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

spectral stability of traveling water waves eigenvalue collision singularities and direct numerical simulation abstract in a recent paper nicholls 2009 1 the author conjectured upon the connection between the onset of dynamic spectral instability of periodic traveling water waves and singularities present in taylor series representations of spectral data for the linearized water wave equations more specifically he proposed that the onset of instability is always coincident with encountering the smallest singularity in these taylor series in this paper we study this connection via a new direct numerical simulation algorithm derived from the surface formulation of the water wave problem due to zakharov 1968 5 and craig sulem 1993 6 we find compelling evidence that the conjecture is true in the case of deep as compared to benjamin feir s 1967 7 critical depth h 1 363 water but false for shallow depths as it significantly underpredicts the onset of instability the utility of the singularity identification strategy advocated in 1 while somewhat lessened in the shallow water case is nonetheless upheld due to its ability to reliably identify a lower bound of stability and its extremely favorable computational complexity

statistical approach to normalization of feature vectors and clustering of mixed datasets normalization of feature vectors of datasets is widely used in a number of fields of data mining in particular in cluster analysis where it is used to prevent features with large numerical values

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

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

stability of water waves we apply some general results for hamiltonian systems depending on the notion of signature of eigenvalues to determine the circumstances under which collisions of imaginary eigenvalue for the linearized problem about a travelling water wave of permanent form are avoided or lead to loss of stability up to non degeneracy assumptions a new superharmonic instability is predicted and verified

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

enhanced neural gas network for prototype based clustering in practical cluster analysis tasks an efficient clustering algorithm should be less sensitive to parameter configurations and tolerate the existence of outliers based on the neural gas ng network framework we propose an efficient prototype based clustering pbc algorithm called enhanced neural gas eng network several problems associated with the traditional pbc algorithms and original ng algorithm such as sensitivity to initialization sensitivity to input sequence ordering and the adverse influence from outliers can be effectively tackled in our new scheme in addition our new algorithm can establish the topology relationships among the prototypes and all topology wise badly located prototypes can be relocated to represent more meaningful regions experimental resultson synthetic and uci datasets show that our algorithm possesses superior performance in comparison to several pbc algorithms and their improved variants such as hard c means fuzzy c means ng fuzzy possibilistic c means credibilistic fuzzy c means hard fuzzy robust clustering and alternative hard fuzzy c means in static data clustering tasks with a fixed number of prototypes

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

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

fuzzy shell clustering algorithms in image processing fuzzy c rectangular and 2 rectangular shells objective function based clustering has been generalized recently to detect contours of circles and ellipses or even hyperbolas in a set of binary data vectors although there are special algorithms to discover lines the detection of rectangles needs further treatment a simple line detection algorithm is not sufficient for rectangles since for identifying four lines as one rectangle additional information such as the length of the lines and whether they are parallel or meet at a right angle is necessary in this paper a special fuzzy shell clustering algorithm for rectangular contours is developed the principal idea behind it can be generalized for other polygons so we also derive an algorithm that is capable of detecting rectangles and other polygons as well as approximating circles ellipses and lines

numerical conditioning problems and solutions for nonparametric i i d statistical active contours in this paper we propose an active contour model based on nonparametric independent and identically distributed i i d statistics of the image that can segment an image without any a priori information about the intensity distributions of the region of interest or the background this is not however the first active contour model proposed to solve the segmentation problem under these same assumptions in contrast to prior active contour models based on nonparametric i i d statistics we do not formulate our optimization criterion according to any distance measure between estimated probability densities inside and outside the active contour instead treating the segmentation problem as a pixel wise classification problem we formulate an active contour to minimize the unbiased pixel wise average misclassification probability amp this not only simplifies the problem by avoiding the need to arbitrarily select among many sensible distance measures to measure the difference between the probability densities estimated inside and outside the active contour but it also solves a numerical conditioning problem that arises with such prior active contour models as a result the amp model exhibits faster convergence with higher accuracy and robustness when compared to active contour models previously formulated to solve the same nonparametric i i d statistical segmentation problem via probability distances to discuss this improved numerical behavior more precisely we introduce the notion of x201c conditioning ratio x201d and demonstrate that the proposed amp active contour is numerically better conditioned i e exhibits a much smaller conditioning ratio than prior probability distance based active contours

