Fuzzy Clustering For Symbolic Data

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 6, NO. 2, MAY 1998

195

Fuzzy Clustering for Symbolic Data Yasser El-Sonbaty and M. A. Ismail

Abstract—Most of the techniques used in the literature in clustering symbolic data are based on the hierarchical methodology, which utilizes the concept of agglomerative or divisive methods as the core of the algorithm. The main contribution of this paper is to show how to apply the concept of fuzziness on a data set of symbolic objects and how to use this concept in formulating the clustering problem of symbolic objects as a partitioning problem. Finally, a fuzzy symbolic c-means algorithm is introduced as an application of applying and testing the proposed algorithm on real and synthetic data sets. The results of the application of the new algorithm show that the new technique is quite efficient and, in many respects, superior to traditional methods of hierarchical nature. Index Terms—Fuzzy clustering, hierarchical techniques, partitioning techniques, soft clustering, symbolic objects.

I. INTRODUCTION

T

HE objective of cluster analysis is to group a set of objects into clusters such that objects within the same cluster have a high degree of similarity, while objects belonging to different clusters have a high degree of dissimilarity. The clustering of data set into subsets can be divided into hierarchical and nonhierarchical or partitioning methods. The general rationale behind partitioning methods is to choose some initial partitioning of the data set and then alter cluster memberships so as to obtain better partitions according to a predefined objective function. Hierarchical clustering procedures can be divided into agglomerative methods, which progressively merge the objects according to some distance measure in such a way that whenever two objects belong to the same cluster at some level they remain together at all higher levels and divisive methods, which progressively subdivide the data set [1]. Objects to be clustered usually come from an experimental study of some phenomenon and are described by a specific set of features selected by the data analyst. The feature values may be measured on different scale and these can be continuous numeric, symbolic, or structured. Continuous numeric data are well known as a classical data type and many algorithms for clustering this type of data using partitioning or hierarchical techniques can be found in the literature [2]. Meanwhile, there is some research dealing with symbolic objects [3]–[9] and this is due to the nature of such objects, which is simple in construction but hard in processing. Besides, the values taken by the features of Manuscript received March 18, 1996; revised February 24, 1997. Y. El-Sonbaty is with the Department of Electrical and Computer Engineering, Arab Academy for Science and Technology, Alexandria, 1029 Egypt. M. A. Ismail is with the Department of Computer Science, Faculty of Engineering, Alexandria, 21544 Egypt. Publisher Item Identifier S 1063-6706(98)00806-6.

symbolic objects may include one or more elementary objects and the data set may have a variable number of features [4]. Structured objects have higher complexity than continuous and symbolic objects because of their structure, which is much more complex, and their representation, which needs higher data structures to permit the description of relations between elementary object components and facilitate hierarchical object models that describe how an object is built up from the primitives. A survey of different representations and proximity measures of structured objects can be found in [10]. Diday [3] and Gowda and Diday [4], [5] presented dissimilarity and similarity measures based on position, span, and content of symbolic objects. The distance measure is used in the area of conventional hierarchical clustering of symbolic data. More work can be found in the field of conceptual hierarchical clustering of symbolic data. Fisher [6] introduced a top-down incremental conceptual clustering using a category utility metric called the COBWEB. Cheng and Fu [7] developed HUATUO which produced intermediate conceptual structures for rule-based systems. Michalski and Stepp [8] proposed CLUSTER/2, a conjunctive conceptual clustering in which descriptive concepts are conjunctive statements involving relations on selected objects features and optimized according to certain criterion of clustering quality. Ralambondrainy [9] presented a conceptual version of the Kmeans algorithm for numeric and discrete data based on coding symbolic data numerically and using a mix of Euclidean and Chi-square distances to calculate the distance between the hybrid types of data that are represented using predicates as a group of attribute-value tuples joined by logical operators. A survey of different techniques of conceptual hierarchical clustering for symbolic data can be found in [10]. From the above, it is clear that most of the algorithms available in the literature for clustering symbolic objects, are based on either conventional or conceptual hierarchical techniques using agglomerative or divisive methods as the core of the algorithm [3]–[8], [10]. Although a partitioning technique was introduced in [9], it has many drawbacks like: it coded symbolic data numerically (which distorted the original data); it cannot handle interval type of data; the suggested distance has two weights that their values are very difficult to be chosen; and the structure of the predicates selected in representing the objects are hard in processing. The drawbacks in using hierarchical techniques are well known in the field of data clustering. Memory size, updating the membership matrix, complexity per iteration of calculating distance function, overall complexity of the algorithm, and giving a nondetermined classification of the patterns to name a few of these difficulties faced when using any hierarchical based technique [2].

