on relational possibilistic clustering this paper initially describes the relational counterpart of possibilistic c means pcm algorithm called relational pcm or rpcm rpcm is then improved to better handle arbitrary dissimilarity data first a re scaling of the pcm membership function is proposed in order to obtain zero membership values when the distance to prototype equals the maximum value allowed in bounded dissimilarity measures second a heuristic method of reference distance initialisation is provided which diminishes the known pcm tendency of producing coincident clusters finally rpcm improved with our initialisation strategy is tested on both synthetic and real data sets with satisfactory results

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

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

a fuzzy approach for multitype relational data clustering mining interrelated data among multiple types of objects or entities is important in many real world applications despite extensive study on fuzzy clustering of vector space data very limited exploration has been made on fuzzy clustering of relational data that involve several object types in this paper we propose a new fuzzy clustering approach for multitype relational data fc mr in fc mr different types of objects are clustered simultaneously an object is assigned a large membership with respect to a cluster if its related objects in this cluster have high rankings in each cluster an object tends to have a high ranking if its related objects have large memberships in this cluster the fc mr approach is formulated to deal with multitype relational data with various structures the objective function of fc mr is locally optimized by an efficient iterative algorithm which updates the fuzzy membership matrix and the ranking matrix of one type at once while keeping those of other types constant we also discuss the simplified fc mr for multitype relational data with two special structures namely star structure and extended star structure experimental studies are conducted on benchmark document datasets to illustrate how the proposed approach can be applied flexibly under different scenarios in real world applications the experimental results demonstrate the feasibility and effectiveness of the new approach compared with existing ones

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

function approximation with polynomial membership functions and alternating cluster estimation abstract nonlinear functions are often approximated using local linear models sets of local linear models can be represented as a first order takagi sugeno ts system triangular and trapezoidal left hand side membership functions are not compatible with ts systems because they lead to non differentiable input output characteristics we develop a method to determine parameters of piecewise quadratic membership functions to obtain characteristics which exactly match the rule centers and the corresponding slopes the right hand side parameters are obtained using i fuzzy c elliptotypes alternating optimization and ii alternating cluster estimation in our experiments the smoothest and most accurate approximations are obtained with piecewise quadratic membership functions and alternating cluster estimation

on the quasi isometries of harmonic quasiconformal mappings abstract we prove versions of the ahlfors schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the poincare metric

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

a possibilistic approach to latent component analysis for symmetric fuzzy data in many situations the available amount of data is huge and can be intractable when the data set is single valued latent component models are recognized techniques which provide a useful compression of the information this is done by considering a regression model between observed and unobserved latent variables in this paper an extension of latent component analysis to deal with fuzzy data is proposed our extension follows the possibilistic approach widely used both in the cluster and regression frameworks in this case the possibilistic approach involves the formulation of a latent component analysis for fuzzy data by optimization specifically a non linear programming problem in which the fuzziness of the model is minimized is introduced in order to show how our model works the results of two applications are proposed

alternating cluster estimation a new tool for clustering and function approximation many clustering models define good clusters as extrema of objective functions optimization of these models is often done using an alternating optimization ao algorithm driven by necessary conditions for local extrema we abandon the objective function model in favor of a generalized model called alternating cluster estimation ace ace uses an alternating iteration architecture but membership and prototype functions are selected directly by the user virtually every clustering model can be realized as an instance of ace out of a large variety of possible instances of non ao models we present two examples 1 an algorithm with a dynamically changing prototype function that extracts representative data and 2 a computationally efficient algorithm with hyperconic membership functions that allows easy extraction of membership functions we illustrate these non ao instances on three problems a simple clustering of plane data where we show that creating an unmatched ace algorithm overcomes some problems of fuzzy c means fcm ao and possibilistic c means pcm ao b functional approximation by clustering on a simple artificial data set and c functional approximation on a 12 input 1 output real world data set ace models work pretty well in all three cases

constrained clustering as an optimization method a deterministic annealing approach to clustering is derived on the basis of the principle of maximum entropy this approach is independent of the initial state and produces natural hierarchical clustering solutions by going through a sequence of phase transitions it is modified for a larger class of optimization problems by adding constraints to the free energy the concept of constrained clustering is explained and three examples are are given in which it is used to introduce deterministic annealing the previous clustering method is improved by adding cluster mass variables and a total mass constraint the traveling salesman problem is reformulated as constrained clustering yielding the elastic net en approach to the problem more insight is gained by identifying a second lagrange multiplier that is related to the tour length and can also be used to control the annealing process the open path constraint formulation is shown to relate to dimensionality reduction by self organization in unsupervised learning a similar annealing procedure is applicable in this case as well

