fast and robust fuzzy c means clustering algorithms incorporating local information for image segmentation fuzzy c means fcm algorithms with spatial constraints fcm s have been proven effective for image segmentation however they still have the following disadvantages 1 although the introduction of local spatial information to the corresponding objective functions enhances their insensitiveness to noise to some extent they still lack enough robustness to noise and outliers especially in absence of prior knowledge of the noise 2 in their objective functions there exists a crucial parameter a used to balance between robustness to noise and effectiveness of preserving the details of the image it is selected generally through experience and 3 the time of segmenting an image is dependent on the image size and hence the larger the size of the image the more the segmentation time in this paper by incorporating local spatial and gray information together a novel fast and robust fcm framework for image segmentation i e fast generalized fuzzy c means fgfcm clustering algorithms is proposed fgfcm can mitigate the disadvantages of fcm s and at the same time enhances the clustering performance furthermore fgfcm not only includes many existing algorithms such as fast fcm and enhanced fcm as its special cases but also can derive other new algorithms such as fgfcm s1 and fgfcm s2 proposed in the rest of this paper the major characteristics of fgfcm are 1 to use a new factor s i j as a local both spatial and gray similarity measure aiming to guarantee both noise immunity and detail preserving for image and meanwhile remove the empirically adjusted parameter a 2 fast clustering or segmenting image the segmenting time is only dependent on the number of the gray levels q rather than the size n q of the image and consequently its computational complexity is reduced from o nci 1 to o qci 2 where c is the number of the clusters i 1 and i 2

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

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

fuzzy clustering for data time arrays with inlier and outlier time trajectories in many knowledge discovery and data mining tasks fuzzy clustering is one of the most common tools for data partitioning in this paper dynamic fuzzy clustering models for classifying a set of multivariate time trajectories time series sequences are developed in particular by adopting an exploratory approach based on a geometric algebraic formulation of the data time array different kinds of dynamic fuzzy clustering models based on cross sectional and longitudinal aspects are suggested furthermore a modified version of the previous clustering models that can be seen as a generalization of these models is proposed by utilizing these models we can obtain beneficial effects in the clustering process when anomalous trajectories trajectories with anomalous positions and slopes are present in the dataset in fact the models are suitable for detecting structures of time trajectories with anomalous patterns that are not uniformly distributed over the structure s domains and are characterized by strange slopes in these models the disruptive effect of the anomalous trajectories is neutralized and smoothed and the information on the influence of individual time trajectories on the detected groups is given furthermore some remarks on dynamic three way extensions of a few robust fuzzy clustering models for two way data are suggested demonstrative examples are shown and a comparison assessment based on artificial multivariate time varying data is carried out

estimation of a fuzzy regression model using fuzzy distances regression analysis is a powerful statistical tool that has many applications in different areas the problem of regression analysis under a fuzzy environment has been treated in the literature from different points of view and considering a variety of input output data crisp or fuzzy however we realize that in general most research papers have a conflict between the solution of the fuzzy regression problem using crisp distances minimizing a real error function and the interpretation of fuzzy data as possibility distributions the main aim of this paper is to develop a methodology to solve this problem introducing a fuzzy partial order and a family of fuzzy distance measures on the whole set of fuzzy numbers the new approach allows us to obtain linear and nonlinear models that reach the lowest fuzzy error the estimation process in general can be considered easier to apply in practice and it is not limited to triangular fuzzy numbers numerical examples are provided to illustrate the usefulness and applicability of these results and comparisons with existing methodologies show that the performance of the proposed solution is very satisfactory

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

garch based robust clustering of time series in this paper we propose different robust fuzzy clustering models for classifying heteroskedastic volatility time series following the so called model based approach to time series clustering and using a partitioning around medoids procedure the proposed models are based on a garch parametric modeling of the time series i e the unconditional volatility and the time varying volatility garch representation of the time series we first suggest a timid robustification of the fuzzy clustering then we propose three robust fuzzy clustering models belonging to the so called metric noise and trimmed approaches respectively each model neutralizes the negative effects of the outliers in the clustering process in a different manner in particular the first robust model based on the metric approach achieves its robustness with respect to outliers by taking into account a robust distance measure the second based on the noise approach achieves its robustness by introducing a noise cluster represented by a noise prototype the third based on the trimmed approach achieves its robustness by trimming away a certain fraction of outlying time series the usefulness and effectiveness of the proposed clustering models is illustrated by means of a simulation study and two applications in finance and economics

