conservation laws for conformally invariant variational problems we succeed in writing 2 dimensional conformally invariant non linear elliptic pde harmonic map equation prescribed mean curvature equations etc in divergence form these divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces from this form we can recover without the use of moving frame all the classical regularity results known for 2 dimensional conformally invariant non linear elliptic pde see hel it enables us also to establish new results in particular we solve a conjecture by e heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by s hildebrandt hil1 claiming that critical points of continuously differentiable elliptic conformally invariant lagrangian in two dimensions are continuous

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

on t norms based measures of specificity this paper gives a general expression for families of measures of specificity of a fuzzy set or a possibility distribution based on three t norms and a negation other known measures of specificity are particular cases of this expression and new examples are provided

on quantification of different facets of uncertainty abstract with a brief introduction to three major types of uncertainties randomness nonspecificity and fuzziness we discuss various attempts to quantify them we also discuss several attempts to quantify total uncertainty in a system we then talk about some new facets of uncertainty like higher order fuzzy entropy hybrid entropy and conflict in a body of evidence in conclusion we indicate some other aspects of uncertainty that need to be modeled and quantified

visual assessment of clustering tendency for rectangular dissimilarity matrices we have an m times n matrix d and assume that its entries correspond to pair wise dissimilarities between m row objects or and n column objects oc which taken together as a union comprise a set o of n m n objects this paper develops a new visual approach that applies to four different cluster assessment problems associated with o the problems are the assessment of cluster tendency pi amongst the row objects or p2 amongst the column objects oc p3 amongst the union of the row and column objects or u oc and p4 amongst the union of the row and column objects that contain at least one object of each type co clusters the basis of the method is to regard d as a subset of known values that is part of a larger unknown n times n dissimilarity matrix and then impute the missing values from d this results in estimates for three square matrices dr dc druc that can be visually assessed for clustering tendency using the previous vat or svat algorithms the output from assessment of druc ultimately leads to a rectangular covat image which exhibits clustering tendencies in d five examples are given to illustrate the new method two important points i because vat is scalable by svat to data sets of arbitrary size and because covat depends explicitly and only on vat this new approach is immediately scalable to say the scovat model which works for even very large unloadable data sets without alteration and ii vat svat and covat are autonomous parameter free models no hidden values are needed to make them work

weak limit and blowup of approximate solutions to h systems abstract let h r 3 r be a continuous function such that h p h 0 r as p fixing a domain in r 2 we study the behaviour of a sequence u n of approximate solutions to the h system u 2 h u u x u y in assuming that sup p r 3 h p h 0 p 1 we show that the weak limit of the sequence u n solves the h system and u n u strongly in h 1 apart from a countable set s made by isolated points moreover if in addition h p h 0 o 1 p as p then in correspondence of each point of s we prove that the sequence u n blows either an h bubble or an h 0 sphere

the jacobian determinant revisited this paper is devoted to the study of the jacobian determinant of a map g from a smooth bounded open subset of rn into rn n 2 more generally could be a smooth bounded open subset of an n dimensional manifold starting with the seminal work of c b morrey 27 y reshetnyak 32 and j ball 1 it has been known that one can define the distributional jacobian determinant det g under fairly weak assumption on g in particular it is defined for all maps g w 1 n2 n 1 and also for all

a periodic solution of a differential equation with state dependent delay we study a differential equation for delayed negative feedback which models a situation where the delay depends on the present state and becomes effective in the future the main result is existence of a periodic solution in case the equilibrium is linearly unstable the proof employs the ejective fixed point principle on a compact convex set k0 c h 0 r of lipschitz continuous functions and uses that the equation generates a smooth semiflow on an infinite dimensional submanifold of the space c1 h 0 r

