be implemented as a real clinical decision support system that is useful for medical practitioners

a mutually recurrent interval type 2 neural fuzzy system mrit2nfs with self evolving structure and parameters in this paper a mutually recurrent interval type 2 neural fuzzy system mrit2nfs is proposed for the identification of nonlinear and time varying systems the mrit2nfs uses type 2 fuzzy sets in order to enhance noise tolerance of the system in the mrit2nfs the antecedent part of each recurrent fuzzy rule is defined using interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang type with interval weights the antecedent part of mrit2nfs forms a local internal feedback and interaction loop by feeding the rule firing strength of each rule to others including itself the consequent is a linear combination of exogenous input variables the learning of mrit2nfs starts with an empty rule base and all rules are learned online via structure and parameter learning the structure learning of mrit2nfs uses online type 2 fuzzy clustering for parameter learning the consequent part parameters are tuned by rule ordered kalman filter algorithm to reinforce parameter learning ability the type 2 fuzzy sets in the antecedent and weights representing the mutual feedback are learned by the gradient descent algorithm after the training a weight elimination scheme eliminates feedback connections that do not have much effect on the network behavior this method can efficiently remove redundant recurrence and interaction weights finally the mrit2nfs is used for system identification under both noise free and noisy environments for this we consider both time series prediction and nonlinear plant modeling compared with type 1 recurrent fuzzy neural networks simulation results show that our approach produces smaller root mean squared errors using the same number of iterations

a triangular type 2 multi objective linear programming model and a solution strategy abstract we consider a multi objective linear programming model with type 2 fuzzy objectives the considered model has the flexibility for the user to specify the more general membership functions for objectives to reflect the inherent fuzziness while being simple and practical we develop two solution strategies with reasonable computing costs the additional cost as compared to the type 1 fuzzy model is indeed insignificant these two algorithms compute pareto optimal solutions of the type 2 problems one being based on a maxmin approach and the other on aggregating the objectives finally applying the proposed algorithms we work out two illustrative examples

on constructing parsimonious type 2 fuzzy logic systems via influential rule selection type 2 fuzzy systems are increasing in popularity and there are many examples of successful applications while many techniques have been proposed for creating parsimonious type 1 fuzzy systems there is a lack of such techniques for type 2 systems the essential problem is to reduce the number of rules while maintaining the system s approximation performance in this paper four novel indexes for ranking the relative contribution of type 2 fuzzy rules are proposed which are termed r values c values omega1 values and omega2 values the r values of type 2 fuzzy rules are obtained by applying a qr decomposition pivoting algorithm to the firing strength matrices of the trained fuzzy model the c values rank rules based on the effects of rule consequents while the omega1 values and omega2 values consider both the rule base structure via firing strength matrices and the output contribution of fuzzy rule consequents two procedures for utilizing these indexes in fuzzy rule selection termed ldquoforward selectionrdquo and ldquobackward eliminationrdquo are described experiments are presented which demonstrate that by using the proposed methodology the most influential type 2 fuzzy rules can be effectively retained in order to construct parsimonious type 2 fuzzy models

a fast geometric method for defuzzification of type 2 fuzzy sets generalized type 2 fuzzy logic systems cannot currently be used for practical problems because the amount of computation required to defuzzify a generalized type 2 fuzzy set is too large this paper presents a new method for defuzzifing a type 2 fuzzy set the new much faster technique is based on geometric representations and operations the results of a real world example contained in this paper show this new approach to be over 200 000 times faster than type reduction we present a new method for assessing the accuracy of the membership function of a type 2 fuzzy set this method is used to show that the new representation used by the defuzzifier is not detrimental to the accuracy of the set we also discuss the differences between the new approach and type reduction identifying the origin of this massive improvement in execution speed

