a stability result for linear parameter varying systems abstract a negative real part eigenvalue assumption on a linear parameter varying system implies exponential stability under a condition that includes either slowly varying parameters or slowly accelerating parameters

optimality tests for the fuzzy c means algorithm abstract because of lack of convexity of the fuzzy c means fcm objective functionals the fcm algorithm may converge to a local minimizer or a saddle point in this paper we present a new scheme for testing the optimality of the fixed points of the fcm algorithm one which needs much less computation than kim et al s scheme pattern recognition 21 651 663 1988 we also point out that kim et al s scheme is partly incorrect and propose a repair of it numerical experiments are presented for several sets of data they compare kim et al s scheme the repaired scheme and our scheme

alternative c means clustering algorithms abstract in this paper we propose a new metric to replace the euclidean norm in c means clustering procedures on the basis of the robust statistic and the influence function we claim that the proposed new metric is more robust than the euclidean norm we then create two new clustering methods called the alternative hard c means ahcm and alternative fuzzy c means afcm clustering algorithms these alternative types of c means clustering have more robustness than c means clustering numerical results show that ahcm has better performance than hcm and afcm is better than fcm we recommend afcm for use in cluster analysis recently this afcm algorithm has successfully been used in segmenting the magnetic resonance image of ophthalmology to differentiate the abnormal tissues from the normal tissues

singularity conditions for the non existence of a common quadratic lyapunov function for pairs of third order linear time invariant dynamic systems abstract the condition that a finite collection of stable matrices a 1 a m has no common quadratic lyapunov function cqlf is formulated as a hierarchy of singularity conditions for block matrices involving a number of unknown parameters these conditions are applied to the case of two stable 3 3 matrices where they are used to derive necessary and sufficient conditions for the non existence of a cqlf

a fuzzy algorithm for color quantization of images in this paper we review a number of techniques for fuzzy color quantization we show that the fuzzy membership paradigm is particularly suited to color quantization where color cluster boundaries are not well defined we propose a new fuzzy color quantization technique which incorporates a term for partition index this algorithm produces better results than fuzzy c means at a reduced computational cost we test the results of the fuzzy algorithms using quality metrics which model the perception of the human visual system and illustrate that substantial quality improvements are achieved

hierarchies consistency analysis for nontransitive problems abstract by assuming different hierarchies or different hierarchy structures for the same problem should yield the same aggregated performance index an approach can be developed for evaluating the performance of multiple level and multiple criteria problems this proposed approach has the advantage that it is completely based on a rational index and no requirement of transitivity is needed furthermore the analyses of different criteria structures give further insights about the problem being considered the disadvantage is of course we must assume that different hierarchies must yield the same aggregate performance index which is not generally true for all problems under all conditions

estimation of parameters in latent class models using fuzzy clustering algorithms abstract a mixture approach to clustering is an important technique in cluster analysis a mixture of multivariate multinomial distributions is usually used to analyze categorical data with latent class model the parameter estimation is an important step for a mixture distribution described here are four approaches to estimating the parameters of a mixture of multivariate multinomial distributions the first approach is an extended maximum likelihood ml method the second approach is based on the well known expectation maximization em algorithm the third approach is the classification maximum likelihood cml algorithm in this paper we propose a new approach using the so called fuzzy class model and then create the fuzzy classification maximum likelihood fcml approach for categorical data the accuracy robustness and effectiveness of these four types of algorithms for estimating the parameters of multivariate binomial mixtures are compared using real empirical data and samples drawn from the multivariate binomial mixtures of two classes the results show that the proposed fcml algorithm presents better accuracy robustness and effectiveness overall the fcml algorithm has the superiority over the ml em and cml algorithms thus we recommend fcml as another good tool for estimating the parameters of mixture multivariate multinomial models

bias correction fuzzy clustering algorithms fuzzy clustering is generally an extension of hard clustering and it is based on fuzzy membership partitions in fuzzy clustering the fuzzy c means fcm algorithm is the most commonly used clustering method numerous studies have presented various generalizations of the fcm algorithm however the fcm algorithm and its generalizations are usually affected by initializations in this paper we propose a bias correction term with an updating equation to adjust the effects of initializations on fuzzy clustering algorithms we first propose the so called bias correction fuzzy clustering of the generalized fcm algorithm we then construct the bias correction fcm bias correction gustafson and kessel clustering and bias correction inter cluster separation algorithms we compared the proposed bias correction fuzzy clustering algorithms with other fuzzy clustering algorithms by using numerical examples we also applied the bias correction fuzzy clustering algorithms to real data sets the results indicated the superiority and effectiveness of the proposed bias correction fuzzy clustering methods

rank inclusion in criteria hierarchies this paper presents a method called rank inclusion in criteria hierarchies rich for the analysis of incomplete preference information in hierarchical weighting models in rich the decision maker is allowed to specify subsets of attributes which contain the most important attribute or more generally to associate a set of rankings with a given set of attributes such preference statements lead to possibly non convex sets of feasible attribute weights allowing decision recommendations to be obtained through the computation of dominance relations and decision rules an illustrative example on the selection of a subcontractor is presented and the computational properties of rich are considered

optimality test for generalized fcm and its application to parameter selection in cluster analysis the fuzzy c means fcm clustering algorithm is the best known and most widely used method it was proven to converge to either a local minimum or saddle points by bezdek et al wei and mendel produced efficient optimality tests for fcm fixed points recently a weighting exponent selection for fcm was proposed by yu et al inspired by these results we unify several alternative fcm algorithms into one model called the generalized fuzzy c means gfcm this gfcm model presents a wide variation of fcm algorithms and can easily lead to new and interesting clustering algorithms moreover we construct a general optimality test for gfcm fixed points this is applied to theoretically choose the parameters in the gfcm model the experimental results demonstrate the precision of the theoretical analysis