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

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

on special zeros of p adic l functions of hilbert modular forms let e be a modular elliptic curve over a totally real number field f we prove the weak exceptional zero conjecture which links a higher derivative of the p adic l function attached to e to certain p adic periods attached to the corresponding hilbert modular form at the places above p where e has split multiplicative reduction under some mild restrictions on p and the conductor of e we deduce the exceptional zero conjecture in the strong form i e where the automorphic p adic periods are replaced by the mathcal l invariants of e defined in terms of tate periods from a special case proved earlier by mok crucial for our method is a new construction of the p adic l function of e in terms of local data

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

a contribution to convergence theory of fuzzy c means and derivatives in this paper we revisit the convergence and optimization properties of fuzzy clustering algorithms in general and the fuzzy c means fcm algorithm in particular our investigation includes probabilistic and a slightly modified implementation of possibilistic memberships which will be discussed under a unified view we give a convergence proof for the axis parallel variant of the algorithm by gustafson and kessel that can be generalized to other algorithms more easily than in the usual approach using reformulated fuzzy clustering algorithms we apply banach s classical contraction principle and establish a relationship between saddle points and attractive fixed points for the special case of fcm we derive a sufficient condition for fixed points to be attractive allowing identification of them as local minima of the objective function excluding the possibility of a saddle point

a time domain constrained fuzzy clustering method and its application to signal analysis this paper introduces a new fuzzy clustering method with time domain constraints which is used to signal analysis proposed method makes it possible to include natural constraints for signal analysis using fuzzy clustering that is the neighboring samples of signal belong to the same cluster this method can be called time domain constrained fuzzy clustering this paper introduces two approaches to include the above kind of constraints the first approach leads to the time domain constrained fuzzy c regression models method the second approach leads to the insensitive version of the above method which results in additional robustness for outliers and non gaussian noise finally simulations on synthetic as well as real life signals are realized to evaluate the performance of the time domain constrained fuzzy clustering methods a comparison with the traditional fuzzy c regression models is also made large scale simulations demonstrate the competitiveness of the proposed methods for signal analysis with respect to the traditional fuzzy clustering methods

deterministic annealing gustafson kessel fuzzy clustering algorithm abstract the gustafson and kessel gk fuzzy clustering algorithm proposed by gustafson and kessel in 1979 was the first important extension to the fuzzy c means fcm algorithm up to now the gk algorithm had become one of the most commonly used fuzzy clustering algorithms where the mahalanobis distance is used as a dissimilarity measure to provide more effectiveness and robustness than the fcm algorithm recently chaomurilige et al 2015 proposed a theoretical analysis on the parameter selection for the gk algorithm in which they indicated that the parameter of the fuzziness index heavily influences the performance of the gk algorithm in this paper we propose a novel gk fuzzy clustering algorithm based on the deterministic annealing approach for decreasing the effect of parameters we first consider maximizing the shannon s entropy of membership functions to the gk objective function and then use deterministic annealing to adjust the annealing parameter we also mathematically provide a theoretical initialization lower bound for the annealing parameter of the proposed deterministic annealing gk da gk algorithm comparisons between the da gk algorithm and other methods are made the computational complexity of the proposed method is also provided experimental results and comparisons actually verify theoretical results and also indicate the superiority and effectiveness of the proposed da gk algorithm

shintani cocycles and the order of vanishing of p adic hecke l series at s 0 let chi be a hecke character of finite order of a totally real number field f by using hill s shintani cocycle we provide a cohomological construction of the p adic l series l p chi s associated to chi this is used to show that l p chi s has a trivial zero at s 0 of order at least equal to the number of places of f above p where the local component of chi is trivial