warped k means an algorithm to cluster sequentially distributed data many devices generate large amounts of data that follow some sort of sequentiality e g motion sensors e pens eye trackers etc and often these data need to be compressed for classification storage and or retrieval tasks traditional clustering algorithms can be used for this purpose but unfortunately they do not cope with the sequential information implicitly embedded in such data thus we revisit the well known k means algorithm and provide a general method to properly cluster sequentially distributed data we present warped k means wkm a multi purpose partitional clustering procedure that minimizes the sum of squared error criterion while imposing a hard sequentiality constraint in the classification step we illustrate the properties of wkm in three applications one being the segmentation and classification of human activity wkm outperformed five state of the art clustering techniques to simplify data trajectories achieving a recognition accuracy of near 97 which is an improvement of around 66 over their peers moreover such an improvement came with a reduction in the computational cost of more than one order of magnitude

space structure and clustering of categorical data learning from categorical data plays a fundamental role in such areas as pattern recognition machine learning data mining and knowledge discovery to effectively discover the group structure inherent in a set of categorical objects many categorical clustering algorithms have been developed in the literature among which k modes type algorithms are very representative because of their good performance nevertheless there is still much room for improving their clustering performance in comparison with the clustering algorithms for the numeric data this may arise from the fact that the categorical data lack a clear space structure as that of the numeric data to address this issue we propose in this paper a novel data representation scheme for the categorical data which maps a set of categorical objects into a euclidean space based on the data representation scheme a general framework for space structure based categorical clustering algorithms sbc is designed this framework together with the applications of two kinds of dissimilarities leads two versions of the sbc type algorithms to verify the performance of the sbc type algorithms we employ as references four representative algorithms of the k modes type algorithms experiments show that the proposed sbc type algorithms significantly outperform the k modes type algorithms

dual fuzzy possibilistic coclustering for categorization of documents in this paper we develop a new soft model dual fuzzy possibilistic coclustering dfpc for document categorization the proposed model targets robustness to outliers and richer representations of coclusters dfpc is inspired by an existing algorithm called possibilistic fuzzy c means pfcm that hybridizes fuzzy and possibilistic clustering it has been shown that pfcm can perform effectively for low dimensional data clustering to achieve our goal we expand this existing idea by introducing a novel pfcm like coclustering model the new algorithm dfpc preserves the desired properties of pfcm in addition as a coclustering algorithm dfpc is more suitable for our intended high dimensional application document clustering besides the coclustering mechanism enables dfpc to generate together with document clusters fuzzy possibilistic word memberships these word memberships which are absent in the existing pfcm model can play an important role in generating useful descriptions of document clusters we detail the formulation of the proposed model and provide an extensive analytical study of the algorithm dfpc experiments on an artificial dataset and various benchmark document datasets demonstrate the effectiveness and potential of dfpc

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

applying the possibilistic c means algorithm in kernel induced spaces in this paper we study a kernel extension of the classic possibilistic c means in the proposed extension we implicitly map input patterns into a possibly high dimensional space by means of positive semidefinite kernels in this new space we model the mapped data by means of the possibilistic clustering algorithm we study in more detail the special case where we model the mapped data using a single cluster only since it turns out to have many interesting properties the modeled memberships in kernel induced spaces yield a modeling of generic shapes in the input space we analyze in detail the connections to one class support vector machines and kernel density estimation thus suggesting that the proposed algorithm can be used in many scenarios of unsupervised learning in the experimental part we analyze the stability and the accuracy of the proposed algorithm on some synthetic and real datasets the results show high stability and good performances in terms of accuracy

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

clustering by competitive agglomeration we present a new clustering algorithm called competitive agglomeration ca which minimizes an objective function that incorporates the advantages of both hierarchical and partitional clustering the ca algorithm produces a sequence of partitions with a decreasing number of clusters the initial partition has an over specified number of clusters and the final one has the optimal number of clusters the update equation in the ca algorithm creates an environment in which clusters compete for feature points and only clusters with large cardinalities survive the algorithm can incorporate different distance measures in the objective function to f find an unknown number of clusters of various shapes

