autocorrelation based fuzzy clustering of time series the traditional approaches to clustering a set of time series are generally applicable if there is a fixed underlying structure to the time series so that each will belong to one cluster or the other however time series often display dynamic behaviour in their evolution over time this dynamic behaviour should be taken into account when attempting to cluster time series for instance during a certain period a time series might belong to a certain cluster afterwards its dynamics might be closer to that of another cluster in this case the traditional clustering approaches are unlikely to find and represent the underlying structure in the given time series this switch from one time state to another which is typically vague can be naturally treated following a fuzzy approach this paper proposes a fuzzy clustering approach based on the autocorrelation functions of time series in which each time series is not assigned exclusively to only one cluster but it is allowed to belong to different clusters with various membership degrees

segmentation and classification of range images the recognition of objects in three dimensional space is a desirable capability of a computer vision system range images which directly measure 3 d surface coordinates of a scene are well suited for this task in this paper we report a procedure to detect connected planar convex and concave surfaces of 3 d objects this is accomplished in three stages the first stage segments the range image into surface patches by a square error criterion clustering algorithm using surface points and associated surface normals the second stage classifies these patches as planar convex or concave based on a non parametric statistical test for trend curvature values and eigenvalue analysis in the final stage boundaries between adjacent surface patches are classified as crease or noncrease edges and this information is used to merge compatible patches to produce reasonable faces of the object s this procedure has been successfully applied to a large number of real and synthetic images four of which we present in this paper

output feedback direct adaptive fuzzy controller based on frequency domain methods this paper describes a design method that guarantees the stability of an adaptive fuzzy control system the system consists of an unstable plant with an irrational transfer function a takagi sugeno fuzzy adaptive controller a nonlinear reference model and an adaptation mechanism the gradient based adaptation mechanism changes the consequents of the controller rules in such a way that the closed loop system behaves like the reference model the proposed design procedure utilizes frequency domain methods in the forms of the nyquist and circle criteria it is assumed that the function of the fuzzy controller is a nonlinearity described by a sector condition this condition means that the controller function lies between two lines passing through the origin in this method the function of the fuzzy controller is verified during online adaptation therefore it stays in the allowed sector this method is illustrated by an example of a control system containing an unstable plant with a transport delay the main contribution of this paper is to use frequency methods combined with adaptive fuzzy control these methods have previously been used for nonadaptive fuzzy control but in this paper they are used for the first time for adaptive fuzzy control

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

a clustering fuzzification algorithm based on alm abstract in this paper we propose a fuzzification method for clusters produced from a clustering process based on active learning method alm alm is a soft computing methodology which is based on a hypothesis claiming that human brain interprets information in pattern like images the proposed fuzzification method is applicable to all non fuzzy clustering algorithms as a post process the most outstanding advantage of this method is the ability to determine the membership degrees of each data to all clusters based on the density and shape of the clusters it is worth mentioning that for existing fuzzy clustering algorithms such as fcm the membership degree is usually determined as a function of distance to the center of the clusters in our proposed method all data points of a cluster will play a role in order to determine the membership degrees consequently the obtained membership degrees will depend on all of the data points of clusters the amount of data points and the density distribution of the clusters simulations prove the advantages of the proposed method

the existence of solitary wave solutions of delayed camassa holm equation via a geometric approach abstract this paper is concerned with the camassa holm equation which is a model for shallow water waves we first establish the existence of solitary wave solutions for the equation without delay and then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system especially the geometric singular perturbation theory and invariant manifold theory according to the relationship between solitary wave and homoclinic orbit the camassa holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution it is proved that the equation with disturbance also possesses homoclinic orbit and there exists solitary wave solution of the delayed camassa holm equation