1063–6706/98$10.00  1998 IEEE

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The main contribution of this paper is to show how to apply the concept of fuzziness on a data set of symbolic objects. Also, how to use this concept in formulating the clustering problem of symbolic objects as a partitioning problem. Finally, a modified fuzzy c-means algorithm is introduced as an application of applying and testing the proposed algorithm on real and synthetic data sets. The proposed algorithm eliminates most of the drawbacks found in hierarchical techniques by formulating the given symbolic clustering problem as an optimization problem with a specific objective function subject to a group of constraints. The concept of fuzziness is applied here to give more meaning and easier interpretation of the results obtained from the proposed algorithm. In the following, the details of the proposed algorithm are given together with the description of symbolic objects and their distance measure. Section II discusses the definition and the distance measure for symbolic objects. Section III describes the proposed algorithm. Applications and analysis of experimental results are shown in Sections IV and V. II. SYMBOLIC OBJECTS Various definitions and descriptions of symbolic objects and distance measures are found in the literature. Here, we follow those given by Diday [3] and Gowda and Diday [4], [5] and in the following two sections these definitions and descriptions are demonstrated. A. Feature Types

If some of the features could exhibit a much profound effect on calculating the total dissimilarity, the above mentioned formula can be rewritten in the following format:

where represents the weight corresponding to the th feature. The dissimilarity component due to position arises only when the feature type is quantitative. It indicates the relative positions of two feature values on real line. The dissimilarity component due to span indicates the relative sizes of the feature values without referring to common parts between them. The component due to content is a measure of the noncommon , , parts between two feature values. The components are defined such that their values are normalized between zero and one [4]. and : The dissimilarity be1) Quantitative Type of tween two feature values of quantitative type is defined as the dissimilarity of these values due to position, span, and content previously mentioned. The dissimilarity component due to position is lower limit of lower limit of length of maximum interval ( th feature) where length of maximum interval ( th feature) is the difference between highest and lowest values of the th feature over all the objects. The dissimilarity component due to span is length of length of span length of and

can be written as the Cartesian The symbolic object product of specific values of its features ’s as

The feature values may be measured on different scales resulting in the following types: 1) quantitative features, which can be classified into continuous, discrete. and interval values and 2) qualitative features, which can be classified into nominal (unordered), ordinal (ordered), and combinational.

where span length of and is the length of minimum interval containing both and . The dissimilarity component due to content is length of

length of and

of intersection of length of

length span

and

The net dissimilarity between

and

is

B. Dissimilarity Many distance measures are introduced in the literature for symbolic objects [4], [5], [12]. Here we follow the dissimilarity distance measure introduced by Gowda and Diday [4] with a brief explanation of this distance. The dissimilarity between two symbolic objects and is defined as

For the th feature, is defined using the following three components [4]: due to position ; 1) 2) due to span ; due to content c. 3)

Quantitative ratio and absolute type of features are special cases of interval type having the following properties: length of

length of

2) Qualitative Type of and : For qualitative type of features, the dissimilarity component due to position is absent. The two components which contribute to dissimilarity are as follows. The dissimilarity component due to span length of length of span length of and Where the length of qualitative feature value is the number of its elements and the span length of two qualitative feature values is defined as the number of elements in their union.