multivariate modeling and type 2 fuzzy sets this paper explores the link between type 2 fuzzy sets and multivariate modeling elements of a space x are treated as observations fuzzily associated with values in a multivariate feature space a category or class is likewise treated as a fuzzy allocation of feature values possibly dependent on values in x we observe that a type 2 fuzzy set on x generated by these two fuzzy allocations captures imprecision in the class definition and imprecision in the observations in practice many type 2 fuzzy sets are in fact generated in this way and can therefore be interpreted as the output of a classification task we then show that an arbitrary type 2 fuzzy set can be so constructed by taking as a feature space a set of membership functions on x this construction presents a new perspective on the representation theorem of mendel and john the multivariate modeling underpinning the type 2 fuzzy sets can also constrain realizable forms of membership functions because averaging operators such as centroid and subsethood on type 2 fuzzy sets involve a search for optima over membership functions constraining this search can make computation easier and tighten the results we demonstrate how the construction can be used to combine representations of concepts and how it therefore provides an additional tool alongside standard operations such as intersection and subsethood for concept fusion and computing with words

analysis of data generated from multidimensional type 1 and type 2 fuzzy membership functions due to the numerous applications that utilize different types of fuzzy membership functions mfs it may sometimes be difficult to choose an appropriate mf for a particular application in this paper we establish preliminary guidelines to direct this selection by proposing a three stage method in the forward stage different mfs such as crisp mfs type 1 t1 fuzzy mfs and type 2 t2 fuzzy mfs are generated from multidimensional data sets next in the reverse stage data is generated back from these mfs by considering different bin sizes in doing so various data sets may be generated for different applications which require fuzzy data finally for the similarity analysis stage we propose an iterative algorithm that makes use of the results of wilcoxon signed rank wsr and wilcoxon rank sum wrs tests to compare the original data and the generated data from the results of these tests recommendations concerning the suitability of mfs for a specific application may be suggested by observing the accuracy of representation and the requirements of the application with this analysis the objective is to gain insight on when t2 fuzzy sets may be considered to outperform t1 fuzzy sets and vice versa several examples are provided using synthetic and real data to validate the iterative algorithm for data sets in various dimensions

interaction measures for control configuration selection based on interval type 2 takagi sugeno fuzzy model interaction measure determines decentralized and sparse control configurations for a multivariable process control this paper investigates interval type 2 takagi sugeno fuzzy it2tsf model based interaction measures using two different criteria one is controllability and observability gramians the other is relative normalized gain array rnga the main contributions are first a data driven it2tsf modeling method is introduced second explicit formulas to execute the two measures based on it2tsf models are given third two interaction indexes are defined from rnga to select sparse control configuration fourth the calculations to derive sensitivities of the two measures with respect to parametric variations in the it2tsf models are developed and fifth the discussion to compare the two measures is presented three multivariable processes are used as examples to show that the results calculated from it2tsf models are more accurate than that from their type 1 counterparts and compared to gramian based measure rnga selects more reasonable control configurations and is more robust to the parametric uncertainties

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

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

an interval type 2 fuzzy neural network with support vector regression for noisy regression problems this paper proposes an interval type 2 fuzzy neural network with support vector regression it2fnn svr for noisy regression problems the antecedent part in each fuzzy rule of an it2fnn svr uses interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang tsk type the use of interval type 2 fuzzy sets helps improve the network s noise resistance the network inputs may be numerical values or type 1 fuzzy sets with the latter being used for further improvements in robustness it2fnn svr learning consists of both structure learning and parameter learning the structure learning algorithm is responsible for online rule generation the parameters are optimized for structural risk minimization using a two phase linear svr algorithm in order to endow the network with high generalization ability it2fnn svr performance is verified through comparisons with type 1 and type 2 fuzzy logic systems and other regression models on noisy regression problems

critique of footprint of uncertainty for type 2 fuzzy sets 9 abstract in this letter we provide some critiques of the 2014 information sciences paper by mo et al 9 and point out and correct six erroneous statements and false impressions that have been made in this paper

super exponential convergence of the karnik mendel algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval t2 fs is an important operation in a type 2 fuzzy logic system where it is called type reduction but it is also a potentially time consuming operation the karnik mendel km iterative algorithms are widely used for doing this in this paper we prove that these algorithms converge monotonically and super exponentially fast both properties are highly desirable for iterative algorithms and explain why in practice the km algorithms have been observed to converge very fast thereby making them very practical to use