cluster validity index for estimation of fuzzy clusters of different sizes and densities cluster validity indices are used for estimating the quality of partitions produced by clustering algorithms and for determining the number of clusters in data cluster validation is difficult task because for the same data set more partitions exists regarding the level of details that fit natural groupings of a given data set even though several cluster validity indices exist they are inefficient when clusters widely differ in density or size we propose a clustering validity index that addresses these issues it is based on compactness and overlap measures the overlap measure which indicates the degree of overlap between fuzzy clusters is obtained by calculating the overlap rate of all data objects that belong strongly enough to two or more clusters the compactness measure which indicates the degree of similarity of data objects in a cluster is calculated from membership values of data objects that are strongly enough associated to one cluster we propose ratio and summation type of index using the same compactness and overlap measures the maximal value of index denotes the optimal fuzzy partition that is expected to have a high compactness and a low degree of overlap among clusters testing many well known previously formulated and proposed indices on well known data sets showed the superior reliability and effectiveness of the proposed index in comparison to other indices especially when evaluating partitions with clusters that widely differ in size or density

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

canard solutions at non generic turning points this paper deals with singular perturbation problems for vector fields on 2 dimensional manifolds canard solutions are solutions that starting near an attracting normally hyperbolic branch of the singular curve cross a turning point and follow for a while a normally repelling branch of the singular curve following the geometric ideas developed by dumortier and roussarie in 1996 for the study of canard solutions near a generic turning point we study canard solutions near non generic turning points characterization of manifolds of canard solutions is given in terms of boundary conditions their regularity properties are studied and the relation is described with the more traditional asymptotic approach it reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve we also study their contact and its relation to the control curve

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

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

a cluster validity index for fuzzy clustering in this paper a cluster validity index proposed by kim et al 15 is analyzed and a problem is discussed that the validity index faces in situations when there are well separated clusters that themselves include subclusters based on this analysis a new validity index is proposed the new validity index employs a compactness measure and a separation measure the compactness measure combines the fuzziness in the membership matrix u with the geometrical compactness of the representation of the data set x via the prototypes v the separation measure is defined as the average value of the degrees of overlap of all possible pairs of fuzzy clusters in the system the proposed index is tested and validated using several data sets the results of the comparison show the superior effectiveness and reliability of the proposed index in comparison to other indices

interval type 2 credibilistic clustering for pattern recognition this paper presents a new approach to interval type 2 fuzzy clustering in order to consider compactness within the clusters and separation of them simultaneously the objective function of this paper is designed such that it generates both degrees of membership and non membership of each data in each cluster and integrates them using credibility concept also a new approach to separation of clusters is proposed and utilized in designing the objective function in this approach the borders of clusters and therefore their compactness contribute in attaining their separation so the separation of clusters is not assessed only by the distance of their centers the credibility degrees are transformed to interval type 2 form to handle different sources of uncertainty finally a new validity index to characterize the number of clusters based on proposed approach to separation of clusters and choquet integrals are proposed the advantages of paper contributions are illustrated using several examples a new objective function based fuzzy clustering is introduced using credibility concept a new approach to separation of clusters is presented a new validity index is designed by proposed approach to separation and choquet integral the obtained results show that proposed model can handle different types of clusters

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

canard cycles for predator prey systems with holling types of functional response abstract by using the singular perturbation theory developed by dumortier and roussarie and recent work of de maesschalck and dumortier we study the canard phenomenon for predator prey systems with response functions of holling types we first develop a formula for computing the slow divergence integrals by using the formula we prove that for the systems with the response function of holling types iii and iv the cyclicity of any limit periodic set is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the limit periodic set by small perturbations we also indicate the regions in parameter space where the corresponding limit periodic set has cyclicity at most one or at most two

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

bootstrap technique in cluster analysis abstract we define a method to estimate the number of clusters in a data set e using the bootstrap technique this approach involves the generation of several fake data sets by sampling patterns with replacement in e bootstrapping for each number k of clusters a measure of stability of the k cluster partitions over the bootstrap samples is used to characterize the significance of the k cluster partition for the original data set the value of k which provides the most stable partitions is the estimate of the number of clusters in e the performance of this new technique is demonstrated on both synthetic and real data and is applied to the segmentation of range images