robust learning fuzzy c means clustering algorithm with unknown number of clusters abstract in fuzzy clustering the fuzzy c means fcm algorithm is the most commonly used clustering method various extensions of fcm had been proposed in the literature however the fcm algorithm and its extensions are usually affected by initializations and parameter selection with a number of clusters to be given a priori although there were some works to solve these problems in fcm there is no work for fcm to be simultaneously robust to initializations and parameter selection under free of the fuzziness index without a given number of clusters in this paper we construct a robust learning based fcm framework called a robust learning fcm rl fcm algorithm so that it becomes free of the fuzziness index m and initializations without parameter selection and can also automatically find the best number of clusters we first use entropy type penalty terms for adjusting bias with free of the fuzziness index and then create a robust learning based schema for finding the best number of clusters the computational complexity of the proposed rl fcm algorithm is also analyzed comparisons between rl fcm and other existing methods are made experimental results and comparisons actually demonstrate these good aspects of the proposed rl fcm where it exhibits three robust characteristics 1 robust to initializations with free of the fuzziness index 2 robust to without parameter selection and 3 robust to number of clusters with unknown number of clusters

fuzzy and possibilistic shell clustering algorithms and their application to boundary detection and surface approximation ii shell clustering algorithms are ideally suited for computer vision tasks such as boundary detection and surface approximation particularly when the boundaries have jagged or scattered edges and when the range data is sparse this is because shell clustering is insensitive to local aberrations it can be performed directly in image space and unlike traditional approaches it does assume dense data and does not use additional features such as curvatures and surface normals the shell clustering algorithms introduced in part i of this paper assume that the number of clusters is known however which is not the case in many boundary detection and surface approximation applications this problem can be overcome by considering cluster validity we introduce a validity measure called surface density which is explicitly meant for the type of applications considered in this paper we show through theoretical derivations that surface density is relatively invariant to size and partiality incompleteness of the clusters we describe unsupervised clustering algorithms that use the surface density measure and other measures to determine the optimum number of shell clusters automatically and illustrate the application of the proposed algorithms to boundary detection in the case of intensity images and to surface approximation in the case of range images

on analyticity of travelling water waves in this paper we establish the existence and analyticity of periodic solutions of a classical free boundary model of the evolution of three dimensional capillary gravity waves on the surface of an ideal fluid the result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem and it yields analyticity jointly with respect to the perturbation parameter and the spatial variables the travelling waves we find can be interpreted as resulting from the nonlinear interaction of two two dimensional wavetrains giving rise to a periodic travelling pattern our analyticity theorem extends the most sophisticated results known to date in the absence of resonance short crested waves which result from the interaction of two wavetrains with unit amplitude ratio are realized as a special case our method of proof also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical approximation of travelling surface waves indeed we demonstrate that the rather unstable numerical behaviour of these approaches can be attributed to the strong but subtle cancellations in the formulas underlying their classical implementations these observations motivate the derivation and use of an alternative stable formulation which in addition to providing our method of proof suggests new stabilized implementations of boundary perturbation algorithms

stability of high order perturbative methods for the computation of dirichlet neumann operators abstract in this paper we present results on the stability of perturbation methods for the evaluation of dirichlet neumann operators dno defined on domains that are viewed as complex deformations of a basic simple geometry in such cases geometric perturbation methods based on variations of the spatial domains of definition have long been recognized to constitute efficient and accurate means for the approximation of dno and in fact several numerical implementations have been previously proposed inspired by our recent analytical work here we demonstrate that the convergence of these algorithms is quite generally limited by numerical instability indeed we show that these standard perturbative methods for the calculation of dno suffer from significant ill conditioning which is manifest even for very smooth boundaries and whose severity increases with boundary roughness moreover and again motivated by our previous work we introduce an alternative perturbative approach that we show to be numerically stable this approach can be interpreted as a reformulation of classical perturbative algorithms in suitable independent variables and thus it allows for both direct comparison and the possibility of analytic continuation of the perturbation series it can also be related to classical preconditioned spectral approaches and as such it retains in finite arithmetic the spectral convergence properties of classical perturbative methods albeit at a higher computational cost as it does not take advantage of possible dimensional reductions still as we show an alternative approach such as the one we propose may be mandated in cases where substantial information is contained in high order harmonics and or perturbation coefficients of the solution