study on enhanced karnik mendel algorithms initialization explanations and computation improvements computing the centroid of an interval type 2 fuzzy set is an important operation in a type 2 fuzzy logic system and is usually implemented by karnik mendel km iterative algorithms by connecting km algorithms and continuous km algorithms together this paper gives theoretical explanations on the initialization methods of km and enhanced karnik mendel ekm algorithms proposes exact methods for centroid computation of an interval type 2 fuzzy set and extends the enhanced karnik mendel ekm algorithms to three different forms of weighted ekm wekm algorithms it shows that ekm algorithms become a special case of the wekm algorithms when the weights of the latter are constant value it also shows that in general the weighted ekm algorithms have smaller absolute error and faster convergence speed than the ekm algorithms which make them very attractive for real time applications of fuzzy logic system four numerical examples are used to illustrate and analyze the performance of wekm algorithms

general type 2 fuzzy logic systems made simple a tutorial the purpose of this tutorial paper is to make general type 2 fuzzy logic systems gt2 flss more accessible to fuzzy logic researchers and practitioners and to expedite their research designs and use to accomplish this the paper 1 explains four different mathematical representations for general type 2 fuzzy sets gt2 fss 2 demonstrates that for the optimal design of a gt2 fls one should use the vertical slice representation of its gt2 fss because it is the only one of the four mathematical representations that is parsimonious 3 shows how to obtain set theoretic and other operations for gt2 fss using type 1 t1 fs mathematics cuts play a central role 4 reviews mamdani and tsk interval type 2 it2 flss so that their mathematical operations can be easily used in a gt2 fls 5 provides all of the formulas that describe both mamdani and tsk gt2 flss 6 explains why center of sets type reduction should be favored for a gt2 fls over centroid type reduction 7 provides three simplified gt2 flss two are for mamdani gt2 flss and one is for a tsk gt2 fls all of which bypass type reduction and are generalizations from their it2 fls counterparts to gt2 flss 8 explains why gradient based optimization should not be used to optimally design a gt2 fls 9 explains how derivative free optimization algorithms can be used to optimally design a gt2 fls and 10 provides a three step approach for optimally designing flss in a progressive manner from t1 to it2 to gt2 each of which uses a quantum particle swarm optimization algorithm by virtue of which the performance for the it2 fls cannot be worse than that of the t1 fls and the performance for the gt2 fls cannot be worse than that of the it2 fls

generalized extended fuzzy implications this paper is devoted to investigate generalized extended fuzzy implications in accordance with the generalized extension principle the algebraic properties of generalized extended fuzzy implications are discussed on different algebras of fuzzy truth values on this basis fuzzy valued fuzzy implications are constructed from arbitrary fuzzy implications and t norms moreover a fuzzy valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t norm

a central limit theorem for lipschitz killing curvatures of gaussian excursions abstract this paper studies the excursion set of a real stationary isotropic gaussian random field above a fixed level we show that the standardized lipschitz killing curvatures of the intersection of the excursion set with a window converges in distribution to a normal distribution as the window grows to the d dimensional euclidean space moreover a lower bound for the asymptotic variance is derived

answering an open problem on t norms for type 2 fuzzy sets abstract this paper proves that a binary operation on 0 1 ensuring that the binary operation is a t norm or is a t conorm is a t norm where and are special convolution operations defined by f g x sup f y g z y z x f g x sup f y g z y z x for any f g map 0 1 0 1 where and are a continuous t norm and a continuous t conorm on 0 1 answering negatively an open problem posed in 8 besides some characteristics of t norm and t conorm are obtained in terms of the binary operations and

type 2 implications on non interactive fuzzy truth values in this paper we consider algebras of fuzzy truth values equipped with an implication operation generalizing the definition of type 1 fuzzy implications we discuss extended fuzzy s implications and coimplications and also extended residual implications and coimplications in particular to the extended residual of minimum we study the algebraic properties of these operations and their relationships to extended t norms and t conorms in particular to meet and join these investigations are intended to provide a theoretical background for type 2 approximate reasoning applications