limit cycles near hyperbolas in quadratic systems abstract in this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane we study a strip of hyperbolas occurring in some quadratic systems we deal with the cyclicity of the degenerate graphics d i 2 a from the programme set up in f dumortier r roussarie c rousseau hilbert s 16th problem for quadratic vector fields j differential equations 110 1994 86 133 to solve the finiteness part of hilbert s 16th problem for quadratic systems techniques from geometric singular perturbation theory are combined with the use of the bautin ideal we also rely on the theory of darboux integrability

multi layer canard cycles and translated power functions abstract the paper deals with two dimensional slow fast systems and more specifically with multi layer canard cycles these are canard cycles passing through n layers of fast orbits with n 2 the canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n we prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

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

clustering granular data and their characterization with information granules of higher type the study is devoted to the clustering of granular data and an evaluation of the results of such clustering a comprehensive and systematic approach is developed which is composed of three fundamental phases 1 representation of granular data 2 clustering carried out in the representation space of information granules and 3 evaluation of quality of clusters following the reconstruction criterion the reconstruction criterion formed originally for numeric data and leading to an idea of granular prototypes is revisited we show here an emergence of granular information of higher type which are used to implement granular interval prototypes we discuss a way of forming granular data in the context of representation of time series and present clustering of granular time series

multiple canard cycles in generalized lienard equations the paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields the results deal with any number of parameters proofs are based on the techniques introduced in canard cycles and center manifolds f dumortier and r roussarie 1996 mem amer math soc 121 the presentation is limited to generalized lienard equations ex x c x x c 0

a novel cluster validity index for fuzzy clustering based on bipartite modularity a novel cluster validity index whose implementation is based on the membership degrees and improved bipartite modularity of bipartite network is proposed for the validation of partitions produced by the fuzzy c means fcm algorithm fcm algorithm is employed to group the dataset in order to obtain the membership degree of samples then a weighted bipartite network is constructed by samples and centroids of each cluster this allows the introduction of a new measurement for optimizing the numbers of clusters for fuzzy partitions the proposed index utilizes the optimum membership as its global property and the modularity of bipartite network as its local independent property the proposed index is compared with a number of popular validation indices on fifteen datasets the experimental results show that the effectiveness and reliability of the proposal is superior to other 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

the generalized c index for internal fuzzy cluster validity the c index is an internal cluster validity index that was introduced in 1970 as a way to define and identify a best crisp partition on n objects represented by either unlabeled feature vectors or dissimilarity matrix data this index is often one of the better performers among the plethora of internal indices available for this task this paper develops a soft generalization of the c index that can be used to evaluate sets of candidate partitions found by either fuzzy or probabilistic clustering algorithms we define four generalizations based on relational transformations of the soft partition and then compare their performance to eight other popular internal fuzzy cluster indices using two methods of comparison internal best c and internal external i e best match six synthetic datasets and six real world labeled datasets our main conclusion is that the sum min generalization is the second best performer in the best c tests and the best performer in the i e tests on small data

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

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

an efficient approach for unsupervised fuzzy clustering based on grouping evolution strategies fuzzy data clustering plays an important role in practical use and has become a prerequisite step for decision making in fuzzy environment in this paper we propose a new algorithm called fuzzyges for unsupervised fuzzy clustering based on adaptation of the recently proposed grouping evolution strategy ges to adapt ges for fuzzy clustering we devise a fuzzy counterpart of the grouping mutation operator typically used in ges and employ it in a two phase procedure to generate a new clustering solution unlike conventional clustering algorithms which should get the number of clusters as an input fuzzyges tries to determine the true number of clusters as well as providing the optimal cluster centroids after several iterations the proposed approach is validated through using several data sets and results are compared with those of fuzzy c means algorithm particle swarm optimization algorithm pso differential evolution de and league championship algorithm lca we also investigate the performance of fuzzyges through using different cluster validity indices our results indicate that fuzzyges is fast and provides acceptable results in terms of both determining the correct number of clusters and the accurate cluster centroids

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 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