additive spectral method for fuzzy cluster analysis of similarity data including community structure and affinity matrices an additive spectral method for fuzzy clustering is proposed the method operates on a clustering model which is an extension of the spectral decomposition of a square matrix the computation proceeds by extracting clusters one by one which makes the spectral approach quite natural the iterative extraction of clusters also allows us to draw several stopping rules to the procedure this applies to several relational data types differently normalized network structure data the first eigenvector subtracted affinity between multidimensional vectors the pseudo inverse laplacian transformation and conventional relational data including in house data of similarity between research topics according to working of a research center the method is experimentally compared with several classic and recent techniques and shown to be competitive

gaussian clustering method based on maximum fuzzy entropy interpretation abstract a new method of fuzzy clustering is proposed this is a complete gaussian membership function derived by means of the maximum entropy interpretation compared to the traditional fuzzy c means fcm method our approach exhibits the following two advantages 1 having clearer physical meaning and well defined mathematical features 2 having an optimal choice for feature parameter in theory moreover we also review some existing measures of uncertainty of fuzzy sets and redefine fuzzy entropy as analogous to probabilistic entropy

a method of relational fuzzy clustering based on producing feature vectors using fastmap the first stage of organizing objects is to partition them into groups or clusters the clustering is generally done on individual object data representing the entities such as feature vectors or on object relational data incorporated in a proximity matrix this paper describes another method for finding a fuzzy membership matrix that provides cluster membership values for all the objects based strictly on the proximity matrix this is generally referred to as relational data clustering the fuzzy membership matrix is found by first finding a set of vectors that approximately have the same inter vector euclidian distances as the proximities that are provided these vectors can be of very low dimension such as 5 or less fuzzy c means fcm is then applied to these vectors to obtain a fuzzy membership matrix in addition two dimensional vectors are also created to provide a visual representation of the proximity matrix this allows comparison of the result of automatic clustering to visual clustering the method proposed here is compared to other relational clustering methods including nerfcm rouben s method and windhams a p method various clustering quality indices are also calculated for doing the comparison using various proximity matrices as input simulations show the method to be very effective and no more computationally expensive than other relational data clustering methods the membership matrices that are produced by the proposed method are less crisp than those produced by nerfcm and more representative of the proximity matrix that is used as input to the clustering process

analysis of parameter selections for fuzzy c means the weighting exponent m is called the fuzzifier that can influence the performance of fuzzy c means fcm it is generally suggested that email protected 1 5 2 5 on the basis of a robust analysis of fcm a new guideline for selecting the parameter m is proposed we will show that a large m value will make fcm more robust to noise and outliers however considerably large m values that are greater than the theoretical upper bound will make the sample mean a unique optimizer a simple and efficient method to avoid this unexpected case in fuzzy clustering is to assign a cluster core to each cluster we will also discuss some clustering algorithms that extend fcm to contain the cluster cores in fuzzy clusters for a large theoretical upper bound case we suggest the implementation of the fcm with a suitable large m value otherwise we suggest implementing the clustering methods with cluster cores when the data set contains noise and outliers the fuzzifier m 4 is recommended for both fcm and cluster core based methods in a large theoretical upper bound case

interval valued fuzzy relation based clustering with its application to performance evaluation a similarity relation with its partition tree has been applied in the performance evaluation area for obtaining an agglomerative hierarchical clustering these fuzzy relation based methods require a decision maker to perform pair wise comparisons for the similarity among criteria as forming a fuzzy relation matrix the approach is developed based on real membership values of fuzzy relations however interval valued memberships may be better than real membership values to represent higher order imprecision and vagueness for human perception thus in this paper we would like to extend fuzzy relations to interval valued fuzzy relations and then construct interval valued similarity relations for performance evaluation we first give some definitions for these interval valued types of fuzzy relation similarity relation and resolution form we then construct an interval valued fuzzy similarity relation into a hierarchical structure schema it is shown that both of procedures and results for the partition tree derived from interval valued and crisp valued similarity relation matrices have some corresponding relationships and different merits to demonstrate the usefulness of the proposed approach performance evaluations for academic departments of higher education are considered by using actual engineering school data in taiwan

fuzzy random renewal process with queueing applications using extension principle associated with a class of continuous archimedean triangular norms this paper studies a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables some limit theorems in chance measure and in expected value for the sum of fuzzy random variables are proved on the basis of the continuous archimedean triangular norm based arithmetics furthermore we discuss the fuzzy random renewal process based on the obtained limit theorems and derive a fuzzy random elementary renewal theorem for the long run expected renewal rate the renewal theorem obtained in this paper can degenerate to the corresponding classical result in stochastic renewal process finally two case studies of queueing systems are provided to illustrate the application of the fuzzy random elementary renewal theorem

a generalization of distance functions for fuzzy c means clustering with centroids of arithmetic means fuzzy c means fcm is a widely used fuzzy clustering method which allows an object to belong to two or more clusters with a membership grade between zero and one despite the considerable efforts made by the clustering community the common characteristics of distance functions suitable for fcm remain unclear to fill this crucial void in this paper we first provide a generalized definition of distance functions that fit fcm directly the goal is to provide more flexibility to fcm in the choice of distance functions while preserving the simplicity of fcm by using the centroids of arithmetic means indeed we show that any distance function that fits fcm directly can be derived by a continuously differentiable convex function and thus is an instance of the generalized point to centroid distance p2c d by definition in addition we prove that if the membership grade matrix is nondegenerate any instance of the p2c d fits fcm directly finally extensive experiments have been conducted to demonstrate that the p2c d leads to the global convergence of fcm and that the clustering performances are significantly affected by the choices of distance functions

generalized fuzzy c means clustering strategies using l sub p norm distances fuzzy c means fcm is a useful clustering technique modifications of fcm using l sub 1 norm distances increase robustness to outliers object and relational data versions of fcm clustering are defined for the more general case where the l sub p norm p spl ges 1 or semi norm 0 0 in order to facilitate the empirical examination of the object data models both object and relational approaches are included in a numerical study

