Miin-Shen Yang

A cluster validity index for fuzzy clustering

Cluster validity indexes have been used to evaluate the fitness of partitions produced by clustering algorithms. This paper presents a new validity index for fuzzy clustering called a partition coefficient and exponential separation (PCAES) index. It uses the factors from a normalized partition coefficient and an exponential separation measure for each cluster and then pools these two factors to create the PCAES validity index. Considerations involving the compactness and separation measures for each cluster provide different cluster validity merits. In this paper, we also discuss the problem that the validity indexes face in a noisy environment. The efficiency of the proposed PCAES index is compared with several popular validity indexes. More information about these indexes is acquired in series of numerical comparisons and also three real data sets of Iris, Glass and Vowel. The results of comparative study show that the proposed PCAES index has high ability in producing a good cluster number estimate and in addition, it provides a new point of view for cluster validity in a noisy environment.

Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance

This paper presents a new method for similarity measures between intuitionistic fuzzy sets (IFSs). We will present a method to calculate the distance between IFSs on the basis of the Hausdorff distance. We will then use this distance to generate a new similarity measure to calculate the degree of similarity between IFSs. Finally we will prove some properties of the proposed similarity measure and use several examples to compare the proposed similarity measure with existing methods. Numerical results show that the proposed similarity measure is much simpler than existing methods and is well suited to use with linguistic variables.

On a class of fuzzy c -numbers clustering procedures for fuzzy data

This paper describes a class of fuzzy clustering procedures for fuzzy data. Most fuzzy clustering techniques are designed for handling crisp data with their class memberships using the idea of fuzzy set theory. Here we derive new types of fuzzy clustering procedures in dealing with fuzzy data. These procedures are called fuzzy c-numbers (FCN) clusterings. Specially, we construct these FCNs for U-type, triangular, trapezoidal and normal fuzzy numbers. K~JWOY~S; Cluster analysis; Fuzzy clustering; Fuzzy data analysis; Fuzzy numbers; Fuzzy intervals; Fuzzy c-means clustering; Fuzzy c-numbers clustering

Fuzzy entropy on intuitionistic fuzzy sets

In this article we exploit the concept of probability for defining the fuzzy entropy of intuitionistic fuzzy sets (IFSs). We then propose two families of entropy measures for IFSs and also construct the axiom definition and properties. Two definitions of entropy for IFSs proposed by Burillo and Bustince in 1996 and Szmidt and Kacprzyk in 2001 are used. The first one allows us to measure the degree of intuitionism of an IFS, whereas the second one is a nonprobabilistic‐type entropy measure with a geometric interpretation of IFSs used in comparison with our proposed entropy of IFSs in the numerical comparisons. The results show that the proposed entropy measures seem to be more reliable for presenting the degree of fuzziness of an IFS.

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.

Fuzzy clustering algorithms for mixed feature variables

This paper presents fuzzy clustering algorithms for mixed features of symbolic and fuzzy data. El-Sonbaty and Ismail proposed fuzzy c-means (FCM) clustering for symbolic data and Hathaway et al. proposed FCM for fuzzy data. In this paper we give a modified dissimilarity measure for symbolic and fuzzy data and then give FCM clustering algorithms for these mixed data types. Numerical examples and comparisons are also given. Numerical examples illustrate that the modified dissimilarity gives better results. Finally, the proposed clustering algorithm is applied to real data with mixed feature variables of symbolic and fuzzy data.

Cluster analysis based in fuzzy relations

In this paper, cluster analysis based on fuzzy relations is investigated. Tamura’s max-min n-step procedure is extended to all types of max-t compositions. A max-t similarity-relation matrix is obtained by beginning with a proximity-relation matrix based on the proposed max- tn -step procedure. Then a clustering algorithm is created for the max-t similarityrelation matrix. Three critical max-t compositions of max-min, max-prod and max-� are compared. The max-� composition is recommended as the 2rst choice among them. Several examples give more perspectives for di3erent choices of max-t compositions. Finally, the topic of incomplete data via max-t compositions is discussed. Max-t compositions can be e3ectively used to treat the t-connected incomplete data. c 2001 Elsevier Science B.V. All rights reserved.

A similarity measure of intuitionistic fuzzy sets based on the Sugeno integral with its application to pattern recognition

Similarity measures of intuitionistic fuzzy sets are used to indicate the similarity degree between intuitionistic fuzzy sets. Although several similarity measures for intuitionistic fuzzy sets have been proposed in previous studies, no one has considered the use of the Sugeno integral to define them. Since the Sugeno integral provides an expected-value-like operation, it can be a useful tool in defining the expected total similarity degree between two intuitionistic fuzzy sets. In this paper, we propose a new similarity measure formula for intuitionistic fuzzy sets induced by the Sugeno integral. Some examples are illustrated to compare the proposed method with several existing methods. Numerical results show that the proposed similarity measure is more reasonable than those existing methods. On the other hand, measuring the similarity between intuitionistic fuzzy sets is also important in pattern recognition. Finally, the proposed similarity measure uses a robust clustering method to recognize the patterns of intuitionistic fuzzy sets.

Parameter selection for suppressed fuzzy c-means with an application to MRI segmentation

This paper presents an algorithm, called the modified suppressed fuzzy c-means (MS-FCM), that simultaneously performs clustering and parameter selection for the suppressed fuzzy c-means (S-FCM) algorithm proposed by [Fan, J.L., Zhen, W.Z., Xie, W.X., 2003. Suppressed fuzzy c-means clustering algorithm. Pattern Recognition Lett. 24, 1607-1612]. The proposed algorithm is computationally simple, and is able to select the parameter @a in S-FCM with a prototype-driven learning. The parameter selection is based on the exponential separation strength between clusters. Numerical examples will serve to illustrate the effectiveness of the proposed MS-FCM algorithm. Finally, the S-FCM and MS-FCM algorithms are applied in the segmentation of the magnetic resonance image (MRI) of an ophthalmic patient. In our comparisons of S-FCM, MS-FCM, alternative FCM (AFCM) proposed by [Wu, K.L., Yang, M.S., 2002. Alternative c-means clustering algorithms. Pattern Recognition 35, 2267-2278] and similarity-based clustering method (SCM) proposed by [Yang, M.S., Wu, K.L., 2004. A similarity-based robust clustering method. IEEE Trans. Pattern Anal. Machine Intell. 26, 434-448] for these MRI segmentation results, we find that these four techniques provide useful information as an aid to diagnosis in ophthalmology. However, the MS-FCM provides better detection of abnormal tissue than S-FCM, AFCM and SCM when based on a window selection. Overall, the MS-FCM clustering algorithm is more efficient and is strongly recommended as an MRI segmentation technique.

A robust EM clustering algorithm for Gaussian mixture models

Clustering is a useful tool for finding structure in a data set. The mixture likelihood approach to clustering is a popular clustering method, in which the EM algorithm is the most used method. However, the EM algorithm for Gaussian mixture models is quite sensitive to initial values and the number of its components needs to be given a priori. To resolve these drawbacks of the EM, we develop a robust EM clustering algorithm for Gaussian mixture models, first creating a new way to solve these initialization problems. We then construct a schema to automatically obtain an optimal number of clusters. Therefore, the proposed robust EM algorithm is robust to initialization and also different cluster volumes with automatically obtaining an optimal number of clusters. Some experimental examples are used to compare our robust EM algorithm with existing clustering methods. The results demonstrate the superiority and usefulness of our proposed method.