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

stability analysis of control system having pd type of fuzzy controller stability of a control system having pd type of fuzzy controller a two input and single output controller is analyzed in this paper using yakubovich s method in this analysis the controller is treated as a two input and two output controller and the nonlinearity of the controller is characterized as the sector condition absolute stability conditions are derived for the control system as three conditions two are conditions for either one of control modes p or d is independently operating the third is the condition for two modes are simultaneously active the former two conditions are nothing but popov s conditions for two siso systems a new graphical method is proposed to solve the latter condition this method is explained illustratively using an example and the results obtained are compared numerically with those by other graphical methods as well as simulation results

stability analysis of the simplest takagi sugeno fuzzy control system using circle criterion in this paper we present two sufficient conditions in the frequency domain for the global stability analysis of the simplest takagi sugeno t s fuzzy control system based on the circle criterion and its graphical interpretation these conditions are significant in graphically designing the simplest t s fuzzy controller in the frequency domain four numerical examples are provided to demonstrate the efficiency of our two frequency domain based conditions furthermore this t s fuzzy controller and the linear proportional controllers are also compared it is concluded that the simplest t s fuzzy controller can outperform the linear proportional controllers in a noisy environment with external disturbances

relational generalizations of cluster validity indices numerous computational schemes have arisen over the years that attempt to learn information about objects based upon the similarity or dissimilarity of one object to another one such scheme clustering looks for self similar groups of objects to use clustering algorithms an investigator must often have a priori knowledge of the number of clusters i e c to search for in the data moreover it is often convenient to have ways to rank the returned results either for a single value of c a range of c s different clustering methods or any combination thereof however the task of assessing the quality of the results so that c may be determined objectively is currently ill defined for object object relationships to bridge this gap we generalize three well known validity indices the modified hubert s gamma xie beni and the generalized dunn s indices to relational data in doing so we develop a framework to convert many other validity indices to a relational form numerical examples on 12 datasets samples from four normal mixtures four real world object datasets and four real world pure relational datasets using the relational duals of the hard fuzzy and possibilistic c means cluster algorithms are offered to illustrate and evaluate the new indices

comparing fuzzy probabilistic and possibilistic partitions when clustering produces more than one candidate to partition a finite set of objects o there are two approaches to validation i e selection of a best partition and implicitly a best value for c which is the number of clusters in o first we may use an internal index which evaluates each partition separately second we may compare pairs of candidates with each other or with a reference partition that purports to represent the true cluster structure in the objects this paper generalizes many of the classical indices that have been used with outputs of crisp clustering algorithms so that they are applicable for candidate partitions of any type i e crisp or soft with soft comprising the fuzzy probabilistic and possibilistic cases space prevents inclusion of all of the possible generalizations that can be realized this way here we concentrate on the rand index and its modifications we compare our fuzzy rand index with those of campello hullermeier and rifqi and brouwer and show that our extension of the rand index is o n while the other three are all o n2 numerical examples are given to illustrate various facets of the new indices in particular we show that our indices can be used even when the partitions are probabilistic or possibilistic and that our method of generalization is valid for any index that depends only on the entries of the classical i e four pair types contingency table for this problem

singular perturbations and vanishing passage through a turning point abstract the paper deals with planar slow fast cycles containing a unique generic turning point we address the question on how to study canard cycles when the slow dynamics can be singular at the turning point we more precisely accept a generic saddle node bifurcation to pass through the turning point it reveals that in this case the slow divergence integral is no longer the good tool to use but its derivative with respect to the layer variable still is we provide general results as well as a number of applications we show how to treat the open problems presented in artes et al 2009 1 and dumortier and rousseau 2009 13 dealing respectively with the graphics di 2 a and df 1 a from dumortier et al 1994 14

some new qualitative insights into quality of fuzzy rule based models rules in fuzzy rule based models convey essential knowledge about the system under discussion as such they capture the essence of relationships occurring among input and output variables while the quality of such fuzzy models is predominantly related with the accuracy and eventual interpretability of rules although to a limited extent the quality of rules being regarded as generic pieces of knowledge has not been studied so far in this study we formulate and investigate this problem by looking at the quality of rules including aspects of a stability b generalizability and c conflict we identify a concept of rule multiplicity conflict and study an emergence of rule generalization a number of pertinent performance indices are developed and their usage is presented through a series of experimental studies