possibilistic and probabilistic fuzzy clustering unification within the framework of the non extensive thermostatistics fuzzy clustering algorithms are becoming the major technique in cluster analysis in this paper we consider the fuzzy clustering based on objective functions they can be divided into two categories possibilistic and probabilistic approaches leading to two different function families depending on the conditions required to state that fuzzy clusters are a fuzzy c partition of the input data recently we have presented in menard and eboueya fuzzy sets and systems 27 to be published an axiomatic derivation of the possibilistic and maximum entropy inference mei clustering approaches based upon an unifying principle of physics that of extreme physical information epi defined by frieden physics from fisher information a unification cambridge university press cambridge 1999 here using the same formalism we explicitly give a new criterion in order to provide a theoretical justification of the objective functions constraint terms membership functions and weighting exponent m used in the probabilistic and possibilistic fuzzy clustering moreover we propose an unified framework including the two procedures this approach is inspired by the work of frieden and plastino and plastino and miller physics a 235 577 extending the principle of extremal information in the framework of the non extensive thermostatistics then we show how with the help of epi one can propose extensions of the fcm and possibilistic algorithms

the possibilistic c means algorithm insights and recommendations recently the possibilistic c means algorithm pcm was proposed to address the drawbacks associated with the constrained memberships used in algorithms such as the fuzzy c means fcm in this issue barni et al 1996 report a difficulty they faced while applying the pcm and note that it exhibits an undesirable tendency to converge to coincidental clusters the purpose of this paper is not just to address the issues raised by barni et al but to go further and analytically examines the underlying principles of the pcm and the possibilistic approach in general we analyze the data sets used by barni et al and interpret the results reported by them in the light of our findings

clustering construction on a multimodal probability model in 2009 yu et al proposed a multimodal probability model mpm for clustering this paper makes advanced clustering constructions on the mpm we first reconstruct most existing clustering algorithms such as the k means fuzzy c means possibilistic c means mean shift classification maximum likelihood and latent class methods by establishing the relationships between these clustering algorithms and the mpm under our clustering construction we find that the mpm can be seen as a basic probability model for most existing clustering algorithms we then construct new clustering frameworks based on the mpm one of the frameworks develops new penalized type clustering algorithms another one induces entropy type clustering algorithms especially with sample weighted clustering several numerical and real data sets are made for comparisons these experimental results show that our clustering constructions based on the mpm can produce useful and effective clustering algorithms

on linear ordinary differential equations with functionally commutative coefficient matrices a system of linear ordinary differential equations x a t x t i is called semiproper if a t a a a t t i also known as functional commutativity of a t it is known that a semiproper system has a closed form fundamental solution matrix xa t exp ta d where the matrix exponential is defined by the power series exp k 1 kk therefore the problem of solving a semiproper system amounts to that of finding a finite form expression for the matrix exponential based on some recent results obtained by the authors for decomposing semiproper matrix functions a systematic approach is developed for finding a finite form analytical solution for the entire family of semiproper systems this solution is then used to derive a number of important and practical stability criteria for semiproper systems applications of the new results are also discussed

interval type 2 relative entropy fuzzy c means clustering fuzzy set theory especially type 2 fuzzy set theory provides an efficient tool for handling uncertainties and vagueness in real world observations among various clustering techniques type 2 fuzzy clustering methods are the most effective methods in the case of having no prior knowledge about observations while uncertainties in type 2 fuzzy clustering parameters are investigated by researchers uncertainties associated with membership degrees are not very well discussed in the literature in this paper investigating the latter uncertainties is our concern and interval type 2 relative entropy fuzzy c means it2 refcm clustering method is proposed the computational complexity of the proposed method is discussed and its performance is examined using several experiments the obtained results show that the proposed method has a very good ability in detecting noises and assignment of suitable membership degrees to observations

on parameter estimation for normal mixtures based on fuzzy clustering algorithms described here are three approaches to estimate the parameters of a mixture of normal distributions one approach is based on a modification of the expectation maximization algorithm to compute maximum likelihood estimates another makes use of the fuzzy c means clustering algorithms and the third is based on the penalized fuzzy c means clustering algorithms the accuracy and computational efficiency of algorithms of these three types to estimate parameters of normal mixtures are compared with samples drawn from univariate normal mixtures of two classes

establishing performance evaluation structures by fuzzy relation based cluster analysis the purpose of this study is to establish hierarchical structures for the performance evaluation of vague humanistic complicated systems to overcome the difficulties due to partial information and the vagueness of human knowledge and recognition a fuzzy relation based clustering method is proposed to model this evaluation first the effects of different max t i compositions on the formation of clusters are discussed then an improved clustering algorithm is developed to produce several partition trees with different levels and clusters according to different t i norm compositions to demonstrate the usefulness of the proposed algorithm the academic departments of higher education were considered using actual engineering school data in taiwan three performance evaluation structures are established by using max t 1 max t 2 and max t 3 compositions the results show that the proposed fuzzy hierarchical approach is useful and practical for performance evaluations of complicated humanistic systems

generalized fuzzy c shells clustering and detection of circular and elliptical boundaries abstract the fuzzy c shells fcs algorithm and its adaptive generalization called the adaptive fuzzy c shells afcs algorithm are considered for detection of curved boundaries specifically circular and elliptical the fcs algorithms utilize hyper spherical shells as cluster prototypes thus in two dimensions the prototypes are circles the afcs algorithms consider hyper ellipsoidal shells as prototypes hence the ability to characterize elliptical boundaries the generalization is achieved by allowing the distances to be measured through a norm inducing matrix that is symmetric positive definite each cluster is allowed to have a different matrix which is made a variable of optimization the ability of the algorithms to detect circular and elliptical boundaries in two dimensional data is illustrated through several examples

rank from comparisons and from ratings in the analytic hierarchy network processes rank preservation and reversal are important subjects in multi criteria decision making particularly if a theory uses only one of two ways of creating priorities rating alternatives one at a time with respect to an ideal or standard or comparing them in pairs it is known that our minds can do both naturally and without being tutored when rating alternatives they must be assumed to be independent and rank should be preserved when comparing alternatives they must be assumed to be dependent and rank may not always be preserved however even in making comparisons rank can be preserved if one uses idealization instead of normalization with the original set of alternatives and preserves that ideal from then on unless that ideal itself is deleted for some reason so often it is a matter of judgment as to whether it is desirable to force rank preservation or allow rank to adjust as necessary examples are given to illustrate the foregoing ideas