a robust competitive clustering algorithm with applications in computer vision this paper addresses three major issues associated with conventional partitional clustering namely sensitivity to initialization difficulty in determining the number of clusters and sensitivity to noise and outliers the proposed robust competitive agglomeration rca algorithm starts with a large number of clusters to reduce the sensitivity to initialization and determines the actual number of clusters by a process of competitive agglomeration noise immunity is achieved by incorporating concepts from robust statistics into the algorithm rca assigns two different sets of weights for each data point the first set of constrained weights represents degrees of sharing and is used to create a competitive environment and to generate a fuzzy partition of the data set the second set corresponds to robust weights and is used to obtain robust estimates of the cluster prototypes by choosing an appropriate distance measure in the objective function rca can be used to find an unknown number of clusters of various shapes in noisy data sets as well as to fit an unknown number of parametric models simultaneously several examples such as clustering mixture decomposition line plane fitting segmentation of range images and estimation of motion parameters of multiple objects are shown

superconvergent discontinuous galerkin methods for nonlinear elliptic equations based on the analysis of cockburn et al math comp 78 2009 pp 1 24 for a selfadjoint linear elliptic equation we rst discuss su perconvergence results for nonselfadjoint linear elliptic problems using discon tinuous galerkin methods further we have extended our analysis to derive superconvergence results for quasilinear elliptic problems when piecewise polynomials of degree k 1 are used to approximate both the potential as well as the ux it is shown in this article that the error estimate for the discrete ux in l 2 norm is of order k 1 further based on solving a discrete linear elliptic problem at each element a suitable postprocessing of the discrete potential is developed and then it is proved that the resulting post processed potential converges with order of convergence k 2 in l 2 norm these results conrm superconvergent results for linear elliptic problems

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

an efficient accelerator for attribute reduction from incomplete data in rough set framework feature selection attribute reduction from large scale incomplete data is a challenging problem in areas such as pattern recognition machine learning and data mining in rough set theory feature selection from incomplete data aims to retain the discriminatory power of original features to address this issue many feature selection algorithms have been proposed however these algorithms are often computationally time consuming to overcome this shortcoming we introduce in this paper a theoretic framework based on rough set theory which is called positive approximation and can be used to accelerate a heuristic process for feature selection from incomplete data as an application of the proposed accelerator a general feature selection algorithm is designed by integrating the accelerator into a heuristic algorithm we obtain several modified representative heuristic feature selection algorithms in rough set theory experiments show that these modified algorithms outperform their original counterparts it is worth noting that the performance of the modified algorithms becomes more visible when dealing with larger data sets

quantization based clustering algorithm in this paper a quantization based clustering algorithm qbca is proposed to cluster a large number of data points efficiently unlike previous clustering algorithms qbca places more emphasis on the computation time of the algorithm specifically qbca first assigns the data points to a set of histogram bins by a quantization function then it determines the initial centers of the clusters according to this point distribution finally qbca performs clustering at the histogram bin level rather than the data point level we also propose two approaches to improve the performance of qbca further i a shrinking process is performed on the histogram bins to reduce the number of distance computations and ii a hierarchical structure is constructed to perform efficient indexing on the histogram bins finally we analyze the performance of qbca theoretically and experimentally and show that the approach 1 can be easily implemented 2 identifies the clusters effectively and 3 outperforms most of the current state of the art clustering approaches in terms of efficiency

a new credibilistic clustering algorithm abstract this paper focuses on credibilistic clustering approach a data clustering method partitions unlabeled data sets into clusters and labels them for various goals such as computer vision and pattern recognition there are different models for objective function based fuzzy clustering such as fuzzy c means fcm possibilistic c mean pcm and their combinations credibilistic clustering is a new approach in this field in this paper a new credibilistic clustering model is introduced in which credibility measure is applied instead of possibility measure in possibilistic clustering also in objective function the separation of clusters is considered in addition to the compactness within clusters the steps of clustering are designed based on this approach finally the main issues about model are discussed and the results of computational experiments are presented to show the efficiency of the proposed model

the multisynapse neural network and its application to fuzzy clustering in this paper a new neural architecture the multisynapse neural network is developed for constrained optimization problems whose objective functions may include high order logarithmic and sinusoidal forms etc unlike the traditional hopfield networks which can only handle quadratic form optimization meanwhile based on the application of this new architecture a fuzzy bidirectional associative clustering network fbacn which is composed of two layers of recurrent networks is proposed for fuzzy partition clustering according to the objective functional method it is well known that fuzzy c means is a milestone algorithm in the area of fuzzy c partition clustering all of the following objective functional based fuzzy c partition algorithms incorporate the formulas of fuzzy c means as the prime mover in their algorithms however when an application of fuzzy c partition has sophisticated constraints the necessity of analytical solutions in a single iteration step becomes a fatal issue of the existing algorithms the largest advantage of fbacn is that it does not need analytical solutions for the problems on which some prior information is known we bring a combination of part crisp and part fuzzy clustering in the third optimization problem