a convex semi nonnegative matrix factorisation approach to fuzzy c means clustering we propose an alternative approach to fuzzy c means clustering which eliminates the weighting exponent parameter of conventional algorithms it is based on a particular convex factorisation of data matrix the proposed method is invariant under certain linear transformations of the data including principal component analysis we tested its accuracy using both synthetic data and real datasets and compared it to that provided by the usual fuzzy c means algorithm we were able to ascertain that our proposal can be a credible yet easier alternative to this approach to fuzzy clustering moreover it showed no noticeable sensitivity to the initial guess of the partition matrix

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

minimax e stable cluster validity index for type 2 fuzziness in this paper we concentrate on the usage of uncertainty associated with the level of fuzziness in determination of the number of clusters in fcm for any data set we propose a minimax e stable cluster validity index based on the uncertainty associated with the level of fuzziness within the framework of interval valued type 2 fuzziness if the data have a clustered structure the optimum number of clusters may be assumed to have minimum uncertainty under upper and lower levels of fuzziness upper and lower values of the level of fuzziness for fuzzy c mean fcm clustering methodology have been found as m 2 6 and 1 4 respectively in our previous studies our investigation shows that the stability of cluster centers with respect to the level of fuzziness is sufficient for the determination of the number of clusters

determining the number of clusters using information entropy for mixed data in cluster analysis one of the most challenging and difficult problems is the determination of the number of clusters in a data set which is a basic input parameter for most clustering algorithms to solve this problem many algorithms have been proposed for either numerical or categorical data sets however these algorithms are not very effective for a mixed data set containing both numerical attributes and categorical attributes to overcome this deficiency a generalized mechanism is presented in this paper by integrating renyi entropy and complement entropy together the mechanism is able to uniformly characterize within cluster entropy and between cluster entropy and to identify the worst cluster in a mixed data set in order to evaluate the clustering results for mixed data an effective cluster validity index is also defined in this paper furthermore by introducing a new dissimilarity measure into the k prototypes algorithm we develop an algorithm to determine the number of clusters in a mixed data set the performance of the algorithm has been studied on several synthetic and real world data sets the comparisons with other clustering algorithms show that the proposed algorithm is more effective in detecting the optimal number of clusters and generates better clustering results

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

canards heteroclinic and homoclinic orbits for a slow fast predator prey model of generalized holling type iii abstract for a classical ratio dependent predator prey model with the generalized holling type iii functional response it was previously investigated in 20 by hsu and huang for global stability of an equilibrium and in 21 by huang ruan and song for subcritical hopf and bogdanov takens bifurcations here in this model when prey reproduces much faster than predator by using geometric singular perturbation theory we achieve much richer new dynamical phenomena than the existing ones such as the existence of canard cycles canard explosion and relaxation oscillations heteroclinic and homoclinic orbits cyclicity of slow fast cycles and the coexistence of the hopf cycle and the relaxation oscillation on global stability of the equilibrium we also provide less restricted conditions than the existing ones

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

slow divergence integral and its application to classical lienard equations of degree 5 abstract the slow divergence integral is a crucial tool to study the cyclicity of a slow fast cycle for singularly perturbed planar vector fields in this paper we deduce a useful form for this integral in order to apply it to various problems as an example we use it to prove that the slow divergence integral along any non degenerate slow fast cycle for singular perturbations of classical lienard equations of degree 5 has at most one zero and the zero is simple if it exists hence the cyclicity of any non degenerate slow fast cycle in this class of equations is at most 2 up to now there were many interesting results about lienard equations of degree 3 4 and 6 but almost nothing is known about degree 5 the result in this paper can be seen as a first stage to study the uniform property for classical lienard equations of degree 5