EL-SONBATY AND ISMAIL: FUZZY CLUSTERING FOR SYMBOLIC DATA

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sets. Unlike hard clustering, the patterns in fuzzy clustering need not commit to a cluster center indefinitely and this makes it possible to escape from local extreme of the objective function. Hard clustering for symbolic objects is intuitive and can be easily implemented using the concept of Cartesian join, which will be mentioned later in this section. On the other hand, fuzzy clustering of symbolic data is quite involved and is the main concern of this paper.

TABLE I MICROCOMPUTER DATA

A. Fuzzy Clustering for Symbolic Objects Fuzzy c-means clustering for numerical data is the algorithm that attempts to find a solution to the mathematical program [16] Minimize

(1)

The dissimilarity component due to content length of

length of

of intersection of length of

length

and

subject to

span

and

The net dissimilarity between

(2) and

is where number of patterns

Illustrative Example: Assume we would like to calculate the distance between objects number zero (Apple II) and nine (Ohio Sc. II Series) shown in Table I:

number of clusters a scalar,

D(Apple II, Ohio Sc. II Series) D(COLOR TV B&W TV) D(48K, 48K) D(10K, 10K) D(6502, 6502C) D(52, 53–56) D(COLOR TV, B&W TV)

center of cluster degree of membership of pattern cluster centers matrix dimension of the feature space

D(48K, 48K) 0 D(10K, 10K) 0 D(6502, 6502C) D(52, 53–56) where is the maximum length of the interval “Keys” D(Apple II, Ohio Sc. II Series)

membership matrix pattern In applying the above algorithm to symbolic objects, two main problems are encountered. These problems are as follows. Problem 1: The formation of cluster centers. This process differs from numeric objects where the centers are calculated using the formula

This value is different from the equivalent distance published in [4] “0.42” because the distance in [4] is normalized. III. THE PROPOSED ALGORITHM Most of the techniques found in the literature that deal with symbolic objects are hierarchical rather than partitioning techniques and their drawbacks are well known, as mentioned in Section I. In this section, a new algorithm for fuzzy clustering of symbolic objects is presented. The main purpose of this algorithm is to show how to apply the concept of fuzziness [13]–[16] on a data set of symbolic objects. A modified version of the fuzzy c-means algorithm [17]–[19] is introduced here to test the behavior of the proposed algorithm on different data

in cluster

(3) While in symbolic objects, arithmetic operations are completely absent because of the nature of the objects with which we are dealing. The only valid operation is the Cartesian join [4] which states that: If and are two objects, then the composite object resulting and is from merging

where

is a Cartesian join operator.

198


Fig. 1. Structure of a cluster center for data set in Table I.

MEMBERSHIP VALUES

OF

TABLE II PATTERNS IN TABLE I

TO

CLUSTER CENTER

IN

FIG. 1

When the th feature is quantitative or ordinal qualitative, is defined as the minimum interval that includes and ; that is both

(7) (8)

where where and stand for the lower and upper limits of . When the th feature is qualitative nominal, is the union of and ; that is

no. of features no. of events/feature no. of objects

In hard clustering of symbolic objects, the Cartesian join operator can be used for constructing the cluster centers. In this case, the cluster centers can be represented as the merging of all symbolic objects belonging to a specific cluster. On the other hand, the Cartesian join operator cannot be used in solving the problem of fuzzy clustering of symbolic objects because the domain of this problem is completely different from that of hard clustering. To overcome this problem, the following solution is suggested. A cluster center can be formed as a group of features, each feature is a group of ordered pairs, each is of the form , where is the th event of feature and is the degree of association of this event to the feature in cluster center such that (4) is the th feature of the th cluster center. It where is regarded as a fuzzy set of all possible events in the th feature of the patterns “ ” and the associated memberships are a degree of association with each event “ ” (5) (6)

no. of clusters if the event associated with it is not a part of feature . While if there are no events sharing this event in forming the feature. The starting value for is one for all events forming the features because the initial conditions are chosen to be distinct. is updated using the formula The value of