uncertain fuzzy clustering interval type 2 fuzzy approach to c means in many pattern recognition applications it may be impossible in most cases to obtain perfect knowledge or information for a given pattern set uncertain information can create imperfect expressions for pattern sets in various pattern recognition algorithms therefore various types of uncertainty may be taken into account when performing several pattern recognition methods when one performs clustering with fuzzy sets fuzzy membership values express assignment availability of patterns for clusters however when one assigns fuzzy memberships to a pattern set imperfect information for a pattern set involves uncertainty which exist in the various parameters that are used in fuzzy membership assignment when one encounters fuzzy clustering fuzzy membership design includes various uncertainties e g distance measure fuzzifier prototypes etc in this paper we focus on the uncertainty associated with the fuzzifier parameter m that controls the amount of fuzziness of the final c partition in the fuzzy c means fcm algorithm to design and manage uncertainty for fuzzifier m we extend a pattern set to interval type 2 fuzzy sets using two fuzzifiers m1 and m2 which creates a footprint of uncertainty fou for the fuzzifier m then we incorporate this interval type 2 fuzzy set into fcm to observe the effect of managing uncertainty from the two fuzzifiers we also provide some solutions to type reduction and defuzzification i e cluster center updating and hard partitioning in fcm several experimental results are given to show the validity of our method

on fuzzy valued operations and fuzzy valued fuzzy sets this paper is devoted to investigate fuzzy valued operations and the lattice structures of the algebra of fuzzy values the algebra of fuzzy values is proven to be a complete completely distributive lattice and the list of all join irreducible elements is provided fuzzy valued t norms and t conorms are induced by arbitrary t norms and t conorms regardless of their continuity respectively moreover fuzzy valued fuzzy implications are constructed from fuzzy valued t norms induced by left continuous t norms and residuated lattices are constructed on the algebra of fuzzy values as a consequence fuzzy valued fuzzy rough sets and fuzzy valued fuzzy complete lattices are proposed and investigated in fuzzy valued fuzzy sets

computing derivatives in interval type 2 fuzzy logic systems this paper makes type 2 fuzzy logic systems much more accessible to fuzzy logic system designers because it provides mathematical formulas and computational flowcharts for computing the derivatives that are needed to implement steepest descent parameter tuning algorithms for such systems it explains why computing such derivatives is much more challenging than it is for a type 1 fuzzy logic system it provides derivative calculations that are applicable to any kind of type 2 membership functions since the calculations are performed without prespecifying the nature of those membership functions some calculations are then illustrated for specific type 2 membership functions

corrections to an automatic approach for learning and tuning gaussian interval type 2 fuzzy membership functions applied to lung cad classification system the potential of type 2 fuzzy sets to manage high levels of uncertainty in the subjective knowledge of experts or of numerical information has focused on control and pattern classification systems in recent years one of the main challenges in designing a type 2 fuzzy logic system fls is how to estimate the parameters of the type 2 fuzzy membership function t2mf and the footprint of uncertainty fou from imperfect and noisy datasets this paper presents an automatic approach to learn and tune gaussian interval type 2 membership functions it2mfs with application to multidimensional pattern classification problems t2mfs and their fous are tuned according to the uncertainties in the training dataset by a combination of genetic algorithm ga and cross validation techniques in our ga based approach the structure of the chromosome has fewer genes than other ga methods and chromosome initialization is more precise the proposed approach addresses the application of the interval type 2 fuzzy logic system it2fls for the problem of nodule classification in a lung computer aided detection system the designed it2fls is compared with its type 1 fuzzy logic system t1fls counterpart the results demonstrate that the it2fls outperforms the t1fls by more than 30 in terms of classification accuracy

mpeg vbr video traffic modeling and classification using fuzzy technique we present an approach for mpeg variable bit rate vbr video modeling and classification using fuzzy techniques we demonstrate that a type 2 fuzzy membership function i e a gaussian mf with uncertain variance is most appropriate to model the log value of i p b frame sizes in mpeg vbr video the fuzzy c means fcm method is used to obtain the mean and standard deviation std of t p b frame sizes when the frame category is unknown we propose to use type 2 fuzzy logic classifiers flcs to classify video traffic using compressed data five fuzzy classifiers and a bayesian classifier are designed for video traffic classification and the fuzzy classifiers are compared against the bayesian classifier simulation results show that a type 2 fuzzy classifier in which the input is modeled as a type 2 fuzzy set and antecedent membership functions are modeled as type 2 fuzzy sets performs the best of the five classifiers when the testing video product is not included in the training products and a steepest descent algorithm is used to tune its parameters