a kernel fuzzy c means clustering based fuzzy support vector machine algorithm for classification problems with outliers or noises the support vector machine svm has provided higher performance than traditional learning machines and has been widely applied in real world classification problems and nonlinear function estimation problems unfortunately the training process of the svm is sensitive to the outliers or noises in the training set in this paper a common misunderstanding of gaussian function based kernel fuzzy clustering is corrected and a kernel fuzzy c means clustering based fuzzy svm algorithm kfcm fsvm is developed to deal with the classification problems with outliers or noises in the kfcm fsvm algorithm we first use the fcm clustering to cluster each of two classes from the training set in the high dimensional feature space the farthest pair of clusters where one cluster comes from the positive class and the other from the negative class is then searched and forms one new training set with membership degrees finally we adopt fsvm to induce the final classification results on this new training set the computational complexity of the kfcm fsvm algorithm is analyzed a set of experiments is conducted on six benchmarking datasets and four artificial datasets for testing the generalization performance of the kfcm fsvm algorithm the results indicate that the kfcm fsvm algorithm is robust for classification problems with outliers or noises

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

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

improved possibilistic c means clustering algorithms a possibilistic approach was proposed in a previous paper for c means clustering and two algorithms realizing this approach were reported in two previous papers although the possibilistic approach is sound these two algorithms tend to find identical clusters in this paper we modify and improve these algorithms to overcome their shortcoming the numerical results demonstrate that the improved algorithms can determine proper clusters and they can realize the advantages of the possibilistic approach

a novel adaptive possibilistic clustering algorithm in this paper a novel possibilistic c means clustering algorithm called adaptive possibilistic c means is presented its main feature is that its parameters after their initialization are properly adapted during its execution provided that the algorithm starts with a reasonable overestimate of the number of physical clusters formed by the data it is capable in principle to unravel them a long standing issue in the clustering literature this is due to the fully adaptive nature of the proposed algorithm that enables the removal of the clusters that gradually become obsolete in addition the adaptation of all its parameters increases the flexibility of the algorithm in following the variations in the formation of the clusters that occur from iteration to iteration theoretical results that are indicative of the convergence behavior of the algorithm are also provided finally extensive simulation results on both synthetic and real data highlight the effectiveness of the proposed algorithm

a general discontinuous galerkin method for finite hyperelasticity formulation and numerical applications a discontinuous galerkin formulation of the boundary value problem of finite deformation elasticity is presented the primary purpose is to establish a discontinuous galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the dg concept can show its superiority compared to the continuous formulation the method is based on a general hu washizu de veubeke functional allowing for displacement and stress discontinuities in the domain interior it is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation the consistency and linearized stability of the method in the non linear range as well as its convergence rate are proven an implementation in three dimensions is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner in order to demonstrate the versatility accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided copyright 2006 john wiley sons ltd

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

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

k means type algorithms a generalized convergence theorem and characterization of local optimality the k means algorithm is a commonly used technique in cluster analysis in this paper several questions about the algorithm are addressed the clustering problem is first cast as a nonconvex mathematical program then a rigorous proof of the finite convergence of the k means type algorithm is given for any metric it is shown that under certain conditions the algorithm may fail to converge to a local minimum and that it converges under differentiability conditions to a kuhn tucker point finally a method for obtaining a local minimum solution is given

visual stability analysis for model selection in graded possibilistic clustering abstract when compared to standard clustering fuzzy clustering provides more flexible and powerful data representation most fuzzy methods require setting some parameters as is the case for the graded possibilistic c means clustering method which has two parameters in addition to number of centroids however for this model selection task there is no well established criterion available building on our own previous work on fuzzy clustering similarity indexes we introduce a technique to evaluate the stability of clusterings by using the fuzzy jaccard index and use this procedure to select the most suitable values of parameters the experiments indicate that the procedure is effective for fuzzy and possibilistic clustering

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

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

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

on certain quasiconformal and elliptic mappings abstract let d be the closure of the unit disk d in the complex plane c and g be a continuous function in d in this paper we discuss some characterizations of elliptic mappings f satisfying the poisson s equation f g in d and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length respectively the obtained results are the extension of the corresponding classical results