connection between fuzzy theory simulated annealing and convex duality abstract simulated annealing is a well known heuristic method in which the basic notions of statistical mechanics are used as an analogy for treating optimization problems the physical term of energy is regarded as an objective criterion which has to be minimized in this context also the terms of temperature free energy and entropy have been involved the present paper generalizes this formalism in order to make it available as a basic tool for fuzzy theory and multicriteria analysis therefore the formula of free energy is generalized in order to introduce a new aggregation operator the main feature of this operator is the compensatory property of aggregation it fills the gap between the maximum operator the minimum operator and the weighted sum by a coherent mathematical formalism which is connected with a large mathematical background the aggregation operator plays the same peculiar part as the shannon entropy does in information theory this is verified by means of convex duality in this formalism the fuzzy decisions that correspond to the boltzmann distribution are involved from there we derive a fuzzy extension principle which can be applied to developing new algorithms moreover we discover a bridge between the boltzmann distribution and the fuzzy c means approach across this bridge some help arises for both of the two sides thus we derive new and generalized versions of the fuzzy c means algorithm and we establish a fuzzy version of simulated annealing which is more efficient than the original simulated annealing

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

multiple solutions of h systems and rellich s conjecture soit x y r 2 x 2 y 2 1 on considere une fonction u r 3 satisfaisant le systeme h u 2hu x u y sur avec l une des conditions suivantes soit dirichlet u sur soit plateau u x 2 u y 2 u x u y 0 sur u et u non decroissante sur

on cluster validity index for estimation of the optimal number of fuzzy clusters a new cluster validity index is proposed that determines the optimal partition and optimal number of clusters for fuzzy partitions obtained from the fuzzy c means algorithm the proposed validity index exploits an overlap measure and a separation measure between clusters the overlap measure which indicates the degree of overlap between fuzzy clusters is obtained by computing an inter cluster overlap the separation measure which indicates the isolation distance between fuzzy clusters is obtained by computing a distance between fuzzy clusters a good fuzzy partition is expected to have a low degree of overlap and a larger separation distance testing of the proposed index and nine previously formulated indexes on well known data sets showed the superior effectiveness and reliability of the proposed index in comparison to other indexes

fuzzy bayesian validation for cluster analysis of yeast cell cycle data clustering for the analysis of the genes organizes the patterns into groups by the similarity of the dataset and has been used for identifying the functions of the genes in the cluster and analyzing the functions of unknown genes since the genes usually belong to multiple functional families fuzzy clustering methods are more appropriate than the conventional hard clustering methods which assign a sample to only one group in this paper a bayesian like validation method selecting a fuzzy partition is proposed to evaluate the fuzzy partitions effectively the theoretical interpretation of the obtained memberships is beyond the scope of this paper and an empirical evaluation of the proposed method is conducted by comparing to the four representative conventional fuzzy cluster validity measures in four well known datasets analysis of yeast cell cycle data follows to evaluate the proposed method

fcm based model selection algorithms for determining the number of clusters clustering is an important research topic that has practical applications in many fields it has been demonstrated that fuzzy clustering using algorithms such as the fuzzy c means fcm has clear advantages over crisp and probabilistic clustering methods like most clustering algorithms however fcm and its derivatives need the number of clusters in the given data set as one of their initializing parameters the main goal of this paper is to develop an effective fuzzy algorithm for automatically determining the number of clusters after a brief review of the relevant literature we present a new algorithm for determining the number of clusters in a given data set and a new validity index for measuring the goodness of clustering experimental results and comparisons are given to illustrate the performance of the new algorithm

