Fuzzy Clustering Using the Convex Hull as Geometrical Model

Fuzzy Sets and Systems, 1991

A number of hard clustering algorithms have been shown to be derivable from the maximum likelihood principle. The only corresponding fuzzy algorithm are the well known fuzzy k-means or FUZZY 1SODATA of Dunn and its generalizations by Bezdek and by Gustafson and Kessel. The authors show how to generate two other fuzzy algorithms which are the analogous of known hard algorithms: the minimization of the fuzzy determinant and of the product of fuzzy determinants. By comparison between the hard and fuzzy methods it appears that the latter yield more often the global optimum, rather than merely a local optimum. This result and the comparison between the different algorithms, together with their specific domains of application, are illustrated by a few numerical examples.

Automatic Control and Computer Sciences

A new hierarchical approach to the problem of clustering, called the Fuzzy Joint Point, FJP) method is proposed. In the FJP method each element of the clusterized set is considered a fuzzy point of a multidimensional space. The concepts of a fuzzy conical point, fuzzy α-neighborhood, and fuzzy α-joint points are introduced and studies of certain properties relative to these concepts are carried out. The proposed algorithm of the FJP method may be used as a preparatory stage of the Fuzzy c-Means (FCM) algorithm for determining the initial classes and their dimensions and also as an independent clustering algorithm.

Control, Automation, Robotics and Vision …

In fuzzy neural network systems, fuzzy membership functions play a key role in making the fuzzy sets organize the input data knowledge in an appropriate and representative manner. Earlier clustering techniques exploit some uniform, convex algebraic functions, such as Gaussian, Triangular or Trapezoidal to represent the fuzzy sets. However, due to the irregularity of the input data, regular and uniform fuzzy sets may not be able to represent the exact feature information of input data. In order to address this issue, a clustering method called Modified Discrete Clustering Technique (MDCT) is proposed in this paper. MDCT represents non-uniform, and normal fuzzy sets with a set of irregular sampling points. The sampling points learn the knowledge of data feature in an irregular and flexible manner. Thus, the fuzzy membership functions generated using these sampling points can provide a better representation of the actual input data.

Peachfuzz 2000 : 19th International Conference of the North American Fuzzy Information Processing Society - Nafips, 2000

The fuzzy clustering proportional membership (FCPM) proposes a model of how data are generated from a cluster structure to be identi…ed. Clusters' prototypes and membership function are meaningful in the context of the model. In particular, the membership of an entity to a cluster expresses the proportion of the cluster's prototype re ‡ected in the entity (proportional membership). In this work we explore the notion of proportional membership and compare it against the fuzzy c-means (FCM) distance membership. The ability of FCPM to reveal the underlying clustering model of data has been studied and a comparison with FCM had been performed as well.

Fuzzy K-means clustering algorithm is very much useful for exploring the structure of a set of patterns, especially when the clusters are overlapping. K-means algorithm is simple with low time complexity, and can process the large data set quickly. But conventional K-means algorithm cannot get high clustering precise rate, and easily be affected by clustering center random initialized and isolated points. This paper proposes an algorithm to compute initial cluster centers for K-means clustering. A cutting plane is used to partition the data in a cell that divides cell in to two smaller cells. The plane is perpendicular to the data axis with high variance and is intended to reduce the sum squared errors of the two cells while at the same time keeping the two cells apart. Cells are partitioned one at a time till the number of cells equals to the predefined number of clusters, K. The centers of the K cells become the initial cluster centers for K-means. The experimental results suggest that the proposed algorithm is effective, converge to better clustering results than those of the random initialization method. The research also indicated the proposed algorithm would greatly improve the likelihood of every cluster containing some data in it. The research also indicated the proposed algorithm would greatly improve the likelihood of every cluster containing some data in it

The clustering algorithm hybridization scheme has become of research interest in data partitioning applications in recent years. The present paper proposes a Hybrid Fuzzy clustering algorithm (combination of Fuzzy C-means with extension and Subtractive clustering algorithm) for data classifications applications. The fuzzy c-means (FCM) and subtractive clustering (SC) algorithm has been widely discussed and applied in pattern recognitions, machine learning and data classifications. However the FCM could not guarantee unique clustering result because initial cluster number is chosen randomly as the result of the classification is unstable. On the other hand, the SC is a fast, one-pass algorithm for estimating the numbers and center of clusters for a set of data. This paper presents the two different clustering algorithms and their comparison. First clustering algorithm is fuzzy c-means clustering, and second is subtractive clustering. Results show that the SC is better than FCM in respect of speed but not as good in accuracy, so a modified hybrid clustering algorithm is designed with all these parameters. The experiments show that the hybrid clustering algorithm can improve the speed, and reduce the iterative amount. At the same time, the hybrid algorithm can make the results of data partitions are more stable and higher accuracy.

Pattern Recognition, 1999

Fuzzy cluster-validity criterion tends to evaluate the quality of fuzzy c-partitions produced by fuzzy clustering algorithms. Many functions have been proposed. Some methods use only the properties of fuzzy membership degrees to evaluate partitions. Others techniques combine the properties of membership degrees and the structure of data. In this paper a new heuristic method is based on the combination of two functions. The search of good clustering is measured by a fuzzy compactness}separation ratio. The "rst function calculates this ratio by considering geometrical properties and membership degrees of data. The second function evaluates it by using only the properties of membership degrees. Four numerical examples are used to illustrate its use as a validity functional. Its e!ectiveness is compared to some existing cluster-validity criterion.