model transitions in descending flvq fuzzy learning vector quantization flvq also known as the fuzzy kohonen clustering network was developed to improve performance and usability of online hard competitive kohonen vq and soft competitive self organizing map som algorithms flvq s effectiveness seems to depend on the range of change of the weighting exponent m t extreme m t values 1 and spl infin are employed to investigate flvq asymptotic behaviors this analysis shows that when m t tends to either extreme flvq is affected by trivial vq which causes centroids to collapse in the grand mean of the input data set no analytical criterion has been found to improve the heuristic choice of the range of m t change two flvq and som classification experiments of remote sensed data are then presented the two nets are connected in cascade to a supervised second stage based on the delta rule the results confirm that flvq performance can be greatly affected by the user s definition of the range of change of the weighting exponent moreover flvq shows instability when its traditional termination criterion is applied empirical recommendations are proposed for the enhancement of flvq robustness both the analytical and the experimental data reported seem to indicate that the choice of the range of m t change is still open to discussion and that alternative clustering neural network approaches should be developed to pursue during training 1 monotone reduction of the neurons learning rate and 2 monotone reduction of the overlap among neuron receptive fields

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

an integrated approach to fuzzy learning vector quantization and fuzzy c means clustering derives an interpretation for a family of competitive learning algorithms and investigates their relationship to fuzzy c means and fuzzy learning vector quantization these algorithms map a set of feature vectors into a set of prototypes associated with a competitive network that performs unsupervised learning derivation of the new algorithms is accomplished by minimizing an average generalized distance between the feature vectors and prototypes using gradient descent a close relationship between the resulting algorithms and fuzzy c means is revealed by investigating the functionals involved it is also shown that the fuzzy c means and fuzzy learning vector quantization algorithms are related to the proposed algorithms if the learning rate at each iteration is selected to satisfy a certain condition

the approximation theory for the p version finite element method and application to non linear elliptic pdes approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements estimates for the rate of convergence in sobolev spaces w m q omega q in 1 infty are given the results are applied to estimate the rate of convergence when the p version finite element method is used to approximate the alpha laplacian it is shown that the rate of convergence of the p version is always at least that of the h version measured in terms of number of degrees of freedom used if the solution is very smooth then the p version attains an exponential rate of convergence if the solution has certain types of singularity the rate of convergence of the p version is twice that of the h version the analysis generalises the work of babuska and others to the case q not 2 in addition the approximation theoretic results find immediate application for some types of spectral and spectral element methods

representation formula and bi lipschitz continuity of solutions to inhomogeneous biharmonic dirichlet problems in the unit disk abstract the aim of this paper is twofold first we establish the representation formula and the uniqueness of the solutions to a class of inhomogeneous biharmonic dirichlet problems and then prove the bi lipschitz continuity of the solutions

robust clustering methods a unified view clustering methods need to be robust if they are to be useful in practice in this paper we analyze several popular robust clustering methods and show that they have much in common we also establish a connection between fuzzy set theory and robust statistics and point out the similarities between robust clustering methods and statistical methods such as the weighted least squares technique the m estimator the minimum volume ellipsoid algorithm cooperative robust estimation minimization of probability of randomness and the epsilon contamination model by gleaning the common principles upon which the methods proposed in the literature are based we arrive at a unified view of robust clustering methods we define several general concepts that are useful in robust clustering state the robust clustering problem in terms of the defined concepts and propose generic algorithms and guidelines for clustering noisy data we also discuss why the generalized hough transform is a suboptimal solution to the robust clustering problem

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

comments on a possibilistic approach to clustering in this comment we report a difficulty with the application of the possibilistic approach to fuzzy clustering pcm proposed by keller and krishnapuram 1993 in applying this algorithm we found that it has the undesirable tendency to produce coincidental clusters results illustrating this tendency are reported and a possible explanation for the pcm behavior is suggested

soft transition from probabilistic to possibilistic fuzzy clustering in the fuzzy clustering literature two main types of membership are usually considered a relative type termed probabilistic and an absolute or possibilistic type indicating the strength of the attribution to any cluster independent from the rest there are works addressing the unification of the two schemes here we focus on providing a model for the transition from one schema to the other to exploit the dual information given by the two schemes and to add flexibility for the interpretation of results we apply an uncertainty model based on interval values to memberships in the clustering framework obtaining a framework that we term graded possibility we outline a basic example of graded possibilistic clustering algorithm and add some practical remarks about its implementation the experimental demonstrations presented highlight the different properties attainable through appropriate implementation of a suitable graded possibilistic model an interesting application is found in automated segmentation of diagnostic medical images where the model provides an interactive visualization tool for this task

a possibilistic clustering approach toward generative mixture models generative mixture models mms provide one of the most popular methodologies for unsupervised data clustering mms are formulated on the basis of the assumption that each observation derives from belongs to a single cluster however in many applications data may intuitively belong to multiple classes thus rendering the single cluster assignment assumptions of mms irrelevant furthermore even in applications where a single cluster data assignment is required the induced multinomial allocation of the modeled data points to the clusters derived by a mm imposing the constraint that the membership probabilities of a data point across clusters sum to one makes mms very vulnerable to the presence of outliers in the clustered data sets and renders them ineffective in discriminating between cases of equal evidence or ignorance to resolve these issues in this paper we introduce a possibilistic formulation of mms possibilistic clustering is a methodology that yields possibilistic data partitions with the obtained membership values being interpreted as degrees of possibility compatibilities of the data points with respect to the various clusters we provide an efficient maximum likelihood fitting algorithm for the proposed model and we conduct an objective evaluation of its efficacy using benchmark data

