- Email: [email protected]
Research on fuzzy clustering at Chung-Yuan Christain University, Taiwan
Fuzzy Sets and Systems 62 (1994) 379-387 North-Holland
379
Bulletin Editorial The interruption in publication of the Bulletin was caused by an administrative or postal error rather than by any dearth in or paucity of the material sent in. Thank you all for continuing to send me material. I hope you will be patient in waiting for your article to appear. It almost certainly will in due course. We now resume regular publication with some interesting and readable research reports, compelling conference announcements and bubbly book reviews. My particular thanks to Dr. Toth for his three contributions. lan Graham February 1994
Research Activities in the Area of Fuzzy Sets at the Scientific Analysis Group, Metcalfe House, Delhi, India We are a small group of mathematicians specialized basically in albegraic and number theoretic concepts, and are deeply engaged in applicative research on various kinds of problems. One of our main objectives has been to handle uncertainties in various decision support systems where the knowledge may be imprecise but may not be random to fit in the probabilistic model. Proper representation of 'the imprecision' is very much required in such cases and the latest developments Fuzzy Sets and Evidence Theory provides some means to achieve that. In addition to probability theory, Fuzzy Sets and Fuzzy Logic have now proved their applicability in various kinds of problems and have developed now in different dimensions. Be it the generalizations from crisp sets to fuzzy sets, from probability to fuzzy characteristic values or from Boolean logic to fuzzy logic, it has now become an area of interest for scientists of different backgrounds-mathematicians, engineers, computer scientists and so on. With such latest developments of this newly developed discipline, we have developed our interest in the theoretical as well as applicative research on this topic. On the theoretical research side, we have been pursuing the generalizations of various algebraic structures such as groups, fields, linear spaces to fuzzy theoretic concepts and have come up with some interesting results, some of which have already appeared in Fuzzy Sets and Systems and others have been submitted to Fuzzy Sets and Systems. On the applications side, we are working on handling uncertainties through fuzzy sets
in designing knowledge based systems for various kinds of real problems. Our interest lies in fuzzy pattern recognition and clustering, fuzzy isodata and C-means theory of Bezdek, since most of our problems need classifications under a supervised/unsupervised learning environment. More advanced concepts of AI such as hypothesis formation and discovery systems are also under study. We have developed some dissimilarity measures for learning systems which depend upon the setting of parameters on the basis of learning through known examples. In the future we are looking for some real 'hybrid systems' where all kinds of knowledge, precise or imprecise, could be tackled with whatever logic mathematical/fuzzy/Boolean is applicable. We also anticipate some more interesting theoretical results on fuzzy algebraic structures with more thrust on examples which are not so much appearing as they should have been at the beginning of generalizations in this direction. Persons interested in the above areas of Fuzzy Sets are welcome to interact with us at the address given.
References [1] Some A.I. applications in cryptology, Proc. DEFCAP-88 (Restricted) 2 (Sect. III) (1988) 10-20. [2] I.J. Kumar, P.K. Saxena and Pratibha Yadav, Fuzzy normal subgroups and fuzzy quotients, FuzzySets and Systems 46 (1992) 121-132. [3] P.K. Saxena, Fuzzy subgroups as union of two fuzzy subgroups, FuzzySets and Systems57 (1993) 209-218. 14] On some conventions in fuzzy fields and fuzzy linear spaces, submitted. [5] A Rule-Based clear text recognizer for English language, Proc. ICAPRDT-93,28-31 December (1993) 619-627. [6] A dissimilarity measure for learning systems based on fuzzy sets, (submitted). Dr. P.K. Saxena SAG, DRDO Metcalfe House Delhi- 110054, India
Research on Fuzzy Clustering at ChungYuan Christian University, Taiwan The works of Bellman, Kalaba and Zadeh [11 and Ruspini [2] opened the door for research on fuzzy clustering. Now fuzzy clustering has been widely studied and applied in a variety of substantive areas. In the literature of fuzzy clustering, the fuzzy c-means (FCM)
0165-0114/94/$07.00 (~) 1994---Elsevier Science B.V. All rights reserved
Bulletin
380
clustering algorithms defined by Dunn [3] and generated by Bezdek [4] are well-known and powerful methods in cluster analysis. My research interests focuses on this area. Yang and Yu [5, 6] investigated the stochastically asymptotic properties of these FCM's. Their existence and strong consistency were established. We also created their numerical convergence properties in Yang [7]. A class of fuzzy classification maximum likelihood procedures was proposed by Yang [8] which also gave a generalized type of FCM, called the penalized FCM clustering procedure. Yang and Su [9] study and compare the accuracy and efficiency of the parameter estimation for normal mixtures based on EM, FCM and penalized FCM algorithms. The fuzzy k nearest neighbor rule (k-NNR) was also widely studied. Many researchers constructed these algorithms and gave its applications. Recently, Yang and Chen [10, 11] gave its strong consistency and the rate of convergence. These results give well theoretical foundation for the fuzzy k-NNR. Finally, the researchers who are interested in fuzzy clustering may refer to the survey of Yang [12].