(9)

and if the th feature of the th where pattern consists of the th event, otherwise . is the membership of the th pattern in the th cluster. Illustrative Example: Fig. 1 shows an example of the structure of a cluster center for the fuzzy clustering of the data set shown in Table I assuming in Table II the membership values to this cluster center. The data set describes a group of microcomputers [8]. The data consists of 12 objects. Each object has five features. Two of the features are qualitative (display and MP) and the rest are quantitative (RAM, ROM, and keys). For the first feature “DISPLAY,” the structure of this feature in the cluster center can be calculated as follows:


Fig. 2. Average number of iterations for different values of

199

m.

(COLOR, [0.20 0.40 0.15]/3.4), (B&W, [0.50 0.60 0.20 0.15]/3.4), (BUILT-IN, [0.25 0.30 0.05 0.50]/3.4), (TERMINAL, 0.10/3.4) (COLOR, 0.22), (B&W, 0.43), (BUILT-IN, 0.32), (TERMINAL, 0.03) where 3.4 equals the total sum of the ’s shown in Table II. The second problem encountered when applying the fuzzy c-means to symbolic objects is as follows. Problem 2: The calculation of the dissimilarity between patterns and cluster centers. This problem arises due to the changes done in forming of the cluster centers as discussed in Problem 1. This problem is solved using the concept of weighted dissimilarity [20] given by

Fig. 3. Objective function for different values of

m.

Dissimilarity Color TV, Color TV

Color TV,

Color TV, Built-in

B&W TV

Color TV, Terminal 48k, 48k

48k, 32k

48k, 64k where and are the features constructing objects and , . The weights ( ) respectively, for associated with the features are calculated heuristically or using some optimization routines. However, due to the changes proposed in forming cluster centers for symbolic objects, some modifications are done to make the above formula more suitable for the problem of symbolic clustering. The dissimilarity between pattern and cluster center is given by

(10) is the dissimilarity between feature in where ” and event of feature in cluster center pattern “ “ ” and is calculated as mentioned earlier in Section II-B. Illustrative Example: Assume we would like to calculate the dissimilarity distance between the cluster center shown in Fig. 1 and the following pattern:

10k, 1k

RAM

ROM

COLOR TV

48K

10K

MP

KEYS

6502

52

10k, 4k 10k, 8k

10k, 80k

10k, 12k

10k, 14k A

Z

A

HP

C –

– –

where is the distance between and based on their data types. The dissimilarity ( ) is mainly dependent on the selection of the distance measure used in calculating the distance between symbolic data. Review [4], [5], [12] for different distance measures of symbolic data. C. Proof of Correctness objects belonging Assume we have cluster center with degree of membership . For numerical objects, the feature of cluster center is calculated as follows: to

DISPLAY

10k, 10k

10k, 11–16k

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Fig. 4. Results for the proposed algorithm.

Fig. 6. Membership matrix for microcomputer data at

Fig. 5. Membership matrix for microcomputer data at

m

m

= 2:0.

= 1:1.

review (3) Fig. 7. Values of objective function at different initial states for

m

= 2 :0 .

where tion can be rewritten in the following form:

implies

To calculate the distance between feature in object “ ” and feature in cluster center “ ”

Assuming cluster center

’s are the events constructing feature and from (5) and (6)

in

which is equivalent to (10). D. Fuzzy Symbolic C-Means Algorithm (FSCM) For symbolic objects, the arithmetic operation “ ” is not valid for the equation of calculating . This equa-

In this section, a new partitioning algorithm for clustering symbolic data is introduced. This algorithm is a modified


version of fuzzy c-means for numerical data. The main objective of the new algorithm (FSCM) is to show how to apply the concept of fuzziness on symbolic data sets. In each iteration, the membership matrix and cluster centers are updated according to (4)–(9). The initial cluster centers are chosen arbitrarily to allow getting different results and not to stuck to certain one. In case of divergence, the number of iterations exceeds the maximum allowed number of iterations and the algorithm is terminated. Fuzzy symbolic c-means algorithm is shown at the bottom of the page.