adaptive hybrid type 2 intelligent sliding mode control for uncertain nonlinear multivariable dynamical systems a new adaptive hybrid interval type 2 fuzzy neural network fnn controller incorporating sliding mode and lyapunov synthesis approaches is proposed in this paper to handle the training data corrupted by noise or rule uncertainties for a class of uncertain nonlinear multivariable dynamic systems the hybrid adaptive fnn controller the free parameters of which can be tuned on line by an output feedback control law and adaptive laws is a combination of interval type 2 indirect and direct adaptive fnn controllers to meet the requirement of sufficient reach for the sliding mode control a weighting factor which can be adjusted based on the trade off between plant knowledge and control knowledge is included when combining the control efforts of the indirect adaptive fnn controller and the direct adaptive fnn controller the overall adaptive control scheme guarantees the global stability of the resulting closed loop system in the sense that all signals involved are uniformly bounded the mass spring damper nonlinear system is fully illustrated to track sinusoidal signals the resulting adaptive hybrid interval type 2 fnn control system shows better performance than the adaptive hybrid type 1 fnn control system it reduces both the tracking error and the control effort and it is more flexible in the design process

on the calculation of extended max and min operations between convex fuzzy sets of the real line in this paper we will identify the sets of so called sub and pseudo highest intersection points of convex fuzzy sets of the real line and will explore their properties based on the properties of these sets an algorithm for calculating extended max and min operations between two or more convex fuzzy sets of the real line with general membership functions not necessarily continuous is proposed

extended triangular norms the paper is devoted to classical t norms extended to operations on fuzzy quantities in accordance with the generalized zadeh extension principle such extended t norms are used for calculating intersection of type 2 fuzzy sets analytical expressions for membership functions of some extended t norms are derived assuming special classes of fuzzy quantities i e fuzzy truth intervals or fuzzy truth numbers the possibility of applying these results in the construction of type 2 adaptive network fuzzy inference systems is illustrated on several examples

type 2 fuzzy hidden markov models and their application to speech recognition this paper presents an extension of hidden markov models hmms based on the type 2 t2 fuzzy set fs referred to as type 2 fuzzy hmms t2 fhmms membership functions mfs of t2 fss are three dimensional and this new third dimension offers additional degrees of freedom to evaluate the hmms fuzziness therefore t2 fhmms are able to handle both random and fuzzy uncertainties existing universally in the sequential data we derive the t2 fuzzy forward backward algorithm and viterbi algorithm using t2 fs operations in order to investigate the effectiveness of t2 fhmms we apply them to phoneme classification and recognition on the timit speech database experimental results show that t2 fhmms can effectively handle noise and dialect uncertainties in speech signals besides a better classification performance than the classical hmms

an interval type 2 neural fuzzy classifier learned through soft margin minimization and its human posture classification application this paper proposes an interval type 2 neural fuzzy classifier learned through soft margin minimization it2nfc smm and applies it to human body posture classification the it2nfc smm consists of interval type 2 zero order takagi sugeno t s fuzzy rules established through online structure learning the antecedent part of the it2nfc smm uses interval type 2 fuzzy sets to decrease the number of rules and manage noisy data for parameter learning the consequent parameters are learned through a linear support vector machine svm for soft margin minimization to improve the generalization ability the proposed svm based learning addresses the problem that the orders of the fuzzy rules in computing the outputs of an interval type 2 fuzzy system depend on the consequent values that are unknown in advance to address this problem the it2nfc smm uses weighted bound set boundaries to simplify the type reduction operation and a novel crisp to interval linear svm learning algorithm based on the soft margin minimization the antecedent parameters are tuned using the gradient descent algorithm the it2nfc smm is applied to a vision based human body posture classification system the system uses two cameras and novel classification features extracted from a silhouette of the human body to classify the four postures of standing bending sitting and lying the classification performance of the it2nfc smm is verified through results in clean and noisy classification examples and through the posture classification problem as well as through comparisons with various type 1 and type 2 fuzzy classifiers the overall result shows that the it2nfc smm achieves higher classification rates with a smaller or similar model size than the classifiers used for comparison especially for noisy classification problems

dirac general covariance and tetrads i clifford and lie bundles and torsion the isomorphic map of clifford and lie bundles to arbitrary coordinate atlases using a global orthonormal tetrad field on a parallelizable space time is used to construct a fully covariant dirac spinor theory the klein gordon equation exhibits a natural spin torsion coupling of the einstein cartan form the torsion coming from the tetrad field the tetrad connection coefficients are explicitly derived in addition to their relationship to the usual levi civita coefficients various topological conditions for vanishing torsion are given the dirac and adjoint dirac equations are obtained from a simple lagrangian and the structure of the adjoint equation is discussed