penalized preimage learning in kernel principal component analysis finding the preimage of a feature vector in kernel principal component analysis kpca is of crucial importance when kpca is applied in some applications such as image preprocessing since the exact preimage of a feature vector in the kernel feature space normally does not exist in the input data space an approximate preimage is learned and encouraging results have been reported in the last few years however it is still difficult to find a good estimation of preimage as estimation of preimage in kernel methods is ill posed how to guide the preimage learning for a better estimation is important and still an open problem to address this problem a penalized strategy is developed in this paper where some penalization terms are used to guide the preimage learning process to develop an efficient penalized technique we first propose a two step general framework in which a preimage is directly modeled by weighted combination of the observed samples and the weights are learned by some optimization function subject to certain constraints compared to existing techniques this would also give advantages in directly turning preimage learning into the optimization of the combination weights under this framework a penalized methodology is developed by integrating two types of penalizations first to ensure learning a well defined preimage of which each entry is not out of data range convexity constraint is imposed for learning the combination weights more insight effects of the convexity constraint are also explored second a penalized function is integrated as part of the optimization function to guide the preimage learning process particularly the weakly supervised penalty is proposed discussed and extensively evaluated along with laplacian penalty and ridge penalty it could be further interpreted that the learned preimage can preserve some kind of pointwise conditional mutual information finally kpca with preimage learning is applied on face image data sets in the aspects of facial expression normalization face image denoising recovery of missing parts from occlusion and illumination normalization experimental results show that the proposed preimage learning algorithm obtains lower mean square error mse and better visual quality of reconstructed images

general c means clustering model partitional clustering is an important part of cluster analysis based on various theories numerous clustering algorithms have been developed and new clustering algorithms continue to appear in the literature it is known that occam s razor plays a pivotal role in data based models and partitional clustering is categorized as a data based model however no relation had previously been discovered between occam s razor and partitional clustering as we discuss in this paper the three main contributions of this paper can be summarized as follows 1 according to a novel definition of the mean a unifying generative framework for partitional clustering algorithms called a general c means clustering model gcm is presented and studied and 2 based on the local optimality test of the gcm the connection between occam s razor and partitional clustering is established for the first time as its application a comprehensive review of the existing objective function based clustering algorithms is presented based on gcm 3 under a common assumption about partitional clustering a theoretical guide for devising and implementing clustering algorithm is discovered these conclusions are verified by numerical experimental results

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

a robust em clustering algorithm for gaussian mixture models clustering is a useful tool for finding structure in a data set the mixture likelihood approach to clustering is a popular clustering method in which the em algorithm is the most used method however the em algorithm for gaussian mixture models is quite sensitive to initial values and the number of its components needs to be given a priori to resolve these drawbacks of the em we develop a robust em clustering algorithm for gaussian mixture models first creating a new way to solve these initialization problems we then construct a schema to automatically obtain an optimal number of clusters therefore the proposed robust em algorithm is robust to initialization and also different cluster volumes with automatically obtaining an optimal number of clusters some experimental examples are used to compare our robust em algorithm with existing clustering methods the results demonstrate the superiority and usefulness of our proposed method

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

fuzzy c means clustering with local information and kernel metric for image segmentation in this paper we present an improved fuzzy c means fcm algorithm for image segmentation by introducing a tradeoff weighted fuzzy factor and a kernel metric the tradeoff weighted fuzzy factor depends on the space distance of all neighboring pixels and their gray level difference simultaneously by using this factor the new algorithm can accurately estimate the damping extent of neighboring pixels in order to further enhance its robustness to noise and outliers we introduce a kernel distance measure to its objective function the new algorithm adaptively determines the kernel parameter by using a fast bandwidth selection rule based on the distance variance of all data points in the collection furthermore the tradeoff weighted fuzzy factor and the kernel distance measure are both parameter free experimental results on synthetic and real images show that the new algorithm is effective and efficient and is relatively independent of this type of noise

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

tangent hyperplane kernel principal component analysis for denoising kernel principal component analysis kpca is a method widely used for denoising multivariate data using geometric arguments we investigate why a projection operation inherent to all existing kpca denoising algorithms can sometimes cause very poor denoising based on this we propose a modification to the projection operation that remedies this problem and can be incorporated into any of the existing kpca algorithms using toy examples and real datasets we show that the proposed algorithm can substantially improve denoising performance and is more robust to misspecification of an important tuning parameter

perverse sheaves koszul ic modules and the quiver for the category mathcal o for a stratified topological space we introduce the category of ic modules which are linear algebra devices with the relations described by the equation d 2 0 we prove that the category of mixed ic modules is equivalent to the category of mixed perverse sheaves for flag varieties as an application we describe an algorithm calculating the quiver underlying the bgg category mathcal o for arbitrary simple lie algebra thus answering a question which goes back to i m gelfand