IV. EXPERIMENTAL RESULTS In this section, the performance of the proposed algorithm is tested and evaluated using some test data reported in the literature. The data sets used in these experiments are synthetic or real data and their classification is known from other clustering techniques [4], [5], [8], [12]. A comparison between results obtained from the proposed algorithm and other techniques is given. Every experiment is repeated for . The experiments used different values of , are explained in Sections IV-A and B. A. The Problem of Determining a Classification of Microcomputers The data set of microcomputers [8] shown in Table I is used in this experiment. As previously mentioned in Section III-A,

[ [ [ [

201

the data consists of 12 objects; each object has five features. Two of the features are qualitative and the rest are quantitative. Fig. 2 shows the variations of the average number of iterations needed to reach a solution for different values of . Fig. 3 shows the distribution of the average objective function . The final membership matrix of the over proposed algorithm tends to follow the distribution shown in Fig. 4 and these results are completely the same as the results reported in [4]. Figs. 5 and 6 show samples of the and , final membership matrices obtained for respectively. The rest of the memberships for different values of are omitted due to the scope of the paper. Fig. 7 shows the distribution of the objective function over different initial states.

B. The Problem of Determining a Classification of the Fat–Oil Data The data set [4], [5], [21], [22] used for this problem is shown in Table III. It consists of data of fats and oils having four quantitative features of interval type and one qualitative feature. Fig. 8 shows the variations of the average number of iterations needed to reach a solution for different values of . Fig. 9 shows the distribution of the . The final average objective function over membership matrix of the proposed algorithm tends to follow the distribution shown in Fig. 10 and these results are better than the results reported in [4] where the classification

] ] Choose the initial cluster centers arbitrarily. ] Select the value of the exponent . ] ] Calculate the Membership Matrix W; from the formula

where

review (10) [FSCM-3-2] Calculate new cluster centers from: no. of features where

is calculated as follows: review (4)–(9)

(Convergence or maximum number of iterations is exceeded) End [FSCM]

202


TABLE III FAT–OIL DATA

Fig. 10. Results for the proposed algorithm.

Fig. 8. Average number of iterations for different values of

Fig. 9. Objective function for different values of

m

m

.

.

was . The optimum solution for this is , as mentioned in problem for [5], [21], and [22]. The distribution obtained from the proposed algorithm has always lower value of objective function than are explained later in those in [4]. The results for Section V. Figs. 11 and 12 show samples of the final memberand , respectively. The ship matrices obtained for are omitted rest of the memberships for different values of due to the scope of the paper. Fig. 13 shows the effect of initial states on the value of the objective function.

Fig. 11. Membership matrix for fat–oil data at

m

= 1:1.

Fig. 12. Membership matrix for fat–oil data at

m

= 2:0.

V. ANALYSIS OF PROPOSED ALGORITHM AND EXPERIMENTAL RESULTS A new approach for the clustering of symbolic data based on the principle of fuzziness is introduced. Solutions to the problems of the formation of cluster center and calculation of distance measure between objects and cluster centers, are proposed. Fuzzy symbolic c-means algorithm (FSCM) is then introduced as a tool of applying and testing the behavior of the proposed algorithm on different data sets. From the experimental results the following points can be concluded.


Fig. 13.

Values of objective function at different initial states for

m

203

= 2:0.

Fig. 16. Fuzzy and soft clustering for fat–oil data. (a)

(b) Fig. 14. Results of the proposed algorithm on (a) microcomputer data and (b) fat–oil data using the similarity measure reported in [5].

Fig. 15.

Fuzzy and soft clustering for microcomputer data.