fuzzy c ordered means clustering fuzzy clustering helps to find natural vague boundaries in data the fuzzy c means method is one of the most popular clustering methods based on minimization of a criterion function however one of the greatest disadvantages of this method is its sensitivity to the presence of noise and outliers in data this paper introduces a new robust fuzzy clustering method named fuzzy c ordered means fcom clustering this method uses both the huber s m estimators and the yager s owa operators to obtain its robustness the proposed method is compared to many other ones e g the fuzzy c means fcm the possibilistic clustering pc the fuzzy noise clustering method ncm the l p norm clustering l p fcm 0 p 1 the l 1 norm clustering l 1 fcm the fuzzy clustering with polynomial fuzzifier pfcm and the e insensitive fuzzy c means fcm to this end experiments on synthetic data with outliers have been performed as well as on data with heavy tailed and overlapping groups of points in background noise

on the unification of possibilistic fuzzy clustering axiomatic development and convergence analysis abstract fuzzy partitional clustering algorithms like fuzzy c means fcm are sensitive to noise points or outliers because of the probabilistic constraint the sum of the membership values of a specific data point to all the clusters is 1 which enables it to classify any noise or outlier in a specific cluster this not only hampers the clustering performance corresponding to that specific point but also leaves a huge impact on the overall clustering process by deviating the cluster centroids through a significant amount in order to improve this weakness possibilistic approach relaxes the probabilistic constraints nevertheless due to lack of constraints imposed on the typicality matrix depending on initialization possibilistic clustering algorithms suffer from coincident clusters because their membership expressions do not consider the distance to other cluster representatives fuzzy possibilistic clustering which puts a linear constraint on the sum of all the typicality values takes care of this problem but if the number of data points is high the generated typicality values are very small due to the linear constraint on their sums in order to do away with that problem the possibilistic fuzzy clustering algorithm was developed in this article we present an axiomatic development and extension of the possibilistic fuzzy clustering algorithm in three directions choice of the dissimilarity measure joint contribution function and the penalty function we provide a thorough convergence analysis of the proposed generalized possibilistic fuzzy clustering algorithm we investigate the relationships of the proposed generalization with the existing variations of the possibilistic c means pcm fuzzy possiblistic c means fpcm and possibilistic fuzzy c means pfcm algorithms in literature to the best of our knowledge this is the first article of its kind to provide a unification to the long list of possibilistic fuzzy possiblistic and possibilistic fuzzy clustering methods

instability and stability of rolls in the swift hohenberg equation we develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations in particular it enables us to detect sideband instabilities we treat in all detail the stability question of roll solutions in the two dimensional swift hohenberg equation and derive a condition on the amplitude and the wave number of the rolls which is necessary and sufficent for stability moreover we characterize the set of those wave vectors which give rise to unstable perturbations

spectral stability of deep two dimensional gravity water waves repeated eigenvalues the spectral stability problem for periodic traveling waves on a two dimensional fluid of infinite depth is investigated via a perturbative approach computing the spectrum as a function of the wave amplitude beginning with a flat surface we generalize our previous results by considering the crucially important situation of eigenvalues with multiplicity greater than one focusing on the generic case of multiplicity two in the flat water configuration we use this extended method of transformed field expansions which now accounts for the resonant spectrum to numerically simulate the evolution of the eigenvalues as the wave amplitude is increased we observe that there are no instabilities that are analytically connected to the flat state the spectrum loses its analyticity at the benjamin feir threshold we complement the numerical results with an explicit calculation of the first nonzero correction to the linear spectrum of resonant deep water waves two countably infinite families of collisions of ei

fuzzy order statistics and their application to fuzzy clustering the median and the median absolute deviation mad are robust statistics based on order statistics order statistics are extended to fuzzy sets to define a fuzzy median and a fuzzy mad the fuzzy c means fcm clustering algorithm is defined for any p norm pfcm including the l sub 1 norm 1fcm the 1fcm clustering algorithm is implemented via the alternating optimization ao method and the clustering centers are shown to be the fuzzy median the resulting ao 1fcm clustering algorithm is called the fuzzy c medians fcmed clustering algorithm an example illustrates the robustness of the fcmed

