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

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

simplified csp analysis of a stiff stochastic ode system abstract we develop a simplified computational singular perturbation csp analysis of a stochastic dynamical system we focus on the case of parametric uncertainty and rely on polynomial chaos pc representations to quantify its impact we restrict our attention to a system that exhibits distinct timescales and that tends to a deterministic steady state irrespective of the random inputs a detailed analysis of eigenvalues and eigenvectors of the stochastic system jacobian is conducted which provides a relationship between the pc representation of the stochastic jacobian and the jacobian of the galerkin form of the stochastic system the analysis is then used to guide the application of a simplified csp formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system two approaches are specifically developed with the resulting simplified csp framework the first uses the stochastic eigenvectors of the uncertain system as csp vectors whereas the second uses the eigenvectors of the nominal system as csp vectors numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis and illustrate the effectiveness of the simplified csp algorithms in addressing the stiffness of the system dynamics

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

relational duals of the c means clustering algorithms abstract the hard and fuzzy c means algorithms are widely used effective tools for the problem of clustering n objects into hard or fuzzy groups of similar individuals when the data is available as object data consisting of a set of n feature vectors in rp however object data algorithms are not directly applicable when the n objects are implicitly described in terms of relational data which consists of a set of n2 measurements of relations between each of the pairs of objects new relational versions of the hard and fuzzy c means algorithms are presented here for the case when the relational data can reasonably be viewed as some measure of distance some convergence properties of the algorithms are given along with a numerical example

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

fixed point iterations in determining a tikhonov regularization parameter in kirsch s factorization method kirsch s factorization method is a fast inversion technique for visualizing the profile of a scatterer from measurements of the far field pattern the mathematical basis of this method is given by the far field equation which is a fredholm integral equation of the first kind in which the data function is a known analytic function and the integral kernel is the measured and therefore noisy far field pattern we present a tikhonov parameter choice approach based on a fast fixed point iteration method which constructs a regularization parameter associated with the corner of the l curve in log log scale the performance of the method is evaluated by comparing our reconstructions with those obtained via the l curve and we conclude that our method yields reliable reconstructions at a lower computational cost

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

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

on the convergence of miehle s algorithm for the euclidean multifacility location problem for the euclidean single facility location problem e weiszfeld proposed a simple iterative algorithm in 1937 later it was proved by numerous authors that it is a convergent descent algorithm w miehle extended weiszfeld s algorithm to solve the euclidean multifacility location problem then l m ostresh proved that miehle s algorithm is a descent algorithm recently f rado modified miehle s algorithm and provided several sets of sufficient conditions for the modified algorithm to converge he also indicated that the convergence of miehle s algorithm was an open problem in this paper the relationship between miehle s multifacility location algorithm and weiszfeld s single facility location algorithm is analyzed counterexamples show that miehle s algorithm may converge to nonoptimal points for well structured problems

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

a highly robust estimator through partially likelihood function modeling and its application in computer vision the authors present a highly robust estimator known as the model fitting mf estimator for general regression they explain that high robustness becomes possible through partially but completely modeling the unknown log likelihood function the partial modeling takes place by taking the bayesian statistical decision rule and a number of important heuristics into consideration while maximizing the log likelihood function applications include the automatic selection of multiple thresholds single rigid motion estimation or multiple rigid motion segmentation and estimation from two perspective views it is believed that the proposed mf estimator will aid in solving many robust estimation problems that demand an estimator that is either highly robust or capable of handling contaminated gaussian mixture models

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

fuzzy pca guided robust k means clustering this paper proposes a new approach to robust clustering in which a robust k means partition is derived by using a noise rejection mechanism based on the noise clustering approach the responsibility weight of each sample for the k means process is estimated by considering the noise degree of the sample and cluster indicators are calculated in a fuzzy principal component analysis pca guided manner where fuzzy pca guided robust k means is performed by considering responsibility weights of samples then the proposed method achieves cluster core estimation in a deterministic way the validity of the derived cluster cores is visually assessed through distance sensitive ordering which considers responsibility weights of samples numerical experiments demonstrate that the proposed method is useful for capturing cluster cores by rejecting noise samples and we can easily assess cluster validity by using cluster crossing curves

generalized weighted conditional fuzzy 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 among many existing modifications of this method conditional or context dependent c means is the most interesting one in this method data vectors are clustered under conditions based on linguistic terms represented by fuzzy sets this paper introduces a family of generalized weighted conditional fuzzy c means clustering algorithms this family include both the well known fuzzy c means method and the conditional fuzzy c means method performance of the new clustering algorithm is experimentally compared with fuzzy c means using synthetic data with outliers and the box jenkins database

on weighting clustering recent papers and patents in iterative unsupervised learning have emphasized a new trend in clustering it basically consists of penalizing solutions via weights on the instance points somehow making clustering move toward the hardest points to cluster the motivations come principally from an analogy with powerful supervised classification methods known as boosting algorithms however interest in this analogy has so far been mainly borne out from experimental studies only this paper is to the best of our knowledge the first attempt at its formalization more precisely we handle clustering as a constrained minimization of a bregman divergence weight modifications rely on the local variations of the expected complete log likelihoods theoretical results show benefits resembling those of boosting algorithms and bring modified weighted versions of clustering algorithms such as k means fuzzy c means expectation maximization em and k harmonic means experiments are provided for all these algorithms with a readily available code they display the advantages that subtle data reweighting may bring to clustering