a type 2 fuzzy c regression clustering algorithm for takagi sugeno system identification and its application in the steel industry this paper proposes a new type 2 fuzzy c regression clustering algorithm for the structure identification phase of takagi sugeno t s systems we present uncertainties with fuzzifier parameter m in order to identify the parameters of interval type 2 fuzzy sets two fuzzifiers m 1 and m 2 are used then by utilizing these two fuzzifiers in a fuzzy c regression clustering algorithm the interval type 2 fuzzy membership functions are generated the proposed model in this paper is an extended version of a type 1 fcrm algorithm 25 which is extended to an interval type 2 fuzzy model the gaussian mixture model is used to create the partition matrix of the fuzzy c regression clustering algorithm finally in order to validate the proposed model several numerical examples are presented the model is tested on a real data set from a steel company in canada our computational results show that our model is more effective for robustness and error reduction than type 1 nfcrm and the multiple regression

implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear riesz space fractional diffusion equation abstract a numerical method with high accuracy both in time and in space is proposed for the two dimensional nonlinear riesz space fractional diffusion equation the main idea is based on a spectral galerkin method in spatial direction and an s stage implicit runge kutta method in temporal direction a rigorous stability and error analysis is performed for the proposed method it is shown that the proposed method is stable and convergent the optimal spatial error estimate is also derived numerical experiments are provided to illustrate the theoretical results

multiple kernel fuzzy clustering while fuzzy c means is a popular soft clustering method its effectiveness is largely limited to spherical clusters by applying kernel tricks the kernel fuzzy c means algorithm attempts to address this problem by mapping data with nonlinear relationships to appropriate feature spaces kernel combination or selection is crucial for effective kernel clustering unfortunately for most applications it is uneasy to find the right combination we propose a multiple kernel fuzzy c means mkfc algorithm that extends the fuzzy c means algorithm with a multiple kernel learning setting by incorporating multiple kernels and automatically adjusting the kernel weights mkfc is more immune to ineffective kernels and irrelevant features this makes the choice of kernels less crucial in addition we show multiple kernel k means to be a special case of mkfc experiments on both synthetic and real world data demonstrate the effectiveness of the proposed mkfc algorithm

validity guided re clustering with applications to image segmentation when clustering algorithms are applied to image segmentation the goal is to solve a classification problem however these algorithms do not directly optimize classification duality as a result they are susceptible to two problems 1 the criterion they optimize may not be a good estimator of true classification quality and 2 they often admit many suboptimal solutions this paper introduces an algorithm that uses cluster validity to mitigate problems 1 and 2 the validity guided re clustering vgc algorithm uses cluster validity information to guide a fuzzy re clustering process toward better solutions it starts with a partition generated by a soft or fuzzy clustering algorithm then it iteratively alters the partition by applying novel split and merge operations to the clusters partition modifications that result in improved partition validity are retained vgc is tested on both synthetic and real world data for magnetic resonance image mri segmentation evaluations by radiologists show that vgc outperforms the unsupervised fuzzy c means algorithm and vgc s performance approaches that of the supervised k nearest neighbors algorithm

a note on the ics algorithm with corrections and theoretical analysis ozdemir and akarun 2001 proposed an intercluster separation ics fuzzy clustering algorithm the ics algorithm is useful in combined quantization and dithering however there are two errors in the update equations for the ics algorithm this correspondence first points out these errors and gives their corrections since the parameters m c and are important factors in the performance of ics we also conduct a theoretical analysis of these ics parameters in order to analyze the parameters in ics we devise a theorem for the calculation of the hessian matrix from the ics objective function we establish the fixed point property of ics based on the decomposition of the hessian matrix and then analyze the effect of the parameters finally we propose a numerical approach in choosing the appropriate parameters m and for ics these experimental results give a better numerical perspective on the effect of parameters in ics and have conclusions consistent with our theoretical analysis

vector quantization in dct domain using fuzzy possibilistic c means based on penalized and compensated constraints in this paper fuzzy possibilistic c means fpcm approach based on penalized and compensated constraints are proposed to vector quantization vq in discrete cosine transform dct for image compression these approaches are named penalized fuzzy possibilistic c means pfpcm and compensated fuzzy possibilistic c means cfpcm the main purpose is to modify the fpcm strategy with penalized or compensated constraints so that the cluster centroids can be updated with penalized or compensated terms iteratively in order to 4nd near global solution in optimal problem the information transformed by dct was separated into dc and ac coe5cients then the ac coe5cients are trained by using the proposed methods to generate better codebook based on vq the compression performances using the proposed approaches are compared with fpcm and conventional vq method from the experimental results the promising performances can be obtained using the proposed approaches 2002 pattern recognition society published by elsevier science ltd all rights reserved

numerical evaluation of the lambert w function and application to generation of generalized gaussian noise with exponent 1 2 we address the problem of synthesizing a generalized gaussian noise with exponent 1 2 by means of a nonlinear memoryless transformation applied to a uniform noise we show that this transformation is expressable in terms of a special function known under the name of the lambert w function we review the main methods for numerical evaluation of the relevant branch of the multivalued lambert w function with controlled accuracy and complement them with an original rational function approximation based on these methods synthesis of the generalized gaussian noise can be performed with arbitrary accuracy we construct a simple and fast evaluation algorithm with prescribed accuracy which is especially suited for monte carlo simulation requiring large numbers of realizations of the generalized gaussian noise

on a class of fuzzy classification maximum likelihood procedures classification maximum likelihood cml procedure is a remarkable mixture of maximum likelihood approach to clustering this has been well documented in the book of mclachlan and basford in this paper we make the fuzzy extension of the cml procedure based on this class of fuzzy cml procedures we extend the fuzzy clustering algorithms of trauwaert kaufman and rousseeuw by adding a penalty term especially we derive a generalized type of fuzzy c means fcm clustering algorithms called the penalized fcm clustering algorithms then we create some asymptotic behaviors of these penalized fcm procedures by doing some numerical examples we find that the penalized fcm is more meaningful and effective than fcm