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

optimization of clustering criteria by reformulation various hard fuzzy and possibilistic clustering criteria objective functions are useful as bases for a variety of pattern recognition problems at present many of these criteria have customized individual optimization algorithms because of the specialized nature of these algorithms experimentation with new and existing criteria can be very inconvenient and costly in terms of development and implementation time this paper shows how to reformulate some clustering criteria so that specialized algorithms can be replaced by general optimization routines found in commercially available software we prove that the original and reformulated versions of each criterion are fully equivalent finally two numerical examples are given to illustrate reformulation

a fuzzy clustering approach toward hidden markov random field models for enhanced spatially constrained image segmentation hidden markov random field hmrf models have been widely used for image segmentation as they appear naturally in problems where a spatially constrained clustering scheme taking into account the mutual influences of neighboring sites is asked for fuzzy c means fcm clustering has also been successfully applied in several image segmentation applications in this paper we combine the benefits of these two approaches by proposing a novel treatment of hmrf models formulated on the basis of a fuzzy clustering principle we approach the hmrf model treatment problem as an fcm type clustering problem effected by introducing the explicit assumptions of the hmrf model into the fuzzy clustering procedure our approach utilizes a fuzzy objective function regularized by kullback leibler divergence information and is facilitated by application of a mean field like approximation of the mrf prior we experimentally demonstrate the superiority of the proposed approach over competing methodologies considering a series of synthetic and real world image segmentation applications

on stability of generalized short crested water waves abstract in this paper we take up the question of dynamic stability of genuinely two dimensional generalized hexagonal traveling wave patterns on the surface of a three dimensional ideal fluid i e stability of generalized short crested wave gscw solutions of the water wave problem we restrict ourselves to a study of spectral stability which considers the linearization of the water wave operator about one of these traveling generalized hexagonal waves and draws conclusions about stability from the spectral data of the resulting linear operator within the class of perturbations we are allowed to study for a wide range of geometrical parameters we find stable traveling waveforms which eventually destabilize with features that depend strongly on the problem configuration in particular we find zones of instability for patterns shaped as symmetric diamonds while such zones are absent for asymmetric configurations furthermore we note that within a geometrical configuration as a generalized scw ratio is varied essentially the character of the linear solution these waves become more unstable as the waves become more asymmetric

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

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

stable computation of the functional variation of the dirichlet neumann operator this paper presents an accurate and stable numerical scheme for computation of the first variation of the dirichlet neumann operator in the context of euler s equations for ideal free surface fluid flows the transformed field expansion methodology we use is not only numerically stable but also employs a spectrally accurate fourier chebyshev collocation method which delivers high fidelity solutions this implementation follows directly from the authors previous theoretical work on analyticity properties of functional variations of dirichlet neumann operators these variations can be computed in a number of ways but we establish via a variety of computational experiments the superior effectiveness of our new approach as compared with another popular boundary perturbation algorithm the method of operator expansions

a size insensitive integrity based fuzzy c means method for data clustering fuzzy c means fcm is one of the most popular techniques for data clustering since fcm tends to balance the number of data points in each cluster centers of smaller clusters are forced to drift to larger adjacent clusters for datasets with unbalanced clusters the partition results of fcm are usually unsatisfactory cluster size insensitive fcm csifcm dealt with cluster size sensitivity problem by dynamically adjusting the condition value for the membership of each data point based on cluster size after the defuzzification step in each iterative cycle however the performance of csifcm is sensitive to both the initial positions of cluster centers and the distance between adjacent clusters in this paper we present a cluster size insensitive integrity based fcm method called siibfcm to improve the deficiency of csifcm the siibfcm method can determine the membership contribution of every data point to each individual cluster by considering cluster s integrity which is a combination of compactness and purity compactness represents the distribution of data points within a cluster while purity represents how far a cluster is away from its adjacent cluster we tested our siibfcm method and compared with the traditional fcm and csifcm methods extensively by using artificially generated datasets with different shapes and data distributions synthetic images real images and escherichia coli dataset experimental results showed that the performance of siibfcm is superior to both traditional fcm and csifcm in terms of the tolerance for distance between adjacent clusters and the flexibility of selecting initial cluster centers when dealing with datasets with unbalanced clusters

