a fuzzy extension of the silhouette width criterion for cluster analysis the present paper proposes a new cluster validity measure as an additional criterion to help the decision making process in fuzzy cluster analysis this measure named fuzzy silhouette is a generalization to the fuzzy case of the average silhouette width criterion originally conceived to assess crisp non fuzzy data partitions the fuzzy silhouette is more appealing than its crisp counterpart in the context of fuzzy cluster analysis since it makes explicit use of the fuzzy partition matrix provided by the clustering algorithm in addition it has been designed to improve performance of the original silhouette criterion in detecting regions with higher data density when the data set involves overlapping clusters the performance of the fuzzy silhouette is evaluated and compared to that of five well known cluster validity measures six data sets are used to illustrate different scenarios in which the proposed fuzzy silhouette performs similar to or better than these other criteria thus becoming eligible to join a pool of measures to be used all together in fuzzy cluster analysis

quantile autocovariances a powerful tool for hard and soft partitional clustering of time series abstract a key issue in cluster analysis is determining a proper dissimilarity measure between two data objects and many pairwise dissimilarities have been proposed to deal with time series assuming that the clustering purpose is to group series according to the underlying dependence structures a detailed study of the behavior in clustering of a dissimilarity based on comparing estimated quantile autocovariance functions qaf is carried out quantile autocovariances provide information about the serial dependence structure that other conventional features are not able to capture which suggests great potential to perform clustering of series the asymptotic behavior of the sample quantile autocovariances is studied and an algorithm to determine optimal combinations of lags and pairs of quantile levels to perform clustering is introduced the proposed metric is used to perform hard and soft partitioning based clustering first a broad simulation study examines the behavior of the proposed metric in crisp clustering with the pam procedure a novel fuzzy c medoids algorithm based on the qaf dissimilarity is then proposed and compared with other fuzzy procedures in a new simulation study conducted to cluster fuzzy scenarios involving ar and garch models in all cases the qaf based procedures outperform or are highly competitive with a range of dissimilarities reported in the literature particularly exhibiting high capability to cluster conditionally heteroskedastic time series and robustness to the distributional form of the errors two specific applications involving air quality data and financial time series illustrate the usefulness of the proposed procedures

a fuzzy hybrid hierarchical clustering method with a new criterion able to find the optimal partition classical fuzzy clustering methods are not able to compute a partition in a set of points when classes have nonconvex shape furthermore we know that in this case the usual criteria of class validity such as fuzzy hypervolume or compactness separability do not allow to find the optimal partition the purpose of our paper is to provide a clustering method able to divide a set of points into nonconvex classes without knowing a priori their number we will show that it is possible to reconcile a fuzzy clustering method with a hierarchical ascending one while maintaining a fuzzy partition by a method called unsupervised fuzzy graph clustering to that effect we shall use the fuzzy c means algorithm to divide the set of points into an overspecified number of subclasses a fuzzy relation is then established between them in order to extract the structure of the set of points it can be represented by a graduated hierarchy finally we present a new criterion to find the cut of the hierarchy giving the optimal regrouping this one allows to find the real classes existing into the set of points the given results are compared with those obtained by other classical cluster validity criteria and we propose to study the influence of the number of initial subclasses on the final computed partition

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

a fuzzy clustering procedure for random fuzzy sets a fuzzy clustering method for random fuzzy sets is proposed the starting point is a p value matrix with elements obtained by comparing the expected values of random fuzzy sets by means of a bootstrap test as such the p value matrix can be viewed as a relational data matrix since the p values represent a kind of similarity between random fuzzy sets for this reason in order to cluster random fuzzy sets fuzzy clustering techniques for relational data can be applied in this context the so called ne frc algorithm is considered one of the most important advantages of the ne frc is that the relational data could not be derived from euclidean distances some simulations are presented to show the behavior of the proposed procedure and two applications to real life situations are also included

a clustering performance measure based on fuzzy set decomposition clustering is primarily used to uncover the true underlying structure of a given data set and for this purpose it is desirable to subject the same data to several different clustering algorithms this paper attempts to put an order on the various partitions of a data set obtained from different clustering algorithms the goodness of each partition is expressed by means of a performance measure based on a fuzzy set decomposition of the data set under consideration several experiments reported in here show that the proposed performance measure puts an order on different partitions of the same data which is consistent with the error rate of a classifier designed on the basis of the obtained cluster labelings