quivers with relations arising from koszul algebras of g invariants abstract let g be a complex simple lie algebra and let be an extremal set of positive roots after chari and greenstein 2009 9 one associates with an infinite dimensional koszul algebra s g which is a graded subalgebra of the locally finite part of end v op s g g where v is the direct sum of all simple finite dimensional g modules we describe the structure of the algebra s g explicitly in terms of an infinite quiver with relations for g of types a and c we also describe several infinite families of quivers and finite dimensional associative algebras arising from this construction

projective modules in the category _ self duality given a parabolic subalgebra ps of a complex semisimple lie algebra there is a naturally defined category s of g modules which includes all the g modules induced from finite dimensional ps modules this paper treats the question of whether certain projective modules in s are isomorphic to their dual modules the projectives in question are the projective covers of those simple modules occurring in the socles of generalized verma modules their self duality is proved in a number of cases and additional information is obtained on the generalized verma modules

a similarity based robust clustering method this paper presents an alternating optimization clustering procedure called a similarity based clustering method scm it is an effective and robust approach to clustering on the basis of a total similarity objective function related to the approximate density shape estimation we show that the data points in scm can self organize local optimal cluster number and volumes without using cluster validity functions or a variance covariance matrix the proposed clustering method is also robust to noise and outliers based on the influence function and gross error sensitivity analysis therefore scm exhibits three robust clustering characteristics 1 robust to the initialization cluster number and initial guesses 2 robust to cluster volumes ability to detect different volumes of clusters and 3 robust to noise and outliers several numerical data sets and actual data are used in the scm to show these good aspects the computational complexity of scm is also analyzed some experimental results of comparing the proposed scm with the existing methods show the superiority of the scm method

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

a selection model for optimal fuzzy clustering algorithm and number of clusters based on competitive comprehensive fuzzy evaluation fuzzy c means fcm and its variants suffer from two problems local minima and cluster validity which have a direct impact on the formation of final clustering there are two strategies optimization and center initialization strategies that address the problem of local minima this paper proposes a center initialization approach based on a minimum spanning tree to keep fcm from local minima with regard to cluster validity various strategies have been proposed on the basis of the fuzzy cluster validity index this paper proposes a selection model that combines multiple pairs of a fuzzy clustering algorithm and cluster validity index to identify the number of clusters and simultaneously selects the optimal fuzzy clustering for a dataset the promising performance of the proposed center initialization method and selection model is demonstrated by experiments on real datasets

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

from moment graphs to intersection cohomology we describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori in terms of structure of the zero and one dimensional orbits the class of varieties to which our formula applies includes schubert varieties in flag varieties and affine flag varieties we also prove a monotonicity result on local intersection cohomology stalks

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

perverse sheaves and actions where t x denotes the action of t on x the set xw is known to be a locally closed c stable algebraic subvariety of x isomorphic to an affine space the pieces xw form a cell decomposition x uwewxw w e w the socalled bialynicki birula decomposition we assume this decomposition to be an algebraic stratification of x the closure of a cell may not be a union of cells in general let zw be the intersection cohomology complex on the closure of a cell xw extended by 0 to the whole of x the hypercohomology h 2w has a natural structure of a graded h x c module further for any x y e w there is a natural morphism of graded spaces

the pre image problem in kernel methods in this paper we address the problem of finding the pre image of a feature vector in the feature space induced by a kernel this is of central importance in some kernel applications such as on using kernel principal component analysis pca for image denoising unlike the traditional method in which relies on nonlinear optimization our proposed method directly finds the location of the pre image based on distance constraints in the feature space it is noniterative involves only linear algebra and does not suffer from numerical instability or local minimum problems evaluations on performing kernel pca and kernel clustering on the usps data set show much improved performance

fuzzy clustering algorithms for mixed feature variables this paper presents fuzzy clustering algorithms for mixed features of symbolic and fuzzy data el sonbaty and ismail proposed fuzzy c means fcm clustering for symbolic data and hathaway et al proposed fcm for fuzzy data in this paper we give a modified dissimilarity measure for symbolic and fuzzy data and then give fcm clustering algorithms for these mixed data types numerical examples and comparisons are also given numerical examples illustrate that the modified dissimilarity gives better results finally the proposed clustering algorithm is applied to real data with mixed feature variables of symbolic and fuzzy data