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

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

adaptive fuzzy c shells clustering and detection of ellipses several generalizations of the fuzzy c shells fcs algorithm are presented for characterizing and detecting clusters that are hyperellipsoidal shells an earlier generalization the adaptive fuzzy c shells afcs algorithm is examined in detail and is found to have global convergence problems when the shapes to be detected are partial new formulations are considered wherein the norm inducing matrix in the distance metric is unconstrained in contrast to the afcs algorithm the resulting algorithm called the afcs u algorithm performs better for partial shapes another formulation based on the second order quadrics equation is considered these algorithms can detect ellipses and circles in 2d data they are compared with the hough transform ht based methods for ellipse detection existing ht based methods for ellipse detection are evaluated and a multistage method incorporating the good features of all the methods is used for comparison numerical examples of real image data show that the afcs algorithm requires less memory than the ht based methods and it is at least an order of magnitude faster than the ht approach

a cluster validity index combining an overlap measure and a separation measure based on fuzzy aggregation operators since a clustering algorithm can produce as many partitions as desired one needs to assess their quality in order to select the partition that most represents the structure in the data if there is any this is the rationale for the cluster validity cv problem and indices this paper presents a cv index that helps to find the optimal number of clusters of data from partitions generated by a fuzzy clustering algorithm such as the fuzzy c means fcm or its derivatives given a fuzzy partition this new index uses a measure of multiple cluster overlap and a separation measure for each data point both based on an aggregation operation of membership degrees experimental results on artificial and benchmark datasets are given to demonstrate the performance of the proposed index as compared with traditional and recent indices

evaluating the numerical instability in fuzzy clustering validation of high dimensional data abstract fuzzy clustering validation of high dimensional datasets is only possible using a reliable cluster validity index cvi a good cvi must correctly recognize a data structure and its validations must be independently of any parameter of a clustering algorithm or data property however some classical fuzzy cvis as partition coefficient pc partition entropy pe and fukuyama sugeno fs have the monotonic tendency in function of the number of clusters although the literature presents extensive investigations about such tendency they were conducted for low dimensional data in which such data property does not affect the clustering behavior in order to investigate how such aspects affect the fuzzy clustering results of high dimensional data in this work we have clustered objects of thirteen real datasets using the fuzzy c means algorithm the fuzzy partitions were validated by pc pe fs and some proposed improvements of them to lead with the monotonic tendency totaling eight fuzzy cvis analyzed besides the analysis made about the number of clusters selected by the cvis the mann kendall test was performed to verify statistically the monotonic trend of the cvis results from the two analysis made the modified partition coefficient and scaled partition entropy indices were successful in respectively improving the pc and pe indices

a clustering performance measure based on fuzzy set decomposition clustering is primarily used to uncover the true underlying structure of a given data set and for this purpose it is desirable to subject the same data to several different clustering algorithms this paper attempts to put an order on the various partitions of a data set obtained from different clustering algorithms the goodness of each partition is expressed by means of a performance measure based on a fuzzy set decomposition of the data set under consideration several experiments reported in here show that the proposed performance measure puts an order on different partitions of the same data which is consistent with the error rate of a classifier designed on the basis of the obtained cluster labelings

the solution manifold and c1 smoothness for differential equations with state dependent delay abstract let h 0 u c 1 h 0 r n open and f u r n continuously differentiable if f satisfies two mild additional smoothness conditions then the set x u 0 f is a c1 submanifold of codimension n in c 1 h 0 r n the maximal solutions x of the initial value problems x t f x t x 0 x define a continuous semiflow f on x and all operators f t are continuously differentiable their derivatives d2f t are given in the usual way by solutions v to the variational equation along x with segments vt in the tangent spaces tf t x the additional conditions on f are motivated by properties of differential equations with state dependent delay and are verified for an example

on the meaning of dunn s partition coefficient for fuzzy clusters abstract a constantly recurring problem in cluster analysis is that of evaluating the number of clusters that are present in a data set the development of fuzzy clustering has brought no definite answer to this question although some new means have been provided offering new ways to tackle this problem one of the promising approaches was dunn s partition coefficient f k u that has been shown to vary between 1 for hard clusters and 1 k for completely fuzzy sets of objects hence the idea that f k u expresses a measure of how far a given fuzzy partition is from a hard one assuming moreover that an optimal partition in an optimal number of clusters will have a harder look than any other partition f k u could be considered as a cluster validity index that this procedure is unjustified is shown by the study of the d f diagram in which by analogy to the partition coefficient f a fuzziness coefficient d is defined an important result of this study is that a higher f k u value does not always correspond to a better allocation than a partition with a lower value this observation contradicts the role of the partition coefficient as cluster validity measurement these results are emply confirmed by various artificial and classical numerical examples