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

alpha plane representation for type 2 fuzzy sets theory and applications this paper 1 reviews the alpha plane representation of a type 2 fuzzy set t2 fs which is a representation that is comparable to the alpha cut representation of a type 1 fs t1 fs and is useful for both theoretical and computational studies of and for t2 fss 2 proves that set theoretic operations for t2 fss can be computed using very simple alpha plane computations that are the set theoretic operations for interval t2 it2 fss 3 reviews how the centroid of a t2 fs can be computed using alpha plane computations that are also very simple because they can be performed using existing karnik mendel algorithms that are applied to each alpha plane 4 shows how many theoretically based geometrical properties can be obtained about the centroid even before the centroid is computed 5 provides examples that show that the mean value defuzzified value of the centroid can often be approximated by using the centroids of only 0 and 1 alpha planes of a t2 fs 6 examines a triangle quasi t2 fuzzy logic system q t2 fls whose secondary membership functions are triangles and for which all calculations use existing t1 or it2 fs mathematics and hence they may be a good next step in the hierarchy of flss from t1 to it2 to t2 and 7 compares t1 it2 and triangle q t2 flss to forecast noise corrupted measurements of a chaotic mackey glass time series

experimental study of intelligent controllers under uncertainty using type 1 and type 2 fuzzy logic uncertainty is an inherent part in control systems used in real world applications the use of new methods for handling incomplete information is of fundamental importance type 1 fuzzy sets used in conventional fuzzy systems cannot fully handle the uncertainties present in control systems type 2 fuzzy sets that are used in type 2 fuzzy systems can handle such uncertainties in a better way because they provide us with more parameters and more design degrees of freedom this paper deals with the design of control systems using type 2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements environmental noise etc the experimental results are divided in two classes in the first class simulations of a feedback control system for a non linear plant using type 1 and type 2 fuzzy logic controllers are presented a comparative analysis of the systems response in both cases was performed with and without the presence of uncertainty for the second class a non linear identification problem for time series prediction is presented based on the experimental results the conclusion is that the best results are obtained using type 2 fuzzy systems

a fuzzy system constructed by rule generation and iterative linear svr for antecedent and consequent parameter optimization this paper proposes a new fuzzy regression model i e the fuzzy system constructed by rule generation and iterative linear support vector regression fs rglsvr for structural risk minimization the fs rglsvr is composed of takagi sugeno ts type fuzzy if then rules these rules are automatically constructed by a self splitting rule generation algorithm that introduces the self splitting technique to the k means clustering algorithm this new algorithm regards a cluster as a fuzzy rule where no preassignment of the cluster rule number is necessary the cost function for parameter learning is defined based on structural risk instead of empirical risk minimization in order to achieve generalizability tuning all of the free parameters in the fs rglsvr using linear support vector regression svr is proposed to minimize the cost function each of the consequent and antecedent part parameters is expressed as a linear combination coefficient in a transformed input space so that the linear svr is applicable this paper introduces iterative linear svr to tune antecedent and consequent parameters this paper demonstrates the capabilities of fs rglsvr by two simulated and four practical regression examples comparisons with fuzzy systems with different types of learning algorithms verify the performance of the fs rglsvr

simplified interval type 2 fuzzy neural networks this paper describes a self evolving interval type 2 fuzzy neural network fnn for various applications as type 1 fuzzy systems cannot effectively handle uncertainties in information within the knowledge base we propose a simple interval type 2 fnn which uses interval type 2 fuzzy sets in the premise and the takagi sugeno kang tsk type in the consequent of the fuzzy rule the tsk type consequent of fuzzy rule is a linear combination of exogenous input variables given an initially empty the rule base all rules are generated with on line type 2 fuzzy clustering instead of the time consuming k m iterative procedure the design factors ql and qr are learned to adaptively adjust the upper and lower positions on the left and right limit outputs using the parameter update rule based on a gradient descent algorithm simulation results demonstrate that our approach yields fewer test errors and less computational complexity than other type 2 fnns