detecting change points for shifts in mean and variance using fuzzy classification maximum likelihood change point algorithms knowing the time of changes called change point cp in a process is crucial for engineers to recognize the root cause fast and accurately since special causes may induce simultaneous changes in mean and variance detecting changes in both at once is required many methodologies in quality control were developed for detecting changes in either mean or variance only and process parameters were assumed known often however they are rarely known exactly and a small estimation error may lead to unfavorable cp estimates fuzzy partitioning is better suited to cases of vague boundaries between two segments which appear very often in reality a new mechanism called fuzzy classification maximum likelihood change point fcml cp algorithm is proposed to detect shifts in mean and variance simultaneously a cp framework is transferred into a mixture model and then a fcml cp algorithm is created through fuzzy classification maximum likelihood procedures the proposed fcml cp can be applied to phase i and ii processes without knowledge of in control process parameters it can estimate multiple cps of process mean or and variance simultaneously the effectiveness and superiority of fcml cp are shown by extensive experiments with numerical and real data sets specifically the proposed fcml cp is superior to the commonly used statistical mixture likelihood approach using expectation maximization em algorithm it is much more time saving especially the remarkable performance of fcml cp in detecting cps for small changes is particularly important and helpful for engineers to recognize the special cause fast and correctly since an out of control signal resulted from small changes is usually delayed long we propose a new algorithm fcml cp to detect change points in a statistical process the mixture likelihood embedded fuzzy c partition is utilized to estimate the change points the method outperforms statistical likelihood approaches various experiments show the effectiveness and practicability of the proposed method excellent performance in detecting small changes is helpful for root cause analyses

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

robust stability analysis of fuzzy control systems abstract robust stability of fuzzy control systems is studied through the popov lyapunov approach first we transform a fuzzy control system into a lur e system with uncertainties or nonlinearities second the lyapunov s direct method is used to guarantee the stability of the perturbed lur e system and then a robustness measurement which gives the bound on allowable uncertainties or nonlinearities is derived also if the bound of the uncertainties nonlinearities of the system is known we can find an estimated region of asymptotic stability in phase space a nonlinear numerical example and a magnetic bearing experiment demonstrating the feasibility of the method are illustrated

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

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

fuzzy rough nearest neighbour classification and prediction nearest neighbour nn approaches are inspired by the way humans make decisions comparing a test object to previously encountered samples in this paper we propose an nn algorithm that uses the lower and upper approximations from fuzzy rough set theory in order to classify test objects or predict their decision value it is shown experimentally that our method outperforms other nn approaches classical fuzzy and fuzzy rough ones and that it is competitive with leading classification and prediction methods moreover we show that the robustness of our methods against noise can be enhanced effectively by invoking the approximations of the vaguely quantified rough set vqrs model which emulates the linguistic quantifiers some and most from natural language

beyond fcm graph theoretic post processing algorithms for learning and representing the data structure we show that when fuzzy c means fcm algorithm is used in an over partitioning mode the resulting membership values can be further utilized for building a connectivity graph that represents the relative distribution of the computed centroids standard graph theoretic procedures and recent algorithms from manifold learning theory are subsequently applied to this graph this facilitates the accomplishment of a great variety of data analysis tasks the definition of optimal cluster number c o the detection of intrinsic geometrical constraints within the data and the faithful low dimensional representation of the original structure are all performed efficiently by working with just a down sampled version comprised of the centroids of the data our approach is extensively demonstrated using synthetic data and actual brain signals

a stability based validity method for fuzzy clustering an important goal in cluster analysis is the internal validation of results using an objective criterion of particular relevance in this respect is the estimation of the optimum number of clusters capturing the intrinsic structure of your data this paper proposes a method to determine this optimum number based on the evaluation of fuzzy partition stability under bootstrap resampling the method is first characterized on synthetic data with respect to hyper parameters like the fuzzifier and spatial clustering parameters such as feature space dimensionality clusters degree of overlap and number of clusters the method is then validated on experimental datasets furthermore the performance of the proposed method is compared to that obtained using a number of traditional fuzzy validity rules based on the cluster compactness to separation criteria the proposed method provides accurate and reliable results and offers better generalization capabilities than the classical approaches