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

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

automatic aspect discrimination in data clustering the attributes describing a data set may often be arranged in meaningful subsets each of which corresponds to a different aspect of the data an unsupervised algorithm scad that simultaneously performs fuzzy clustering and aspects weighting was proposed in the literature however scad may fail and halt given certain conditions to fix this problem its steps are modified and then reordered to reduce the number of parameters required to be set by the user in this paper we prove that each step of the resulting algorithm named ascad globally minimizes its cost function with respect to the argument being optimized the asymptotic analysis of ascad leads to a time complexity which is the same as that of fuzzy c means a hard version of the algorithm and a novel validity criterion that considers aspect weights in order to estimate the number of clusters are also described the proposed method is assessed over several artificial and real data sets

evolutionary fuzzy clustering of relational data this paper is concerned with the computational efficiency of fuzzy clustering algorithms when the data set to be clustered is described by a proximity matrix only relational data and the number of clusters must be automatically estimated from such data a fuzzy variant of an evolutionary algorithm for relational clustering is derived and compared against two systematic pseudo exhaustive approaches that can also be used to automatically estimate the number of fuzzy clusters in relational data an extensive collection of experiments involving 18 artificial and two real data sets is reported and analyzed

an efficient algorithm for the minimum capacity cut problem given a finite undirected graph with nonnegative edge capacities the minimum capacity cut problem consists of partitioning the graph into two nonempty sets such that the sum of the capacities of edges connecting the two parts is minimum among all possible partitionings the standard algorithm to calculate a minimum capacity cut due to gomory and hu 1961 runs in o n4 time and is difficult to implement we present an alternative algorithm with the same worst case bound which is easier to implement and which was found empirically to be far superior to the standard algorithm we report computational results for graphs with up to 2000 nodes

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

fuzzy clustering of human activity patterns in the context of human activity pattern analysis we adopt a fuzzy clustering around medoids approach to classify ordered sequences paths these sequences represent patterns of individual behavior in an actual or virtual space time domain a fuzzy approach is suitable for path data since sequences of human activities are typically characterized by switching behaviors which are likely to produce overlapping clusters we adopt a partitioning around medoids strategy since in human activity patterns analysis it is useful to represent each cluster by means of an observed not fictitious prototype medoid to measure pairwise distances among all sequence pairs we make use of the levenshtein distance which allows for the comparison between sequences of different length and explicitly takes into account the sequential nature of the data we also consider two robust versions of the fuzzy clustering algorithm based respectively on the noise cluster and on the trimming technique robust algorithms deal with noisy observations which are likely to occur in this framework and could provide an improvement to the standard model we show several applications on sequence data regarding different research areas like web usage mining travel behavior tourists and shopping paths

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

a meta learning approach for determining the number of clusters with consideration of nearest neighbors an important and challenging problem in data clustering is the determination of the best number of clusters a variety of estimation methods has been proposed over the years to address this problem most of these methods depend on several nontrivial assumptions about the data structure and such methods may thus fail to discover the true clusters in a dataset that does not satisfy those assumptions we develop a new approach that takes as a starting point the simple and intuitive observation that close objects should fall within the same cluster whereas distant ones should not based on this simple notion we utilize a new measurement of good clustering called disconnectivity as well as existing goodness measurements and we embed these measures into a meta learning approach for estimating the number of clusters a simulation experiment based on 13 representative models and an application to real world datasets are conducted to show the effectiveness of the proposed method

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

adaptive estimation for hawkes processes application to genome analysis the aim of this paper is to provide a new method for the detection of either favored or avoided distances between genomic events along dna sequences these events are modeled by a hawkes process the biological problem is actually complex enough to need a non asymptotic penalized model selection approach we provide a theoretical penalty that satisfies an oracle inequality even for quite complex families of models the consecutive theoretical estimator is

correction to on cluster validity for the fuzzy c means model correspondence validation of partitions produced by the fuzzy c means fcm clustering algorithm was discussed by pal and bezdek 1 two tables in 1 contain erroneous values this errata reports the correct values and notes that the conclusions drawn in 1 based on these simulations remain unchanged

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

fuzzy clustering of fuzzy data based on robust loss functions and ordered weighted averaging abstract in many real cases the data are not expressed in term of single values but are imprecise in all these cases standard clustering methods for single valued data are unable to properly take into account the imprecise nature of the data in this paper by considering the partitioning around medoids pam approach in a fuzzy framework we propose a fuzzy clustering method for imprecise data formalized in a fuzzy manner in particular in order to neutralize the negative effects of possible outlier fuzzy data in the clustering process we proposed a robust fuzzy c medoids clustering method for fuzzy data based on the combination of huber s m estimators and yager s owa ordered weighted averaging operators the proposed method is able to smooth the influence of anomalous data by means of a suitable parameter the so called typicality parameter capable to tune the influence of the outliers the performance of the proposed method has been shown by means of a simulation study composed of experiments on i simple two dimensional dataset ii benchmark datasets and iii the fuzzy art outliers dataset the comparison made with the robust clustering methods known from the literature indicates the competitiveness of the introduced method to others an application of the suggested method to a real dataset is also provided and the results of the method has been compared with other clustering methods suggested in the literature in the application the comparative assessment has shown the informational gain in term of additional information of the proposed method vs the other robust methods