a simultaneous learning framework for clustering and classification traditional pattern recognition generally involves two tasks unsupervised clustering and supervised classification when class information is available fusing the advantages of both clustering learning and classification learning into a single framework is an important problem worthy of study to date most algorithms generally treat clustering learning and classification learning in a sequential or two step manner i e first execute clustering learning to explore structures in data and then perform classification learning on top of the obtained structural information however such sequential algorithms cannot always guarantee the simultaneous optimality for both clustering and classification learning in fact the clustering learning in these algorithms just aids the subsequent classification learning and does not benefit from the latter to overcome this problem a simultaneous learning framework for clustering and classification scc is presented in this paper scc aims to achieve three goals 1 acquiring the robust classification and clustering simultaneously 2 designing an effective and transparent classification mechanism 3 revealing the underlying relationship between clusters and classes to this end with the bayesian theory and the cluster posterior probabilities of classes we define a single objective function to which the clustering process is directly embedded by optimizing this objective function the effective and robust clustering and classification results are achieved simultaneously experimental results on both synthetic and real life datasets show that scc achieves promising classification and clustering results at one time

nerf c means non euclidean relational fuzzy clustering abstract the relational fuzzy c means rfcm algorithm can be used to cluster a set of n objects described by pair wise dissimilarity values if and only if there exist n points in r n 1 whose squared euclidean distances precisely match the given dissimilarity data this strong restriction on the dissimilarity data renders rfcm inapplicable to most relational clustering problems this paper substantially improves rfcm by generalizing it to the case of arbitrary symmetric dissimilarity data the generalization is obtained using a computationally efficient modification of the existing algorithm that is equivalent to applying a spreading transformation to the dissimilarity data while the method given applies specifically to dissimilarity data a simple transformation can be used to convert similarity relations into dissimilarity data so the method is applicable to any numerical relational data that are positive reflexive or anti reflexive and symmetric numerical examples illustrate and compare the present approach to problems that can be studied with alternatives such as the linkage algorithms

towards a robust fuzzy clustering fuzzy clustering helps to find natural vague boundaries in data the fuzzy c means method fcm is one of the most popular clustering methods based on minimization of a criterion function however one of the greatest disadvantages of this method is its sensitivity to presence of noise and outliers in data this paper introduces a new e insensitive fuzzy c means efcm clustering algorithm as a special case this algorithm includes the well known fuzzy c medians method fcmed also methods with insensitivity control named fcm and fcm are introduced performance of the new clustering algorithm is experimentally compared with the fcm method using synthetic data with outliers and heavy tailed and overlapped groups of data in background noise

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

axiomatic generalization of the membership degree weighting function for fuzzy c means clustering for decades practitioners have been using the center based partitional clustering algorithms like fuzzy c means fcm which rely on minimizing an objective function comprising of an appropriately weighted sum of distances of each data point from the cluster representatives numerous generalizations in terms of choice of the distance function have been introduced for fcm since its inception in a stark contrast to this fact to the best of our knowledge there has not been any significant effort to address the issue of convergence of the algorithm under the structured generalization of the weighting function here by generalization we mean replacing the conventional weighting function uijm where uij indicates the membership of data xi to cluster cj and m is the real valued fuzzifier with 1 m with a more general function g uij which can assume a wide variety of forms under some mild assumptions in this article for the first time we present a novel axiomatic development of the general weighting function based on the membership degrees we formulate the fuzzy clustering problem along with the intuitive justification of the technicalities behind the axiomatic approach we develop an alternative optimization ao procedure to solve the main clustering problem by solving the partial optimization problems in an iterative way the novelty of the article lies in the development of an axiomatic generalization of the weighting function of fcm formulation of an ao algorithm for the fuzzy clustering problem with the extension to this general class of weighting functions and a detailed convergence analysis of the proposed algorithm