hierarchical unsupervised fuzzy clustering a recursive algorithm for hierarchical fuzzy partitioning is presented the algorithm has the advantages of hierarchical clustering while maintaining fuzzy clustering rules each pattern can have a nonzero membership in more than one subset of the data in the hierarchy optimal feature extraction and reduction is optionally reapplied for each subset combining hierarchical and fuzzy concepts is suggested as a natural feasible solution to the cluster validity problem of real data the convergence and membership conservation of the algorithm are proven the algorithm is shown to be effective for a variety of data sets with a wide dynamic range of both covariance matrices and number of members in each class

cluster validity for the fuzzy c means clustering algorithrm the uniform data function is a function which assigns to the output of the fuzzy c means fc m or fuzzy isodata algorithm a number which measures the quality or validity of the clustering produced by the algorithm for the preselected number of cluster c the fc m algorithm produces c vectors in the space in which the data lie called cluster centers which represent points about which the data are concentrated it also produces for each data point c membership values numbers between zero and one which measure the similarity of the data points to each of the cluster centers it is these membership values which indicate how the point is classified they also indicate how well the point has been classified in that values close to one indicate that the point is close to a particular center but uniformly low memberships indicate that the point has not been classified clearly the uniform data functional udf combines the memberships in such a way as to indicate how well the data have been classified and is computed as follows for each data point compute the ratio of its smallest membership to its largest and then compute the probability that one could obtain a smaller ratio indicating better classification from a clustering of a standard data set in which there is no cluster structure these probabilities are then averaged over the data set to obtain the values of the udf

singular perturbations of the holling i predator prey system with a focus abstract we study the occurrence of limit cyles in a model for two species predator prey interaction referred to as holling i this model is based on the phenomenological fitting of the so called functional response function to observed data in nature by a continuous piecewise differentiable function of the original models presented by holling it is the only model for which the possible asymptotic behaviour of the prey and predator densities has not been determined yet we extend the work of liu who showed that for certain values of parameters of the holling i system at least two nested limit cycles will occur surrounding a focus his case corresponds to a bifurcation problem where limit cycles are created from a system with a continuum of singularities i e a singular perturbation problem with slow fast solutions we prove that exactly two hyperbolic limit cycles occur after perturbation

fuzzy rule based decision trees this paper presents a new architecture of a fuzzy decision tree based on fuzzy rules fuzzy rule based decision tree frdt and provides a learning algorithm in contrast with traditional axis parallel decision trees in which only a single feature variable is taken into account at each node the node of the proposed decision trees involves a fuzzy rule which involves multiple features fuzzy rules are employed to produce leaves of high purity using multiple features for a node helps us minimize the size of the trees the growth of the frdt is realized by expanding an additional node composed of a mixture of data coming from different classes which is the only non leaf node of each layer this gives rise to a new geometric structure endowed with linguistic terms which are quite different from the traditional oblique decision trees endowed with hyperplanes as decision functions a series of numeric studies are reported using data coming from uci machine learning data sets the comparison is carried out with regard to traditional decision trees such as c4 5 ladtree bftree simplecart and nbtree the results of statistical tests have shown that the proposed frdt exhibits the best performance in terms of both accuracy and the size of the produced trees highlightsthis paper presents a new architecture of a fuzzy decision tree based on fuzzy rules the growth of the proposed tree is realized by expanding an additional node the proposed tree can capture the oblique geometric structure of class regions the main idea is creating a node for each class to be predicted at every level of the tree

normality based validation for crisp clustering we introduce a new validity index for crisp clustering that is based on the average normality of the clusters unlike methods based on inter cluster and intra cluster distances this index emphasizes the cluster shape by using a high order characterization of its probability distribution the normality of a cluster is characterized by its negentropy a standard measure of the distance to normality which evaluates the difference between the cluster s entropy and the entropy of a normal distribution with the same covariance matrix the definition of the negentropy involves the distribution s differential entropy however we show that it is possible to avoid its explicit computation by considering only negentropy increments with respect to the initial data distribution where all the points are assumed to belong to the same cluster the resulting negentropy increment validity index only requires the computation of covariance matrices we have applied the new index to an extensive set of artificial and real problems where it provides in general better results than other indices both with respect to the prediction of the correct number of clusters and to the similarity among the real clusters and those inferred