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

an extension to possibilistic fuzzy cluster analysis we explore an approach to possibilistic fuzzy clustering that avoids a severe drawback of the conventional approach namely that the objective function is truly minimized only if all cluster centers are identical our approach is based on the idea that this undesired property can be avoided if we introduce a mutual repulsion of the clusters so that they are forced away from each other we develop this approach for the possibilistic fuzzy c means algorithm and the possibilistic gustafson kessel algorithm in our experiments we found that in this way we can combine the partitioning property of the probabilistic fuzzy c means algorithm with the advantages of a possibilistic approach w r t the interpretation of the membership degrees

cauchy transform and poisson s equation let u 2 w 2 p 0 1 6 p 6 1 be a solution of the poisson equationu h h 2 l p in the unit disk it is proved that kr uklp 6 apkhklp with sharp constant ap for p 1 and p 1 and that k uklp 6 bpkhklp with sharp constant bp for p 1 p 2 and p 1 in addition is proved that for p 2 jj ujj l1 6 cpkhklp and jjrujj l1 6 cpkhklp with sharp constants cp and cp an extension to smooth jordan domains is given these problems are equivalent to determining the precise value of l p norm of cauchy transform of dirichlet s problem

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

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

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

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

an axiomatic approach to soft learning vector quantization and clustering this paper presents an axiomatic approach to soft learning vector quantization lvq and clustering based on reformulation the reformulation of the fuzzy c means fcm algorithm provides the basis for reformulating entropy constrained fuzzy clustering ecfc algorithms according to the proposed approach the development of specific algorithms reduces to the selection of a generator function linear generator functions lead to the fcm and fuzzy learning vector quantization algorithms while exponential generator functions lead to ecfc and entropy constrained learning vector quantization algorithms the reformulation of lvq and clustering algorithms also provides the basis for developing uncertainty measures that can identify feature vectors equidistant from all prototypes these measures are employed by a procedure developed to make soft lvq and clustering algorithms capable of identifying outliers in the data set this procedure is evaluated by testing the algorithms generated by linear and exponential generator functions on speech data

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

possibilistic fuzzy co clustering of large document collections in this paper we propose a new co clustering algorithm called possibilistic fuzzy co clustering pfcc for automatic categorization of large document collections pfcc integrates a possibilistic document clustering technique and a combined formulation of fuzzy word ranking and partitioning into a fast iterative co clustering procedure this novel framework brings about simultaneously some benefits including robustness in the presence of document and word outliers rich representations of co clusters highly descriptive document clusters a good performance in a high dimensional space and a reduced sensitivity to the initialization in the possibilistic clustering we present the detailed formulation of pfcc together with the explanations of the motivations behind the advantages over other existing works and the algorithm s proof of convergence are provided experiments on several large document data sets demonstrate the effectiveness of pfcc

quasiconformal and harmonic mappings between jordan domains let and 1 be jordan domains let 0 1 and let f omega mapsto omega 1 be a harmonic homeomorphism the object of the paper is to prove the following results a if f is q c and 1 c 1 then f is lipschitz b if f is q c 1 c 1 and 1 is convex then f is bi lipschitz and c if is the unit disk 1 is convex and 1 c 1 then f is quasiconformal if and only if its boundary function is bi lipschitz and the hilbert transform of its derivative is in l these extend the results of pavlovic ann acad sci fenn 27 365 372 2002

a priori error analyses of a stabilized discontinuous galerkin method abstract we introduce a new stabilized discontinuous galerkin method within a new function space setting that is closely related to the discontinuous galerkin formulation by oden babuska and baumann 1 but involves an extra stabilization term on the jumps of the normal fluxes across the element interfaces the formulation satisfies a local conservation property and we prove well posedness of the new formulation a priori error estimates are derived which are verified by 1d and 2d experiments on a reaction diffusion type model problem