effective level set image segmentation with a kernel induced data term this study investigates level set multiphase image segmentation by kernel mapping and piecewise constant modeling of the image data thereof a kernel function maps implicitly the original data into data of a higher dimension so that the piecewise constant model becomes applicable this leads to a flexible and effective alternative to complex modeling of the image data the method uses an active curve objective functional with two terms an original term which evaluates the deviation of the mapped image data within each segmentation region from the piecewise constant model and a classic length regularization term for smooth region boundaries functional minimization is carried out by iterations of two consecutive steps 1 minimization with respect to the segmentation by curve evolution via euler lagrange descent equations and 2 minimization with respect to the regions parameters via fixed point iterations using a common kernel function this step amounts to a mean shift parameter update we verified the effectiveness of the method by a quantitative and comparative performance evaluation over a large number of experiments on synthetic images as well as experiments with a variety of real images such as medical satellite and natural images as well as motion maps

mean shift based clustering in this paper a mean shift based clustering algorithm is proposed the mean shift is a kernel type weighted mean procedure herein we first discuss three classes of gaussian cauchy and generalized epanechnikov kernels with their shadows the robust properties of the mean shift based on these three kernels are then investigated according to the mountain function concepts we propose a graphical method of correlation comparisons as an estimation of defined stabilization parameters the proposed method can solve these bandwidth selection problems from a different point of view some numerical examples and comparisons demonstrate the superiority of the proposed method including those of computational complexity cluster validity and improvements of mean shift in large continuous discrete data sets we finally apply the mean shift based clustering algorithm to image segmentation

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

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data

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

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

fuzzy c ordered medoids clustering for interval valued data fuzzy clustering for interval valued data helps us to find natural vague boundaries in such data the fuzzy c medoids clustering fcmdc method is one of the most popular clustering methods based on a partitioning around medoids approach however one of the greatest disadvantages of this method is its sensitivity to the presence of outliers in data this paper introduces a new robust fuzzy clustering method named fuzzy c ordered medoids clustering for interval valued data fcomdc id the huber s m estimators and the yager s ordered weighted averaging owa operators are used in the method proposed to make it robust to outliers the described algorithm is compared with the fuzzy c medoids method in the experiments performed on synthetic data with different types of outliers a real application of the fcomdc id is also provided highlightsfuzzy clustering for interval valued data helps us to find natural vague boundaries in such data a new robust fuzzy clustering method named fuzzy c ordered medoids clustering for interval valued data fcomdc id is proposedthe method uses both the huber s m estimators and the yager s owa operators to obtain its robustness experiments performed on synthetic data with different types of outliers and a real application are provided

fuzzy c means based clustering for linearly and nonlinearly separable data in this paper we present a new distance metric that incorporates the distance variation in a cluster to regularize the distance between a data point and the cluster centroid it is then applied to the conventional fuzzy c means fcm clustering in data space and the kernel fuzzy c means kfcm clustering in a high dimensional feature space experiments on two dimensional artificial data sets real data sets from public data libraries and color image segmentation have shown that the proposed fcm and kfcm with the new distance metric generally have better performance on non spherically distributed data with uneven density for linear and nonlinear separation

rederivation of the fuzzypossibilistic clustering objective function through bayesian inference unsupervised clustering of a set of datums into homogeneous groups is a primitive operation required in many signal and image processing applications in fact different incarnations and hybrids of fuzzy c means fcm and possibilistic c means pcm have been suggested which address additional requirements such as accepting weighted sets and being robust to the presence of outliers nevertheless arriving at a general framework which is independent of the datum model and the notion of homogeneity of a particular problem class is a challenge however this process has not been followed organically and clustering algorithms are generally based on exogenous objective functions which are heuristically engineered and are believed to lead to the satisfaction of a required behavior these techniques also commonly depend on regularization coefficients which are to be set prudently by the user or through separate processes in contrast in this work we utilize bayesian inference and derive a robustified objective function for a fuzzypossibilistic clustering algorithm by assuming a generic datum model and a generic notion of cluster homogeneity we utilize this model for the purpose of cluster validity assessment as well we emphasize the epistemological importance of the theoretical basis on which the developed methodology rests at the end of this paper we present experimental results to exhibit the utilization of the developed framework in the context of four different problem classes

generalised kernel weighted fuzzy c means clustering algorithm with local information abstract to improve the performance of segmentation for the images corrupted by noise many variants of standard fuzzy c means fcm clustering algorithm have been proposed that incorporate the local spatial neighbourhood information to perform image segmentation among them the kernel weighted fuzzy local information c means kwflicm algorithm gives robust to noise image segmentation results by using local spatial image neighbourhood information it is limited to one dimensional input data i e image intensity in this paper we propose a generalisation of kwflicm gkwflicm that is applicable to m dimensional input data sets the proposed algorithm incorporates neighbourhood information among the m dimensional data which mitigates the disadvantages of the standard fcm clustering algorithm sensitive to noise and outliers poor performance for differently sized clusters and for different density clusters and greatly improves the clustering performance experiments have been performed on several noisy and non noisy data sets as well as natural and real world images to demonstrate the effectiveness efficiency and robustness to noise of the gkwflicm algorithm by comparing it to kernel fuzzy c means kfcm kernel possibilistic fuzzy c means kpfcm fuzzy local information c means flicm and kwflicm