a method of identifying influential data in fuzzy clustering in multivariate statistical methods it is important to identify influential observations for a reasonable interpretation of the data structure in this paper we propose a method for identifying influential data in the fuzzy c means fcm algorithm to investigate such data we consider a perturbation of the data points and evaluate the effect of a perturbation as a perturbation we consider two cases one is the case in which the direction of a perturbation is specified and the other is the case in which the direction of a perturbation is not specified by computing the change in the clustering result of fcm when given data points are slightly perturbed we can look for data points that greatly affect the result also we confirm an efficacy of the proposed method by numerical examples

relative entropy fuzzy c means clustering pattern recognition is a collection of computer techniques to classify various observations into different clusters of similar attributes in either supervised or unsupervised manner application of fuzzy logic to unsupervised classification or clustering methods has resulted in many wildly used techniques such as fuzzy c means fcm method however when the observations are too noisy the performance of such methods might be reduced thus in this paper a new fuzzy clustering method based on fcm is presented and the relative entropy is added to its objective function as a regularization function to maximize the dissimilarity between clusters several examples are provided to examine the performance of the proposed clustering method the obtained results show that the proposed method has a very good ability in detecting noises and assignment of suitable membership degrees to the observations

analysis of a time stepping scheme for time fractional diffusion problems with nonsmooth data this paper establishes the convergence of a time stepping scheme for the time fractional diffusion problems with nonsmooth data we first analyze the regularity of the time fractional diffusion pro

frbc a fuzzy rule based clustering algorithm fuzzy clustering is superior to crisp clustering when the boundaries among the clusters are vague and ambiguous however the main limitation of both fuzzy and crisp clustering algorithms is their sensitivity to the number of potential clusters and or their initial positions moreover the comprehensibility of obtained clusters is not expertized whereupon in data mining applications the discovered knowledge is not understandable for human users to overcome these restrictions a novel fuzzy rule based clustering algorithm frbc is proposed in this paper like fuzzy rule based classifiers the frbc employs a supervised classification approach to do the unsupervised cluster analysis it tries to automatically explore the potential clusters in the data patterns and identify them with some interpretable fuzzy rules simultaneous classification of data patterns with these fuzzy rules can reveal the actual boundaries of the clusters to illustrate the capability of frbc to explore the clusters in data the experimental results on some benchmark datasets are obtained and compared with other fuzzy clustering algorithms the clusters specified by fuzzy rules are human understandable with acceptable accuracy

general type 2 fuzzy c means algorithm for uncertain fuzzy clustering pattern recognition in real world data is subject to various sources of uncertainty that should be appropriately managed the focus of this paper is the management of uncertainty associated with parameters of fuzzy clustering algorithms type 2 fuzzy sets t2 fss have received increased research interest over the past decade primarily due to their potential to model various uncertainties however because of the computational intensity of the processing of general t2 fuzzy sets gt2 fss only their constrained version i e the interval t2 it2 fss were typically used fortunately the recently introduced concepts of planes and zslices allow for efficient representation and computation with gt2 fss following this recent development this paper presents a novel approach for uncertain fuzzy clustering using the general type 2 fuzzy c means gt2 fcm algorithm the proposed method builds on top of the previously published it2 fcm algorithm which is extended via the planes representation theorem the fuzzifier parameter of the fcm algorithm can be expressed using linguistic terms such as small or high which are modeled as t1 fss this linguistic fuzzifier value is then used to construct the gt2 fcm cluster membership functions the linguistic uncertainty is transformed into uncertain fuzzy positions of the extracted clusters the gt2 fcm algorithm was found to balance the performance of t1 fcm algorithms in various uncertain pattern recognition tasks and to provide increased robustness in situations where noisy or insufficient training data are present

reducing the time complexity of the fuzzy c means algorithm in this paper we present an efficient implementation of the fuzzy c means clustering algorithm the original algorithm alternates between estimating centers of the clusters and the fuzzy membership of the data points the size of the membership matrix is on the order of the original data set a prohibitive size if this technique is to be applied to very large data sets with many clusters our implementation eliminates the storage of this data structure by combining the two updates into a single update of the cluster centers this change significantly affects the asymptotic runtime as the new algorithm is linear with respect to the number of clusters while the original is quadratic elimination of the membership matrix also reduces the overhead associated with repeatedly accessing a large data structure empirical evidence is presented to quantify the savings achieved by this new method

hierarchical clustering problems and analysis of fuzzy proximity relation on granular space in this paper based on granular space some hierarchical clustering problems and analysis for fuzzy proximity relation are developed by using rigorous mathematical descriptions and four results are obtained first the granular representation of a fuzzy proximity relation is studied an algorithm is obtained to compute the granular space or hierarchical clustering structure and the min transitive closure derived from the same fuzzy proximity relation on a finite universe and the consistent clustering properties are discussed second a consistency index that is based on granular space is developed and a global optimizing mathematical model is established to obtain the optimal clustering of a fuzzy proximity relation furthermore for given two fuzzy proximity relations on different granulations of the same universe a collaborative clustering method is studied by the intersection operation and this method can provide a better hierarchical clustering structure finally the theory for hierarchical clustering analysis of fuzzy proximity relations on granular space is established for given two fuzzy proximity relations on the same universe two sufficient conditions theorems 6 1 and 6 2 of isomorphism are given and the necessary condition theorem 6 3 and the sufficient condition theorem 6 4 for e similarity are obtained we also illustrate the effectiveness of these theories and methods with some practical examples these results may help form a comprehensive theory and methodology for related potential application and will help us to obtain a deeper understanding of the essence of hierarchical clustering procedures

sufficient conditions for stability of linear time varying systems abstract in this paper we consider sufficient conditions for the exponential stability of linear time varying systems of the form dot x t a t x t t 0 stability guaranteeing upper bounds for different measures of parameter variations are derived