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

an algorithm for data driven bandwidth selection the analysis of a feature space that exhibits multiscale patterns often requires kernel estimation techniques with locally adaptive bandwidths such as the variable bandwidth mean shift proper selection of the kernel bandwidth is however a critical step for superior space analysis and partitioning this paper presents a mean shift based approach for local bandwidth selection in the multimodal multivariate case the method is based on a fundamental property of normal distributions regarding the bias of the normalized density gradient this paper demonstrates that within the large sample approximation the local covariance is estimated by the matrix that maximizes the magnitude of the normalized mean shift vector using this property the paper develops a reliable algorithm which takes into account the stability of local bandwidth estimates across scales the validity of the theoretical results is proven in various space partitioning experiments involving the variable bandwidth mean shift

fuzzy clustering with weighted medoids for relational data the well known k medoids clustering approach groups objects through finding k representative objects based on the pairwise dis similarities of objects in the data set in real applications using only one object to capture or interpret each cluster may not be sufficient enough which in turn could affect the accuracy of the data analysis in this paper we propose a new fuzzy clustering approach called pfc for dis similarity based data or relational data analysis in pfc objects in each fuzzy cluster carry various weights called prototype weights to represent their degrees of representativeness in that cluster this mechanism enables each cluster to be represented by more than one objects compared with existing clustering approaches for relational data pfc is able to capture the underlying structures of the data more accurately and provide richer information for the description of the resulting clusters we introduce the detailed formulation of pfc and provide the analytical as well as experimental studies to demonstrate the merits of the proposed approach

fuzzy clustering with volume prototypes and adaptive cluster merging two extensions to objective function based fuzzy clustering are proposed first the point prototypes are extended to hypervolumes whose size can be fixed or can be determined automatically from the data being clustered it is shown that clustering with hypervolume prototypes can be formulated as the minimization of an objective function second a heuristic cluster merging step is introduced where the similarity among the clusters is assessed during optimization starting with an overestimation of the number of clusters in the data similar clusters are merged in order to obtain a suitable partitioning an adaptive threshold for merging is proposed the extensions proposed are applied to gustafson kessel and fuzzy c means algorithms and the resulting extended algorithm is given the properties of the new algorithm are illustrated by various examples

muckenhoupt weights and lindelof theorem for harmonic mappings abstract we extend the result of lavrentiev which asserts that the harmonic measure and the arc length measure are a equivalent in a chord arc jordan domain by using this result we extend the classical result of lindelof to the class of quasiconformal q c harmonic mappings by proving the following assertion assume that f is a quasiconformal harmonic mapping of the unit disk u onto a jordan domain then the function a z arg f z z where z r e i is well defined and smooth in u z 0 z 1 and has a continuous extension to the boundary of the unit disk if and only if the image domain has c 1 boundary

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

boundary schwarz lemma for solutions to poisson s equation abstract let d be the closure of the unit disk d in c and g a continuous function of d in this paper we establish the schwarz lemma at the boundary for solutions to the poisson s equation f g

vector quantization with complexity costs vector quantization is a data compression method by which a set of data points is encoded by a reduced set of reference vectors the codebook a vector quantization strategy is discussed that jointly optimizes distortion errors and the codebook complexity thereby determining the size of the codebook a maximum entropy estimation of the cost function yields an optimal number of reference vectors their positions and their assignment probabilities the dependence of the codebook density on the data density for different complexity functions is investigated in the limit of asymptotic quantization levels how different complexity measures influence the efficiency of vector quantizers is studied for the task of image compression the wavelet coefficients of gray level images are quantized and the reconstruction error is measured the approach establishes a unifying framework for different quantization methods like k means clustering and its fuzzy version entropy constrained vector quantization or topological feature maps and competitive neural networks

fuzzy relational clustering around medoids a unified view medoid based fuzzy clustering generates clusters of objects based on relational data which records pairwise similarities or dissimilarities among objects compared with single medoid based approaches our recently proposed fuzzy clustering with multiple weighted medoids has shown superior performance in clustering via experimental study in this paper we present a new version of fuzzy relational clustering in this family called fuzzy clustering with multi medoids fmmdd based on the new objective function of fmmdd update equations can be derived more conveniently moreover a unified view of fmmdd and two existing fuzzy relational approaches fuzzy c medoids fcmdd and assignment prototype a p can be established which allows us to conduct further analytical study to investigate the effectiveness and feasibility of the proposed approach as well as the limitations of existing ones the robustness of fmmdd is also investigated our theoretical and numerical studies show that the proposed approach produces good quality of clusters with rich cluster based information and it is less sensitive to noise