1) The proposed algorithm succeeds to apply the concept of fuzziness on symbolic objects. 2) The behavior of the proposed fuzzy symbolic c-means for symbolic objects is consistent with that of conventional fuzzy c-means for numerical vectors. 3) The final results of the proposed algorithm are consistent with the results published in the literature [4] for solving the test problems. 4) From the experimental results, it was found that the most was in the range: . appropriate value of 5) Although, we can find in the literature different results for the test problems [4], [5], [8], [12], [21], [22], the results obtained from the proposed algorithm are completely logical and consistent with those in [4] since the same distance measure is used. The variations of the results in the literature are due to the use of different distance measures and different methodologies. 6) When applying the proposed algorithm on microcomputer data and fat–oil data shown in Tables I and III at and , respectively, using the similarity measure introduced in [5], the results shown in Fig. 14(a) and (b), respectively, were obtained. The results represent the crisp distribution of the final membership matrix for each experiment. The similarity measure of [5] was used here instead of that in [4] to facilitate comparing the results obtained from the proposed algorithm with those reported in [5], [22] that used the same similarity measure. These results are the same as reported in [5]. When applying the single-linkage method given in [20], we got the same results for the fat–oil data and with different merging for the microcomputer data. In [22], the same classification for the fat–oil data was obtained as a result of applying the NERF algorithm [22] on the similarity distance matrix published in [5].

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7) The new algorithm overcomes most of the drawbacks encountered in dealing with hierarchical algorithms for clustering symbolic data by formulating the given symbolic clustering problem as an optimization problem with a specific objective function subject to a group of constraints. The concept of fuzziness is applied here to give more meaning and easier interpretation of the results obtained from the proposed algorithm. 8) The results obtained from the proposed algorithm are mainly dependent on , initial conditions and the similarity/dissimilarity measure used for calculating the distance between the symbolic objects. near 1.1, the behavior of the 9) For small values of new algorithm is almost the same as that of hard clustering and this can be observed from the membership matrix. This result is consistent with the behavior of conventional fuzzy-c-means for numerical data. increases the required number 10) Increasing the factor of iterations needed to reach a solution. This can be , the degree of explained as in the range fuzzy memberships assigned at the levels of event-tofeature and pattern-to-cluster increases in addition to the high percentage of overlapping between the feature values. 11) The nature of symbolic data and the overlapping between the feature values increases the number of iterations needed to reach a solution. 12) The number of iterations needed to reach a solution can be reduced by applying the concept of soft clustering as follows: [23]–[25] and that by changing or

where is a threshold value and is equal to where . The default value of is 0.5. The new values for the number of iterations are shown in Figs. 15 and 16. REFERENCES [1] K. C. Gowda and G. G. Krishna, “Dissaggregative clustering using the concept of mutual nearest neighborhood,” IEEE Trans. Syst., Man, Cybern., vol. 8, pp. 883–895, Dec. 1978. [2] A. K. Jain and R. C. Dubes, Algorithms for Clustering Data. Englewood Cliffs, NJ: Prentice Hall, 1988. [3] E. Diday, The Symbolic Approach in Clustering, Classification and Related Methods of Data Analysis, H. H. Bock, Ed. Amsterdam, The Netherlands: Elsevier, 1988. [4] K. C. Gowda and E. Diday, “Symbolic clustering using a new dissimilarity measure,” Pattern Recogn., vol. 24, no. 6, pp. 567–578, 1991. , “Symbolic clustering using a new similarity measure, IEEE [5] Trans. Syst., Man, Cybern., vol. 22, pp. 368–378, Feb. 1992. [6] D. H. Fisher, “Knowledge acquisition via incremental conceptual clustering,” Mach. Learning, no. 2, pp. 103–138, 1987. [7] Y. Cheng and K. S. Fu, “Conceptual clustering in knowledge organization,” IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-7, pp. 592–598, 1985. [8] R. Michalski and R. E. Stepp, “Automated construction of classifications: Conceptual clustering versus numerical taxonomy,” PAMI, no. 5, pp. 396–410, 1983.