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

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

density weighted fuzzy c means clustering in this short paper a unified framework for performing density weighted fuzzy c means fcm clustering of feature and relational datasets is presented the proposed approach consists of reducing the original dataset to a smaller one assigning each selected datum a weight reflecting the number of nearby data clustering the weighted reduced dataset using a weighted version of the feature or relational data fcm algorithm and if desired extending the reduced data results back to the original dataset several methods are given for each of the tasks of data subset selection weight assignment and extension of the weighted clustering results the newly proposed weighted version of the non euclidean relational fcm algorithm is proved to produce the identical results as its feature data analog for a certain type of relational data artificial and real data examples are used to demonstrate and contrast various instances of this general approach

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

fuzzy clustering with high contrast in a fuzzy clustering an object typically receives strictly positive memberships to all clusters even when the object clearly belongs to one particular cluster consequently each cluster s estimated center and scatter matrix are influenced by many objects that have small positive memberships to it this effect may keep the fuzzy method from finding the true clusters we analyze the cause and propose a remedy which is a modification of the objective function and the corresponding algorithm the resulting clustering has a high contrast in the sense that outlying and bridging objects remain fuzzy whereas the other objects become crisp the enhanced version of fuzzy k means is illustrated with an example as well as the enhanced version of the fuzzy minimum volume method

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

a comparison of fuzzy shell clustering methods for the detection of ellipses in this paper we introduce a shell clustering algorithm for ellipsoidal clusters based on the so called radial distance which can be easily extended to superquadric clusters we compare our algorithm with other algorithms in the literature that are based on the algebraic distance the approximate distance the normalized radial distance and the exact distance we evaluate the performance of each algorithm on two dimensional data sets containing scattered ellipses partial ellipses outliers and ellipses of disparate sizes and summarize the relative strengths and weaknesses of each algorithm

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

fuzzy set technology in knowledge discovery abstract fuzzy models are constructs relying heavily both on a qualitative domain knowledge and diverse optimization techniques what makes them different from other models is their inherent embedding in the context of nonnumeric set or fuzzy set oriented information one can also look at the development of the fuzzy models from the perspective of data mining a prudent and user oriented sifting of data qualitative observations and calibration of commonsense rules in an attempt to establish meaningful and useful relationships between system s variables the role of fuzzy sets in knowledge discovery has not been visible even though they are inherently inclined towards coping with linguistic domain knowledge the paper re examines the key issues of knowledge discovery by putting them in the context of the technology of fuzzy sets subsequently we reveal several interesting links between fuzzy data mining and fuzzy sets the study is also geared toward a knowledge oriented and context based modification of well known fuzzy clustering

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

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

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

a least biased fuzzy clustering method a new operational definition of cluster is proposed and a fuzzy clustering algorithm with minimal biases is formulated by making use of the maximum entropy principle to maximize the entropy of the centroids with respect to the data points clustering entropy the authors make no assumptions on the number of clusters or their initial positions for each value of an adimensional scale parameter spl beta the clustering algorithm makes each data point iterate towards one of the cluster s centroids so that both hard and fuzzy partitions are obtained since the clustering algorithm can make a multiscale analysis of the given data set one can obtain both hierarchy and partitioning type clustering the relative stability with respect to spl beta of each cluster structure is defined as the measurement of cluster validity the authors determine the specific value of spl beta which corresponds to the optimal positions of cluster centroids by minimizing the entropy of the data points with respect to the centroids clustered entropy examples are given to show how this least biased method succeeds in getting perceptually correct clustering results

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

fitting an unknown number of lines and planes to image data through compatible cluster merging abstract a compatible cluster merging algorithm is presented that is specially designed to find the optimum number of linear planar and hyperplanar clusters i e clusters that lie in a subspace of the original space when an upper bound on the number of clusters present is known this algorithm is shown to be superior to more traditional validity measure based approaches the effectiveness and advantages of the proposed technique in 2d and 3d applications is demonstrated with both synthetic and real data the proposed applications include character recognition obtaining straight line descriptions of intensity edge images and obtaining planar approximations of 3d range data