a unified continuous optimization framework for center based clustering methods center based partitioning clustering algorithms rely on minimizing an appropriately formulated objective function and different formulations suggest different possible algorithms in this paper we start with the standard nonconvex and nonsmooth formulation of the partitioning clustering problem we demonstrate that within this elementary formulation convex analysis tools and optimization theory provide a unifying language and framework to design analyze and extend hard and soft center based clustering algorithms through a generic algorithm which retains the computational simplicity of the popular k means scheme we show that several well known and more recent center based clustering algorithms which have been derived either heuristically or and have emerged from intuitive analogies in physics statistical techniques and information theoretic perspectives can be recovered as special cases of the proposed analysis and we streamline their relationships

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

numerical simulation of gravity waves abstract we present a new spectral method to simulate numerically the waterwave problem in a channel for a fluid of finite or infinite depth it is formulated in terms of the free surface elevation and the velocity potential the numerical method is based on the reduction of this problem to a lower dimensional computation involving surface variables alone to accomplish this we describe the taylor expansion of the dirichlet neumann operator in homogeneous powers of the surface elevation each term is a concatenation of fourier multipliers with powers of and its derivatives and is valid uniformly in wavenumber these are easily calculated using the fast fourier transform the method is illustrated by computing the long time evolution of modulated wave packets and of approximations to the stokes steady wave train by imposing a surface pressure we observe surface steepening in large amplitude evolution and wake and bow wave development for flows with a close to critical froude number finally we give an example of nonlinear evolution of the distribution of energy among normal modes

on possibilistic clustering with repulsion constraints for imprecise data in possibilistic clustering objects are assigned to clusters according to the so called membership degrees taking values in the unit interval differently from fuzzy clustering it is not required that the sum of the membership degrees of an object to all clusters is equal to one this is very helpful in the presence of outliers which are usually assigned to the clusters with membership degrees close to zero unfortunately a drawback of the possibilistic approach is the tendency to produce coincident clusters a remedy is to add a repulsion term among prototypes in the loss function forcing the prototypes to be far enough from each other here a possibilistic clustering algorithm with repulsion constraints for imprecise data managed in terms of fuzzy sets is introduced applications to synthetic and real fuzzy data are considered in order to analyze how the proposed clustering algorithm works in practice

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

fuzzy c varieties elliptotypes clustering in reproducing kernel hilbert space abstract fuzzy clustering algorithms are successfully applied to a wide variety of problems such as pattern recognition image analysis modeling and so on the fuzzy c means fcm method is one of the most popular clustering methods based on the minimization of a criterion function however the performance of the fcm method is good only when a data set contains clusters that are approximately the same size and shape in this paper a simple idea will be used to overcome this problem the original input data space will be mapped into the high possibly infinite dimensional feature space f through some nonlinear mapping in this space the data structures will be modeled by the linear varieties or elliptotypes this method is called kernel fuzzy c varieties elliptotypes clustering algorithm performance of the new clustering algorithm is experimentally compared with fcm and fuzzy c varieties elliptotypes methods using synthetic datasets and real life datasets

l 1 norm based fuzzy clustering abstract the paper presents the l1 version of the well known fuzzy clustering method namely fuzzy isodata proposed by bezdek and dunn due to their robustness l1 norm based methods gained much attention in statistics the presented fuzzy clustering problem uses the distance between observations and location parameter vectors which is based on the l1 norm instead of the inner product induced norm used in classical fuzzy isodata two alternative methods to solve the l1 fuzzy clustering problem are derived in practice both membership grades and location parameter vectors are unknown the paper presents two iterative algorithms each the implementation of the derived method finally numerical examples are presented one of them refers to famous iris data

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