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

detection and separation of ring shaped clusters using fuzzy clustering a new fuzzy clustering algorithm designed to detect and characterize ring shaped clusters and combinations of ring shaped and compact spherical clusters has been developed this fkr algorithm includes automatic search for proper initial conditions in the two cases of concentric and excentric intersected combinations of clusters validity criteria based on total fuzzy area and fuzzy density are used to estimate the optimal number of substructures in the data set the fkr algorithm has been tested on a variety of simulated combinations of ring shaped and compact spherical clusters and its performance proved to be very good both in identifying the input shapes and in recovering the input parameters application of the fkr algorithm to an mri image of the heart s left ventricle was used to investigate the possibility of using this algorithm as an aid in image processing

a k order fuzzy or operator for pattern classification with k order ambiguity rejection in pattern recognition the membership of an object to classes is often measured by labels this article mainly deals with the mathematical foundations of labels combination operators built on t norms that extend previous ambiguity measures of objects by dealing not only with two classes ambiguities but also with k classes k lying between 1 and the number of classes c mathematical properties of this family of combination operators are established and a weighted extension is proposed allowing to give more or less importance to a given class a classifier with reject options built on the proposed measure is presented and applied on synthetic data a critical analysis of the results led to derivate some new operators by aggregating previous measures a modified classifier is proposed and applied to synthetic data as well as to standard real data

boundary layer phenomena for differential delay equations with state dependent time lags i in this paper we begin a study of the differential delay equation mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0jc9yq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpeps0he9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiabew7aljqadi hagaqbaiaacicacawg0bgaaiykaiabg2da9iabgkhitiaadihacagg oagaamidaiaacmcacqghrawkcawgmbgaaiikaiaadihacaggoagaam idaiabgkhitiaadkhacaggpagaaiykaiaacycacaqggagaaeiiaiaa dkhacqgh9aqpcawgybgaaiikaiaadihacaggoagaamidaiaacmcaca ggpaaaaa 5192 varepsilon x t x t f x t r r r x t we prove the existence of periodic solutions for 0 0 where 0 is an optimal positive number we investigate regularity and monotonicity properties of solutions x t which are defined for all t and of associated functions like t t r x t we begin the development of a poincare bendixson theory and phase plane analysis for such equations in a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as approaches zero

morse decompositions for delay differential equations abstract the attractor for the scalar delay differential equation x t f x t x t 1 is shown to possess a morse decomposition under the hypothesis of a negative feedback condition in the delay term using this decomposition we prove results on the asymptotic rate of oscillation of solutions of the initial value problem existence of both slowly and rapidly oscillating periodic solutions is also proved

a study of some fuzzy cluster validity indices genetic clustering and application to pixel classification in this article the effectiveness of variable string length genetic algorithm along with a recently developed fuzzy cluster validity index pbmf has been demonstrated for clustering a data set into an unknown number of clusters the flexibility of a variable string length genetic algorithm vga is utilized in conjunction with the fuzzy indices to determine the number of clusters present in a data set as well as a good fuzzy partition of the data for that number of clusters a comparative study has been performed for different validity indices namely pbmf xb pe and pc the results of the fuzzy vga algorithm are compared with those obtained by the well known fcm algorithm which is applicable only when the number of clusters is fixed a priori moreover another genetic clustering scheme that also requires fixing the value of the number of clusters is implemented the effectiveness of the pbmf index as the optimization criterion along with a genetic fuzzy partitioning technique is demonstrated on a number of artificial and real data sets including a remote sensing image of the city of kolkata