fuzzy partition based soft subspace clustering and its applications in high dimensional data as one of the most popular clustering techniques for high dimensional data soft subspace clustering ssc algorithms have been receiving a great deal of attention in recent years unfortunately most existing works do not cluster high dimensional sparse data and noisy data in an effective manner in this study a novel soft subspace clustering algorithm called pi ssc is proposed by introducing a partition index pi into the objective function a novel soft subspace clustering algorithm that combines the concepts of hard and fuzzy clustering is proposed furthermore the robust property of pi ssc is analyzed from the viewpoint of e insensitive distance a convergence theorem for pi ssc is also established by applying zangwill s convergence theorem the results of the experiment demonstrate the effectiveness of the proposed algorithm in high dimensional sparse text data and noisy texture data

fuzzy algorithms for combined quantization and dithering color quantization reduces the number of the colors in a color image while the subsequent dithering operation attempts to create the illusion of more colors with this reduced palette in quantization the palette is designed to minimize the mean squared error mse however the dithering that follows enhances the color appearance at the expense of increasing the mse we introduce three joint quantization and dithering algorithms to overcome this contradiction the basic idea is the same in two of the approaches introducing the dithering error to the quantizer in the training phase the fuzzy c means fcm and the fuzzy learning vector quantization flvq algorithms are used to develop two combined mechanisms in the third algorithm we minimize an objective function including an inter cluster separation ics term to obtain a color palette which is more suitable for dithering the goal is to enlarge the convex hull of the quantization colors to obtain the illusion of more colors after error diffusion the color contrasts of images are also enhanced with the proposed algorithm we test the results of these three new algorithms using quality metrics which model the perception of the human visual system and illustrate that substantial improvements are achieved after dithering

sample weighted clustering methods although there have been many researches on cluster analysis considering feature or variable weights little effort has been made regarding sample weights in clustering in practice not every sample in a data set has the same importance in cluster analysis therefore it is interesting to obtain the proper sample weights for clustering a data set in this paper we consider a probability distribution over a data set to represent its sample weights we then apply the maximum entropy principle to automatically compute these sample weights for clustering such method can generate the sample weighted versions of most clustering algorithms such as k means fuzzy c means fcm and expectation maximization em etc the proposed sample weighted clustering algorithms will be robust for data sets with noise and outliers furthermore we also analyze the convergence properties of the proposed algorithms this study also uses some numerical data and real data sets for demonstration and comparison experimental results and comparisons actually demonstrate that the proposed sample weighted clustering algorithms are effective and robust clustering methods

on answering the question where do i start in order to solve a new problem involving interval type 2 fuzzy sets this paper which is tutorial in nature demonstrates how the embedded sets representation theorem rt for a general type 2 fuzzy set t2 fs when specialized to an interval i t2 fs can be used as the starting point to solve many diverse problems that involve it2 fss the problems considered are set theoretic operations centroid uncertainty measures similarity inference engine computations for mamdani it2 fuzzy logic systems linguistic weighted average person membership function approach to type 2 fuzzistics and interval approach to type 2 fuzzistics each solution obtained from the rt is a structural solution but is not a practical computational solution however the latter are always found from the former it is this author s recommendation that one should use the rt as a starting point whenever solving a new problem involving it2 fss because it has had such great success in solving so many such problems in the past and it answers the question where do i start in order to solve a new problem involving it2 fss

a simple statistics based nearest neighbor cluster detection algorithm we propose a new method for autonomously finding clusters in spatial data the proposed method belongs to the so called nearest neighbor approaches for finding clusters it is a repetitive technique which produces changing averages and deviations of nearest neighbor distance parameters and results in a final set of clusters the proposed technique is capable of eliminating background noise outliers and detection of clusters with different densities in a given data set using a wide variety of data sets we demonstrate that the proposed cluster seeking algorithm performs at least as well as various other currently popular algorithms and in several cases surpasses them in performance highlightsa new clustering algorithm based on simple statistics and lattice metrics is given mathematical rationale is explained in detail and theorem proofs are provided performance classification of the ssnn algorithm is illustrated with 2d datasets jain s benchmark dataset is used to show the ssnn cluster finding capability high dimensional image patterns are included as additional clustering examples

fuzzy c means optimality of solutions and effective termination of the algorithm abstract in this paper the solutions produced by the fuzzy c means algorithm for a general class of problems are examined and a method to test for the local optimality of such solutions is established an equivalent mathematical program is defined for the c means problem utilizing a generalized norm then the properties of the resulting optimization problem are investigated it is shown that the gradient of the resulting objective function at the solution produced by the c means algorithm in this case takes a special structure which can be used in terminating the algorithm moreover the local optimality of the solution obtained is checked utilizing the hessian of the criterion function the solution is a local minimum point if the hessian matrix at this point is positive semidefinite simple rules are proposed to help in checking the definiteness of the matrix

the harmonic consistency index for the analytic hierarchy process a new consistency measure the harmonic consistency index is obtained for any positive reciprocal matrix in the analytic hierarchy process we show how this index varies with changes in any matrix element a tight upper bound is provided for this new consistency measure when the entries of matrix are at most 9 as is often recommended using simulation the harmonic consistency index is shown to give numerical values similar to the standard consistency index but it is easier to compute and interpret in addition new properties of the column sums of reciprocal matrices are obtained

a generalized fuzzy clustering regularization model with optimality tests and model complexity analysis in this paper we propose a generalized fuzzy clustering regularization gfcr model and then study its theoretical properties gfcr unifies several fuzzy clustering algorithms such as fuzzy c means fcm maximum entropy clustering mec fuzzy clustering based on fermi dirac entropy and fuzzy bidirectional associative clustering network etc the proposed gfcr becomes an alternative model of the generalized fcm gfcm that was recently proposed by yu and yang to advance theoretical study we have the following three considerations 1 we give an optimality test to monitor if gfcr converges to a local minimum 2 we relate the gfcr optimality tests to occam s razor principle and then analyze the model complexity for fuzzy clustering algorithms 3 we offer a general theoretical method to evaluate the performance of fuzzy clustering algorithms finally some numerical experiments are used to demonstrate the validity of our theoretical results and complexity analysis