generating a hierarchical fuzzy rule based model abstract this study proposes a novel methodology for the extraction of a hierarchical takagi sugeno fuzzy rule based architecture from data this architecture reduces the number and complexity of involved fuzzy rules and in many cases improves the predictive performance of the model the proposed hierarchical architecture takes the form of a cascading topology in which the predicted result computed at the previous layer is considered in the output part of the fuzzy rules we propose a well defined general methodology for the extraction of this hierarchical topology from data and discuss strategies for feature selection and choosing the number of rules at each level the performance of the proposed methodology is demonstrated through extensive experiments including case studies outlining specific behaviors and parameterizations and comparative experiments showing the performance of the proposed architecture compared to a standard flat fuzzy rule based system

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

a possibilistic fuzzy c means clustering algorithm in 1997 we proposed the fuzzy possibilistic c means fpcm model and algorithm that generated both membership and typicality values when clustering unlabeled data fpcm constrains the typicality values so that the sum over all data points of typicalities to a cluster is one the row sum constraint produces unrealistic typicality values for large data sets in this paper we propose a new model called possibilistic fuzzy c means pfcm model pfcm produces memberships and possibilities simultaneously along with the usual point prototypes or cluster centers for each cluster pfcm is a hybridization of possibilistic c means pcm and fuzzy c means fcm that often avoids various problems of pcm fcm and fpcm pfcm solves the noise sensitivity defect of fcm overcomes the coincident clusters problem of pcm and eliminates the row sum constraints of fpcm we derive the first order necessary conditions for extrema of the pfcm objective function and use them as the basis for a standard alternating optimization approach to finding local minima of the pfcm objective functional several numerical examples are given that compare fcm and pcm to pfcm our examples show that pfcm compares favorably to both of the previous models since pfcm prototypes are less sensitive to outliers and can avoid coincident clusters pfcm is a strong candidate for fuzzy rule based system identification

asymptotic lower bounds on hilbert numbers using canard cycles abstract in this work we give an asymptotic lower bound for the hilbert number for real planar polynomial differential systems this lower bound equals up to leading order to the best existing one but the method we provide is new and involves slow fast systems the construction strongly relies on generalized lienard systems

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 fuzzy c spherical shells algorithm a new approach the fuzzy c spherical shells fcss algorithm is specially designed to search for clusters that can be described by circular arcs or generally by shells of hyperspheres a new approach to the fcss algorithm is presented this algorithm is computationally and implementationally simpler than other clustering algorithms that have been suggested for this purpose an unsupervised algorithm which automatically finds the optimum number of clusters is not known it uses a cluster validity measure to identify good clusters merges all compatible clusters and eliminates spurious clusters to achieve the final results experimental results on several data sets are presented

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

design of state feedback adaptive fuzzy controllers for second order systems using a frequency stability criterion this paper proposes a design method for second order systems that guarantees the stability of an adaptive fuzzy controller with state feedback the system consists of a linear unstable plant the takagi sugeno controller a nonlinear reference model and an adaptation mechanism in this paper gradient based adaptation is used to change the consequents of the controller rules so that the closed loop system behaves like the reference model the proposed method utilizes a frequency domain criterion in the form of the modified circle theorem the controller function is assumed to be a nonlinearity described by a sector condition which means that the function lies between two planes during the process of adaptation this function is verified so it stays in the sector guaranteeing stability the method described here is illustrated by an example of a control system containing an unstable plant with an unknown pole in the right half plane the motivation of this paper is to propose a frequency domain method for the design of state feedback adaptive fuzzy controllers comparing with time domain methods that require advanced software tools the proposed method offers simple graphical interpretation on the nyquist plane