determination of cluster number in clustering microarray data the general purpose of clustering analysis of microarray data is to organize the data into meaningful groups based on their closeness although various algorithms have been proposed for the clustering of microarray data the main difficulty remains to be the determination of the optimal number of clusters to complicate the problem further meaningful groups or closeness cannot be well defined due to the fuzziness nature of the data this paper proposes a dynamic validity index to overcome this problem the proposed index in addition of the dynamic aspects also takes care of both the intra and the inter distances of the clusters an algorithm based on the proposed dynamic validity index and the traditional k means method was developed to make the proposed dynamic validity index more flexible a modulating parameter c is introduced this parameter can be used to take care of noisy data and balance the importance between compactness and separateness in the clusters to illustrate the effectiveness of the approach a numerical example by using the human serum data from the literature was solved and the sensitivity and robustness of the approach are examined

a cluster validation index for gk cluster analysis based on relative degree of sharing in this paper the problem of traditional validity indices when applied to the gustafson kessel gk clustering are reviewed a new cluster validity index for the gk algorithm is proposed this validity index is defined as the average value of the relative degrees of sharing of all possible pairs of fuzzy clusters in the system it computes the overlap of each pair of fuzzy clusters by considering the degree of sharing of each data point in the overlap the optimal number of clusters is obtained by minimizing the validity index experiments in which the proposed validity index and several traditional validity indices were applied to 6 data sets highlight the superior qualities of the proposed index the results indicate that the proposed validity index is very reliable

unsupervised possibilistic clustering in fuzzy clustering the fuzzy c means fcm clustering algorithm is the best known and used method since the fcm memberships do not always explain the degrees of belonging for the data well krishnapuram and keller proposed a possibilistic approach to clustering to correct this weakness of fcm however the performance of krishnapuram and keller s approach depends heavily on the parameters in this paper we propose another possibilistic clustering algorithm pca which is based on the fcm objective function the partition coefficient pc and partition entropy pe validity indexes the resulting membership becomes the exponential function so that it is robust to noise and outliers the parameters in pca can be easily handled also the pca objective function can be considered as a potential function or a mountain function so that the prototypes of pca can be correspondent to the peaks of the estimated function to validate the clustering results obtained through a pca we generalized the validity indexes of fcm this generalization makes each validity index workable in both fuzzy and possibilistic clustering models by combining these generalized validity indexes an unsupervised possibilistic clustering is proposed some numerical examples and real data implementation on the basis of the proposed pca and generalized validity indexes show their effectiveness and accuracy

validity index for crisp and fuzzy clusters abstract in this article a cluster validity index and its fuzzification is described which can provide a measure of goodness of clustering on different partitions of a data set the maximum value of this index called the pbm index across the hierarchy provides the best partitioning the index is defined as a product of three factors maximization of which ensures the formation of a small number of compact clusters with large separation between at least two clusters we have used both the k means and the expectation maximization algorithms as underlying crisp clustering techniques for fuzzy clustering we have utilized the well known fuzzy c means algorithm results demonstrating the superiority of the pbm index in appropriately determining the number of clusters as compared to three other well known measures the davies bouldin index dunn s index and the xie beni index are provided for several artificial and real life data sets

cluster validity for fuzzy clustering algorithms abstract the proportion exponent is introduced as a measure of the validity of the clustering obtained for a data set using a fuzzy clustering algorithm it is assumed that the output of an algorithm includes a fuzzy nembership function for each data point we show how to compute the proportion of possible memberships whose maximum entry exceeds the maximum entry of a given membership function and use these proportions to define the proportion exponent its use as a validity functional is illustrated with four numerical examples and its effectiveness compared to other validity functionals namely classification entropy and partition coefficient