fuzzy linear discriminant analysis guided maximum entropy fuzzy clustering algorithm linear discriminant analysis lda is a classical statistical approach for supervised feature extraction and dimensionality reduction hard c means hcm is a classical unsupervised learning algorithm for clustering based on the analysis of the relationship between lda and hcm linear discriminant analysis guided adaptive subspace hard c means clustering algorithm lda hcm had been proposed lda hcm combines lda and hcm into a coherent framework and can adaptively reduce the dimension of data while performing data clustering simultaneously seeing that lda hcm is still a hard clustering algorithm we consider the fuzzy extension version of lda hcm in this paper to this end firstly we propose a new optimization criterion of fuzzy linear discriminant analysis flda by extending the value of membership function in classical lda from binary 0 or 1 into closed interval 0 1 in the meantime we present an efficient algorithm for the proposed flda secondly we show the close relationship between flda and maximum entropy fuzzy clustering algorithm mefca they both are maximizing fuzzy between class scatter and minimizing within class scatter simultaneously finally based on the above analysis combining flda and mefca into a joint framework we propose fuzzy linear discriminant analysis guided maximum entropy fuzzy clustering algorithm flda mefca lda mefca is a natural and effective fuzzy extension of lda hcm due to the introduction of soft decision strategy flda mefca can yield fuzzy partition of data set and is more flexible than lda hcm we also give the convergence proof of flda mefca extensive experiments on a collection of benchmark data sets are presented to show the effectiveness of the proposed algorithm

a novel fuzzy clustering algorithm with between cluster information for categorical data in this paper we present a new fuzzy clustering algorithm for categorical data in the algorithm the objective function of the fuzzy k modes algorithm is modified by adding the between cluster information so that we can simultaneously minimize the within cluster dispersion and enhance the between cluster separation for obtaining the local optimal solutions of the modified objective function the corresponding update formulas of the membership matrix and the cluster prototypes are strictly derived the convergence of the proposed algorithm under the optimization framework is proved on several real data sets from uci the performance of the proposed algorithm is studied the experimental results illustrate that the algorithm is effective and suitable for categorical data sets

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

upper and lower values for the level of fuzziness in fcm the level of fuzziness is a parameter in fuzzy system modeling which is a source of uncertainty in order to explore the effect of this uncertainty one needs to investigate and identify effective upper and lower boundaries of the level of fuzziness for this purpose fuzzy c means fcm clustering methodology is investigated to determine the effective upper and lower boundaries of the level of fuzziness in order to capture the uncertainty generated by this parameter in this regard we propose to expand the membership function around important information points of fcm these important information points are cluster centers and the mass center at these points it is known that the level of fuzziness has no effect on the membership values in this way we identify the counter intuitive behavior of membership function near these particular information points it will be shown that the upper and lower values of the level of fuzziness can be identified hence the uncertainty generated by this parameter can be encapsulated

comparing hierarchical and nonhierarchical weighting methods for eliciting multiattribute value models a value tree relating general values and concerns to specific value relevant attributes was constructed to compare three energy options nuclear coal and a combined geothermal and conservation package thirty seven nonexpert subjects provided judgments of the relative importance of attributes in the tree using both hierarchical and nonhierarchical weighting procedures and they rated the energy options on all attributes and all levels of the tree from these importance weights and ratings several additive multiattribute value models were constructed and compared with holistic rankings and ratings of the three options the experiment had three basic findings first hierarchical weights were steeper higher weight ratios than nonhierarchical weights second groups that differed in their holistic first choice nevertheless showed substantial agreement in their assessment of attribute weights third attribute level ratings of the relative desirability of energy options also agreed rather well across groups although there was a tendency for each group to favor their holistic first choice this convergence of multiattribute value model parameters resulted in a common model that was most consistent with holistic evaluations of the pro conservation group and generally inconsistent with those of the pro nuclear group this third finding of differential consistency between model composites and holistic evaluations is interpreted as a result of weight parameter distortions due to social desirability and or a neglect to consider attribute value ranges when making weight judgments

approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory ii in this paper the representations of fuzzy concepts based on raw data have been investigated within the framework of afs axiomatic fuzzy set theory first a brief review of afs theory is presented and a completely distributive lattice the e i algebra is proposed secondly two kinds of e i algebra representations of fuzzy concepts are derived in detail in order to represent the membership functions of fuzzy concepts in the interval 0 1 the norm of afs algebra is defined and studied finally the relationships of various representations with their advantages and drawbacks are analyzed

highlights and critical points in the theory and application of the analytic hierarchy process abstract this paper provides a detailed discussion with references on the fundamentals of the analytic hierarchy process and in particular of relative measurement the points discussed are grouped under the following categories structure in the ahp hierarchies and networks scales of measurement judgments consistency and the eigenvector synthesis and normative vs descriptive the paper also includes a discussion of rank and a number of citations of rank reversals attributed to a variety of factors ranging from intransitivity to procedure invariance that are thought to be unexplained by utility theory with its underlying principle to always preserve rank it is shown that when there is synergy due to the number of elements the ahp can be used to both preserve rank when it is desired to preserve it and allow it to reverse when it should reverse

optimality tests for fixed points of the fuzzy c means algorithm abstract the fuzzy c means fcm clustering algorithms are known to converge to either local minima or saddle points of the objective function which defines the fcm method the object of this paper is to derive efficient numerical tests for local extrema of the fcm functional that enable one to identify each candidate as a local minimum or saddle point numerical examples of the theory derived illustrate that the tests proposed cover all possible cases

introduction to a new weighting method hierarchy consistency analysis abstract this paper proposes a new methodology hierarchy consistency analysis hca the approach to this methodology was developed owing to the fact that in the analysis of large scale problems a different viewpoint or specific points emphasis will result in different hierarchy structures on the basis of different structures being created the methodology is designed to combine these different structures and estimate their compromised criterion weights so that all different hierarchy structures will have the same aggregated index the hca methodology will thus provide us with a comprehensive framework for solving hierarchy evaluation problems its merits include amongst others providing a simpler method for calculating the weight of criteria providing a consistency index for monitoring dm judgment consistency and delivering more types of hierarchy structure schemata to aid the decision maker in understanding and auditing the performance status of systems