[9] H. Ralambondrainy, “A conceptual version of the K-means algorithm,” Pattern Recogn. Lett., no. 16, pp. 1147–1157, 1995. [10] D. H. Fisher and P. Langley, “Approaches to conceptual clustering,” in Proc. 9th Int. Joint Conf. Artificial Intell., Los Angeles, CA, 1985, pp. 691–697. [11] Y. A. El-Sonbaty, M. S. Kamel, and M. A. Ismail, Representations and Proximity Measures of Structured Features, to be published. [12] K. C. Gowda and T. V. Ravi, “Divisive clustering of symbolic objects using the concepts of both similarity and dissimilarity,” Pattern Recogn., vol. 28, no. 8, pp. 1277–1282, 1995. [13] M. P. Windham, “Cluster for fuzzy clustering algorithms,” Fuzzy Sets Syst., vol. 5, pp. 177–185, 1981. [14] E. R. Ruspini, “Numerical methods for fuzzy clustering,” Inform. Sci., vol. 2, pp. 318–350, 1970. [15] M. Roubens, “Fuzzy clustering algorithms and their cluster validity,” Eur. J. Operat. Res., vol. 10, pp. 294–301, 1982. [16] , “Pattern classification problems and fuzzy sets,” Fuzzy Sets Syst., vol. 1, pp. 239–253, 1978. [17] M. A. Ismail and S. Z. Selim, “Fuzzy C-means: Optimality of solutions and effective termination of the algorithm,” Pattern Recogn., vol. 19, no. 6, pp. 481–485, 1986. [18] J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function. New York: Plenum, 1981. [19] R. J. Hathaway and J. C. Bezdek, “Local convergence of the fuzzy c-means algorithms,” Pattern Recogn., vol. 19, no. 6, pp. 477–480, 1986. [20] R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis. New York: Wiley, 1973. [21] M. Ichino, “General metrics for mixed features—The Cartesian space theory for pattern recognition,” in Proc. IEEE Int. Conf. Syst., Man, Cybern., Atlanta, GA, Oct. 1988, pp. 14–17. [22] R. J. Hathaway and J. C. Bezdek, “NERF C-means: Non-Euclidean relational fuzzy clustering,” Pattern Recogn., vol. 27, no. 3, pp. 429–437, 1994. [23] M. A. Ismail, “Soft clustering: Algorithms and validity of solutions,” Fuzzy Computing. Amsterdam, The Netherlands: Elsevier, 1988, pp. 445–471. [24] Y. A. El-Sonbaty, “Fuzzy and soft clustering for symbolic data,” M.Sc. thesis, Alexandria Univ., Egypt, 1993. [25] S. Z. Selim and M. A. Ismail, “Soft clustering of multidimensional data: A semi-fuzzy approach,” Pattern Recogn., vol. 17, no. 5, pp. 559–568, 1984.

Yasser El-Sonbaty received the B.Sc. (honors) and M.Sc. degrees in computer science from the University of Alexandria, Egypt. He is currently working toward the Ph.D. degree at the same university. His research interests include object representation and recognition, machine vision, and pattern recognition.

M. A. Ismail received the B.Sc. (honors) and M.Sc. degrees in computer science from the University of Alexandria, Egypt, in 1970 and 1974, respectively, and the Ph.D. degree in electrical engineering from the University of Waterloo, Canada, in 1980. He is a Professor of computer science in the Department of Computer Science, Alexandria University, Egypt. He has taught computer science and engineering at the University of Waterloo, Canada, University of Petroleum and Minerals (UPM), Saudi Arabia, the University of Windsor, Canada, and the University of Michigan, Ann Arbor. His research interests include pattern analysis and machine intelligence, data structures and analysis, medical computer science, and nontraditional databases.