kernelized fuzzy attribute c means clustering algorithm a novel kernelized fuzzy attribute c means clustering algorithm is proposed in this paper since attribute means clustering algorithm is an extension of fuzzy c means algorithm with weighting exponent m 2 and fuzzy attribute c means clustering is a general type of attribute c means clustering with weighting exponent m 1 we modify the distance in fuzzy attribute c means clustering algorithm with kernel induced distance and obtain kernelized fuzzy attribute c means clustering algorithm kernelized fuzzy attribute c means clustering algorithm is a natural generalization of kernelized fuzzy c means algorithm with stable function experimental results on standard iris database and tumor normal gene chip expression data demonstrate that kernelized fuzzy attribute c means clustering algorithm with gaussian radial basis kernel function and cauchy stable function is more effective and robust than fuzzy c means fuzzy attribute c means clustering and kernelized fuzzy c means as well

a fuzzy k partitions model for categorical data and its comparison to the gom model the grade of membership gom model uses fuzzy sets as memberships of each individual to extreme profiles or classes on the likelihood function of multivariate multinomial distributions the gom clustering algorithm derived from the gom model is used in cluster analysis for categorical data but it is iterated with complicated calculations in this paper we create another approach termed a fuzzy k partitions fkp model which is also based on the likelihood function of multivariate multinomial distributions however the calculations of the fkp algorithm for clustering categorical data derived from the proposed fkp model are simpler the proposed fkp clustering algorithm is not only easier in calculation than the gom but also has more accuracy and computation efficiency to verify it we employ real empirical data and also some simulation data we find that fkp has superior results to gom we then apply these two algorithms to classification of pathology the results show the superiority of the fkp clustering algorithm moreover the proposed fkp algorithm can be used as a fuzzy clustering algorithm for categorical data some comparisons between fkp and two popular algorithms fuzzy k modes and fuzzy centroids are made these results show that the fkp clustering algorithm can be another useful tool in analyzing categorical data

a cluster validity index for fuzzy clustering a new cluster validity index is proposed for the validation of partitions of object data produced by the fuzzy c means algorithm the proposed validity index uses a variation measure and a separation measure between two fuzzy clusters a good fuzzy partition is expected to have a low degree of variation and a large separation distance testing of the proposed index and nine previously formulated indices on well known data sets shows the superior effectiveness and reliability of the proposed index in comparison to other indices and the robustness of the proposed index in noisy environments

a clustering method to identify representative financial ratios when companies evaluate their performance it is impractical to take all of their financial ratios into consideration to evaluate the financial performance of a company only a fraction of the available financial ratios are considered and selected as evaluation criteria in general financial ratios presented as sequences or called financial ratio sequences are first clustered and then a representative indicator is chosen from each cluster to serve as an evaluation criterion to cluster financial ratios we propose a clustering method in which the financial ratios of different companies with similar variations are partitioned into the same cluster in other words a fuzzy relation is proposed to represent the similarity between the financial ratios and a cluster validation index is also provided to determine the number of clusters once the financial ratios are clustered the representative indicator for each cluster will be identified

fuzzy change point algorithms for regression models change point cp regression models have been widely applied in various fields where detecting cps is an important problem detecting the location of cps in regression models could be equivalent to partitioning data points into clusters of similar individuals in the literature fuzzy clustering has been widely applied in various fields but it is less used in locating cps in cp regression models in this paper a new method called fuzzy cp fcp algorithm is proposed to detect the cps and simultaneously estimate the parameters of regression models the fuzzy c partitions concept is first embedded into the cp regression models any possible collection of all cps is considered as a partitioning of data with a fuzzy membership we then transfer these memberships into the pseudomemberships of data points belonging to each individual cluster and therefore we can obtain the estimates for model parameters by the fuzzy c regressions method subsequently we use the fuzzy c means clustering to obtain the new iterates of the cp collection memberships by minimizing an objective function concerning the deviations between the predicted response values and data values we illustrate the new approach with several numerical examples and real datasets experimental results actually show that the proposed fcp is an effective and useful cp detection algorithm for cp regression models and can be applied to various fields such as econometrics medicine quality control and signal processing

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