rule base self generation and simplification for data driven fuzzy models data driven fuzzy modeling has been used in a wide variety of applications however in fuzzy rule based models acquired from numerical data redundancy often exists in the form of redundant rules or similar fuzzy sets this results in unnecessary structural complexity and decreases the interpretability of the system in this paper a rule base self extraction and simplification method is proposed to establish interpretable fuzzy models from numerical data a fuzzy clustering technique associated with the proposed fuzzy partition validity index is used to extract the initial fuzzy rule base and find out the optimal number of fuzzy rules to reduce the complexity of fuzzy models while keeping good model accuracy some approximate similarity measures are presented and a parameter fine tuning mechanism is introduced to improve the accuracy of the simplified model using the proposed similarity measures the redundant fuzzy rules are removed and similar fuzzy sets are merged to create a common fuzzy set in the rule base the simplified rule base is computationally efficient and linguistically interpretable the approach has been successfully applied to fuzzy models of non linear function approximation dynamical system modeling and mechanical property prediction for hot rolled steels

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

a new fuzzy clustering validity index with a median factor for centroid based clustering determining the number of clusters which is usually approved by domain experts or evaluated by clustering validity indexes is an important issue in clustering analysis this study discusses the effectiveness of clustering validity indexes for centroid based partitional clustering algorithms most general purpose clustering validity indexes take the minimum maximum distance between a pair of data objects a pair of cluster centroids or an object and a centroid as an important evaluation factor however they may present unstable results especially when two centroids are allocated closely to alleviate this problem a new clustering validity index which is termed the wu and li index wli is proposed in this paper our proposed wli partially allows to some extent the existence of closely allocated centroids in the clustering results by considering not only the minimum but the median distances between a pair of centroids as well therefore possessing better stability the performances of wli and some existing clustering validity indexes are evaluated and compared by running the fuzzy c means algorithm for clustering various types of datasets including artificial datasets uci datasets and images experimental results have shown that wli has the more accurate and satisfactory performance than other indexes

on fuzzy cluster validity indices cluster analysis aims at identifying groups of similar objects and helps to discover distribution of patterns and interesting correlations in large data sets especially fuzzy clustering has been widely studied and applied in a variety of key areas and fuzzy cluster validation plays a very important role in fuzzy clustering this paper introduces the fundamental concepts of cluster validity and presents a review of fuzzy cluster validity indices available in the literature we conducted extensive comparisons of the mentioned indices in conjunction with the fuzzy c means clustering algorithm on a number of widely used data sets and make a simple analysis of the experimental results

a new cluster validity for fuzzy clustering abstract fuzzy cluster validity criterion tends to evaluate the quality of fuzzy c partitions produced by fuzzy clustering algorithms many functions have been proposed some methods use only the properties of fuzzy membership degrees to evaluate partitions others techniques combine the properties of membership degrees and the structure of data in this paper a new heuristic method is based on the combination of two functions the search of good clustering is measured by a fuzzy compactness separation ratio the first function calculates this ratio by considering geometrical properties and membership degrees of data the second function evaluates it by using only the properties of membership degrees four numerical examples are used to illustrate its use as a validity functional its effectiveness is compared to some existing cluster validity criterion

correction to on cluster validity for the fuzzy c means model correspondence validation of partitions produced by the fuzzy c means fcm clustering algorithm was discussed by pal and bezdek 1 two tables in 1 contain erroneous values this errata reports the correct values and notes that the conclusions drawn in 1 based on these simulations remain unchanged

the poincare bendixson theorem for a class of delay equations with state dependent delay and monotonic feedback abstract we consider the real valued differential equation x t f x t x t d x t with state dependent delay where f is strictly monotonic in its second argument we describe a class of such equations for which a version of the poincare bendixson theorem holds

a partitional clustering algorithm validated by a clustering tendency index based on graph theory applying graph theory to clustering we propose a partitional clustering method and a clustering tendency index no initial assumptions about the data set are requested by the method the number of clusters and the partition that best fits the data set are selected according to the optimal clustering tendency index value