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

hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems in this paper we have analyzed a one parameter family of hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems nabla cdot rm a u nabla u f u nabla u 0 with dirichlet boundary conditions these methods depend on the values of the parameter theta in 1 1 where 1 corresponds to the nonsymmetric and 1 corresponds to the symmetric interior penalty methods when rm a u nabla u nabla u and f u u f that is for the poisson problem the error estimate in the broken h 1 norm which is optimal in h mesh size and suboptimal in p degree of approximation is derived using piecewise polynomials of degree p 2 when the solution u in h 5 2 omega in the case of linear elliptic problems also this estimate is optimal in h and suboptimal in p further optimal error estimate in the l 2 norm when 1 is derived numerical experiments are presented to illustrate the theoretical results

a robust automatic merging possibilistic clustering method krishnapuram and keller ieee trans fuzzy syst vol 1 no 2 pp 98 110 may 1993 proposed a possibilistic c means pcm clustering by relaxing the constraint in fuzzy c means fcms that the memberships of a data point across classes sum to 1 the pcm algorithm has a tendency to produce coincident clusters this can be a merit of pcm as a good mode seeking algorithm if initials and parameters are suitably chosen however the performance of pcm heavily depends on initializations and parameter selection in this paper we propose a mechanism of robust automatic merging we then create an automatic merging possibilistic clustering method am pcm where the proposed algorithm does not only solve these parameter selection and initialization problems but also obtains an optimal cluster number the proposed am pcm algorithm first uses all data points as initial cluster centers and then automatically merges these surrounding points around each cluster mode such that it can self organize data groups according to the original data structure the am pcm can exhibit the robustness to parameter noise cluster number different volumes and initializations the computational complexity of am pcm is also analyzed comparisons between am pcm and other clustering methods are made some numerical data and real datasets are used to show these good aspects of am pcm experimental results and comparisons actually demonstrate that the proposed am pcm is an effective and parameter free robust clustering algorithm

a coupled electro thermal discontinuous galerkin method abstract this paper presents a discontinuous galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro thermal problems in this paper we discuss the fundamental equations for the transport of electricity and heat in terms of macroscopic variables such as temperature and electric potential a fully coupled nonlinear weak formulation for electro thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients the weak form can thus be formulated as a discontinuous galerkin method the existence and uniqueness of the weak form solution are proved the numerical properties of the nonlinear elliptic problems i e consistency and stability are demonstrated under specific conditions i e use of high enough stabilization parameter and at least quadratic polynomial approximations moreover the prior error estimates in the h 1 norm and in the l 2 norm are shown to be optimal in the mesh size with the polynomial approximation degree

bounded fuzzy possibilistic method abstract this paper introduces the bounded fuzzy possibilistic method bfpm by addressing several issues that previous clustering classification methods have not considered in fuzzy clustering the data object s membership values should sum to 1 hence any data object may obtain full membership in at most one cluster possibilistic clustering methods remove this restriction however bfpm differs from previous fuzzy and possibilistic clustering approaches by allowing the membership function to take larger values with respect to all clusters furthermore in bfpm a data object can have full membership in multiple clusters or even in all clusters bfpm relaxes the boundary conditions restrictions in membership assignment the assigned boundaries satisfy the necessity of obtaining full memberships and overcome the issues with conventional methods overlapping the functionality of the proposed method has been proved in geometry set theory and other disciplines when learning methods should provide the important arithmetic operations on different domains weighted feature distance function wfd is also used to cover diversity validity and comparison indexes have been applied to evaluate the accuracy of bfpm conventional fuzzy and possibilistic methods are compared with bfpm and bfpm wfd in terms of accuracy fuzzification constant different norms and covering diversity and overlapping the results show that the proposed algorithms perform better than other fuzzy and possibilistic methods

on quasiconformal self mappings of the unit disk satisfying poisson s equation let qc k g be a family of k quasiconformal mappings of the open unit disk onto itself satisfying the pde w g g c u w 0 0 it is proved that qc k g is a uniformly lipschitz family moreover if g is small enough then the family is uniformly bi lipschitz the estimations are asymptotically sharp as k 1 and g 0 so w qc k g behaves almost like a rotation for sufficiently small k and g

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

mean shift mode seeking and clustering mean shift a simple interactive procedure that shifts each data point to the average of data points in its neighborhood is generalized and analyzed in the paper this generalization makes some k means like clustering algorithms its special cases it is shown that mean shift is a mode seeking process on the surface constructed with a shadow kernal for gaussian kernels mean shift is a gradient mapping convergence is studied for mean shift iterations cluster analysis if treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data applications in clustering and hough transform are demonstrated mean shift is also considered as an evolutionary strategy that performs multistart global optimization