on the local optimality of the fuzzy isodata clustering algorithm the convergence of the fuzzy isodata clustering algorithm was proved by bezdek 3 two sets of conditions were derived and it was conjectured that they are necessary and sufficient for a local minimum point in this paper we address this conjecture and explore the properties of the underlying optimization problem the notions of reduced objective function and improving and feasible directions are used to examine this conjecture finally based on the derived properties of the problem a new stopping criterion for the fuzzy isodata algorithm is proposed

the k means type algorithms versus imbalanced data distributions k means is a partitional clustering technique that is well known and widely used for its low computational cost the representative algorithms include the hard k means and the fuzzy k means however the performance of these algorithms tends to be affected by skewed data distributions i e imbalanced data they often produce clusters of relatively uniform sizes even if input data have varied cluster sizes which is called the uniform effect in this paper we analyze the causes of this effect and illustrate that it probably occurs more in the fuzzy k means clustering process than the hard k means clustering process as the fuzzy index m increases the uniform effect becomes evident to prevent the effect of the uniform effect we propose a multicenter clustering algorithm in which multicenters are used to represent each cluster instead of one single center the proposed algorithm consists of the three subalgorithms the fast global fuzzy k means best m plot and grouping multicenter algorithms they will be respectively used to address the three important problems 1 how are the reliable cluster centers from a dataset obtained 2 how are the number of clusters which these obtained cluster centers represent determined 3 how is it judged as to which cluster centers represent the same clusters the experimental studies on both synthetic and real datasets illustrate the effectiveness of the proposed clustering algorithm in clustering balanced and imbalanced data

a finite difference scheme on graded meshes for time fractional nonlinear korteweg de vries equation abstract in this paper a finite difference scheme for time fractional nonlinear korteweg de vries kdv problems with the caputo type fractional derivative is presented to deal with the weak singularity caused by the fractional derivative that the solution has in the initial layer the well known l1 scheme on graded meshes has been used for time discretization meanwhile a nonlinear finite difference approximation on uniform meshes is proposed for spatial discretization the existence stability and convergence of the numerical solution are studied by the energy method it is proved that the scheme is of min 2 r order convergence in time and of first order convergence in space where is the order of fractional derivative and r is a parameter of graded meshes furthermore in order to increase computational efficiency the corresponding fast algorithm of the presented scheme has been considered in numerical simulation a valid approach based on linear interpolation is designed to test convergence rate for the scheme on temporal graded meshes numerical results demonstrate sharpness of the theoretical convergence estimate and effectiveness of the fast algorithm

how to make t transitive a proximity relation three ways to approximate a proximity relation r i e a reflexive and symmetric fuzzy relation by a t transitive one where t is a continuous archimedean t norm are given the first one aggregates the transitive closure r macr of r with a maximal t transitive relation b contained in r the second one computes the closest homotecy of r macr or b to better fit their entries with the ones of r the third method uses nonlinear programming techniques to obtain the best approximation with respect to the euclidean distance for t the lukasiewicz or the product t norm the previous methods do not apply for the minimum t norm an algorithm to approximate a given proximity relation by a min transitive relation a similarity is given in the last section of the paper

a necessary condition for quantitative exponential stability of linear state space systems for an arbitrary n nconstant matrix a the two following facts are well known 1 n re tracea maxj 1 nre j a 0 if u is a unitary matrix one can always find a skew hermitian matrix a so that u ea in this note we present the extension of these two facts to the context of linear time varying dynamical systems x a t x t 0 as a by product this result suggests that the notion of slowly varying state space systems commonly used in literature is mathematically not natural to the problem of exponential stability

analysis of a temporal discretization for a semilinear fractional diffusion equation abstract this paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data with appropriately graded temporal grids first order temporal accuracy in the norm of l 2 0 t l 2 is derived by a new discrete gronwall s inequality and then first order temporal accuracy in the norm of l 0 t l 2 is established for 1 2 1 finally numerical results are provided to verify the theoretical results

gradual numbers and their application to fuzzy interval analysis in this paper we introduce a new way of looking at fuzzy intervals instead of considering them as fuzzy sets we see them as crisp sets of entities we call gradual real numbers they are a gradual extension of real numbers not of intervals such a concept is apparently missing in fuzzy set theory gradual numbers basically have the same algebraic properties as real numbers but they are functions a fuzzy interval is then viewed as a pair of fuzzy thresholds which are monotonic gradual real numbers this view enables interval analysis to be directly extended to fuzzy intervals without resorting to alpha cuts in agreement with zadeh s extension principle several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end points of intervals are changed into left and right fuzzy bounds our approach is illustrated on two known problems computing fuzzy weighted averages and determining fuzzy floats and latest starting times in activity network scheduling

cluster analysis based in fuzzy relations in this paper cluster analysis based on fuzzy relations is investigated tamura s max min n step procedure is extended to all types of max t compositions a max t similarity relation matrix is obtained by beginning with a proximity relation matrix based on the proposed max tn step procedure then a clustering algorithm is created for the max t similarityrelation matrix three critical max t compositions of max min max prod and max are compared the max composition is recommended as the 2rst choice among them several examples give more perspectives for di3erent choices of max t compositions finally the topic of incomplete data via max t compositions is discussed max t compositions can be e3ectively used to treat the t connected incomplete data c 2001 elsevier science b v all rights reserved

enhanced soft subspace clustering integrating within cluster and between cluster information while within cluster information is commonly utilized in most soft subspace clustering approaches in order to develop the algorithms other important information such as between cluster information is seldom considered for soft subspace clustering in this study a novel clustering technique called enhanced soft subspace clustering essc is proposed by employing both within cluster and between class information first a new optimization objective function is developed by integrating the within class compactness and the between cluster separation in the subspace based on this objective function the corresponding update rules for clustering are then derived followed by the development of the novel essc algorithm the properties of this algorithm are investigated and the performance is evaluated experimentally using real and synthetic datasets including synthetic high dimensional datasets uci benchmarking datasets high dimensional cancer gene expression datasets and texture image datasets the experimental studies demonstrate that the accuracy of the proposed essc algorithm outperforms most existing state of the art soft subspace clustering algorithms

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