In this paper, a new level-based (hierarchical) approach to the fuzzy clustering problem for spatial data is proposed. In this approach each point of the initial set is handled as a fuzzy point of the multidimensional space. Fuzzy point conical form, fuzzy -neighbor points, fuzzy -joint points are defined and their properties are explored. It is known that in classical fuzzy clustering the matter of fuzziness is usually a possibility of membership of each element into different classes with different positive degrees from [0,1]. In this study, the fuzziness of clustering is evaluated as how much in detail the properties of classified elements are investigated. In this extent, a new Fuzzy Joint Points (FJP) method which is robust through noises is proposed. Algorithm of FJP method is developed and some properties of the algorithm are explored. Also sufficient condition to recognize a hidden optimal structure of clusters is proven. The main advantage of the FJP algorithm is that it combines determination of initial clusters, cluster validity and direct clustering, which are the fundamental stages of a clustering process. It is possible to handle the fuzzy properties with various level-degrees of details and to recognize individual outlier elements as independent classes by the FJP method. This method could be important in biological, medical, geographical information, mapping, etc. problems.

IEEE International Conference on Fuzzy Systems

In our everyday life the number of groups of similar objects that we visually perceive is deeply constrained by how far we are from the objects and also by the direction we are approaching them. Based on this metaphor, in this work we present a generalization of partitional clustering aiming at the inclusion into the clustering process of both distance and direction of the point of observation towards the dataset. This is done by incorporating a new term in the objective function, accounting for the distance between the clusterspsila prototypes and the point of observation. It is a well known fact that the chosen number of partitions has a major effect on the objective function based partitional clustering algorithms, conditioning both the level of granularity of the data grouping and the capability of the algorithm to accurately reflect the underlying structure of the data. Thus the correct choice of the number of clusters is essential for any successful application of such algorit...

Advances in Fuzzy Systems

A novel hybrid clustering method, named KC-Means clustering, is proposed for improving upon the clustering time of the Fuzzy C-Means algorithm. The proposed method combines K-Means and Fuzzy C-Means algorithms into two stages. In the first stage, the K-Means algorithm is applied to the dataset to find the centers of a fixed number of groups. In the second stage, the Fuzzy C-Means algorithm is applied on the centers obtained in the first stage. Comparisons are then made between the proposed and other algorithms in terms of time processing and accuracy. In addition, the mentioned clustering algorithms are applied to a few benchmark datasets in order to verify their performances. Finally, a class of Minkowski distances is used to determine the influence of distance on the clustering performance.

2007

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Kernelized Fuzzy C-Means clustering technique is an attempt to improve the performance of the conventional Fuzzy C-Means clustering technique. Recently this technique where a kernel-induced distance function is used as a similarity measure instead of a Euclidean distance which is used in the conventional Fuzzy C-Means clustering technique, has earned popularity among research community. Like the conventional Fuzzy C-Means clustering technique this technique also suffers from inconsistency in its performance due to the fact that here also the initial centroids are obtained based on the randomly initialized membership values of the objects. Our present work proposes a new method where we have applied the Subtractive clustering technique of Chiu as a preprocessor to Kernelized Fuzzy C-Means clustering technique. With this new method we have tried not only to remove the inconsistency of Kernelized Fuzzy C-Means clustering technique but also to deal with the situations where the number of clusters is not predetermined. We have also provided a comparison of our method with the Subtractive clustering technique of Chiu and Kernelized Fuzzy C-Means clustering technique using two validity measures namely Partition Coefficient and Clustering Entropy.

International journal of performability engineering, 2006

Performance testing of an algorithm is necessary to ascertain its applicability in real data and to evolve software. Clustering of a data set could be either fuzzy (having vague boundaries among the clusters) or crisp (having welldefined fixed boundaries) in nature. The present work is focused on the performance measure of some similarity-based fuzzy clustering algorithms, where three methods and each method having three different approaches are developed. In the first method, cluster centers are decided based on the minimum of entropy (probability) values of different data points [10]. In the second method, cluster centers are selected based on the maximum of total similarity values of data points and in the third method, a ratio of dissimilarity to similarity is considered to determine the cluster centers. Performances of these methods and approaches are compared on three standard data sets, such as IRIS, WINES, and OLITOS. Experimental results show that entropy-based method is able to generate better quality clusters but at the cost of little more computations. Finally, the best sets of clusters are mapped to 2-D using a self-organizing map (SOM) for visualization.

Fuzzy Sets and Systems, 2015

The initial idea of extending the classical k-means clustering technique to an algorithm that uses membership degrees instead of crisp assignments of data objects to clusters led to the invention of a large variety of new fuzzy clustering algorithms. However, most of these algorithms are concerned with cluster shapes or outliers and could have been defined without any problems in the context of crisp assignments of data objects to clusters. In this paper, we demonstrate that the use of membership degrees for these algorithms-although it is not necessary from the theoretical point of view-is essential for these algorithms to function in practice. With crisp assignments of data objects to clusters these algorithms would get stuck most of the time in a local minimum of their underlying objective function, leading to undesired clustering results. In other contributionsm it was shown that the use of membership degrees can avoid this problem of local minima but it also introduces new problems, especially for clus

Computers & Geosciences, 1984

AbstractnThis paper transmits a FORTRAN-IV coding of the fuzzy c-means (FCM) clustering program. The FCM program is applicable to a wide variety of geostatistical data analysis problems. This program generates fuzzy partitions and prototypes for any set of numerical data. These partitions are useful for corroborating known substructures or suggesting substructure in unexplored data. The clustering criterion used to aggregate subsets is a generalized least-squares objective function. Features of this program include a choice of three norms (Euclidean, Diagonal, or Mahalonobis), an adjustable weighting factor that essentially controls sensitivity to noise, acceptance of variable numbers of clusters, and outputs that include several measures of cluster validity.