References [1] R. Bellman, R. Kalaba and L.A. Zadeh, Abstraction and pattern classification, J, Math. Anal. and Appl. 2 (1966) 581-586. I2] E.H. Ruspini, A new approach to clustering, Inform. and Control 15 (1969) 22-32. [3] J.C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact, well-separated clusters, J. Cybernetics 3 (1974) 32-57. [4] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms (Plenum Press, New York, 1981). [5] M.S. Yang and K.F. Yu, On stochastic convergence theorems for the fuzzy c-means clustering procedure, Int. J. of Genera/Systems 16 (1990) 397-411. [6] M.S. Yang and K.F. Yu, On existence and strong consistency of a class of fuzzy c-means clustering procedures, Cybernetics and Systems 23 (1992) 583-602. [7] M.S. Yang, Convergence properties of the generalized fuzzy c-means clustering algorithms, Computers & Math. Appl. 25(12) (1993) 3-11. i8] M.S. Yang, On a class of fuzzy classification maximum likelihood procedures, Fuzzy Sets and Systems 57(3) (1993) 365-375. [9J M.S. Yang and C.F. Su, On parameter estimation for the normal mixtures based on fuzzy clustering algorithms (submitted). [10] M.S. Yang and C.T. Chen, On strong consistency of the fuzzy generalized nearest neighbor rule, Fuzzy Sets and Systems 60 (1993) 273-281. [11] M.S. Young and C.T. Chen, On convergence rate of the fuzzy generalized nearest neighbor rule, Computers & Math. Appl. (to appear). [12] M.S. Yang, A survey of fuzzy clustering, Math. and Computer Modelling (to be published). Miin-Shen Yang Department o f Mathematics Chung-Yuan Christian University Chungli, Taiwan 32023, ROC
Fuzzy Sets in Novi Sad At the Institute of Mathematics, University of Novi Sad, fuzzy sets have been investigated for years. In Fuzzy Sets and Systems 43(3) (1991) 339-340, a survey article about the work on fuzzy sets at the Institute of Mathematics in Novi Sad was published. Recent activities of the research group investigating fuzzy sets from the algebraic point of view, will be presented here. As it is well known, fuzzy set in Zahedh's original definition was a mapping from a nonempty set to the real interval [0, 1]. Later on the codomain of the mapping was taken to be a complete lattice (Goguen, 1967) and a Boolean algebra (Brown, 1971). We have been working on some further generalizations. Our approach is based on a fuzzy set as a function the codomain of which is not only the lattice, but also a partially ordered set and, the most generally, a suitable relational system. From the algebraic point of view, there is an interest in considering the collection of cuts (level functions) of a fuzzy set. By the above mentioned approach, we can deal with some problems in representation of lattices and also in applications of fuzzy sets. It turns out that the results appear to be more readily applicable than those obtained by 'classical' (interval valued) fuzzy sets. (Roughly speaking, the difference between these new conceptions of fuzzy sets and the 'classical' one is comparable to the one between a colour and b / w picture). From the theoretic point of view, up to now, we have been investigating partially ordered, and relational valued fuzzy relations, algebras and congruences, and some other algebraic notions, and these results will appear in Fuzzy Sets and Systems and in Information Sciences. Moreover, we have applied these theoretic results on problems in coding theory, and more recently on problems which arose in biogeography. Namely, in some researches at the institute of Biology in Novi Sad a problem of classification of biogeographical areas appeared. Using partially order valued fuzzy relations, we gave a useful solution of the problem. By this approach, the character and the level of connections of fauna in different biogeographical territories could be seen very clearly and were easy to investigate. People working at the above mentioned problems are Branimir Se~,elja and Andreja Tepav~evi6. The doctoral thesis in which fuzzy sets were applied in biology was 'Taxonomic status and zoographic analysis of genus cheilosia meigen and of related genera (diptera:syrphidae) on Balkan peninsula', by Ante Vuji6.
References I1] B. ~e~elja and A. Tepav~evi6,On a construction of codes by P-fuzzy sets, Review of Research Fac. of Sci, Univ. of Novi Sad 20(2) (1990) 71-80. i2] B. ~e~,eljaand Tepav~;evi6,Relational valued fuzzy sets, Fuzzy Sets and Systems 52 (1992) 217-222. 13] B. ~e~,elja and A. Tepav~evi6, Fuzzy Boolean algebras, Automated Reasoning, IFIP Transactions A-19 (1992) 83-88. [4] B. ~e~elja, A. TepavP,evi6, and G. Vojvodi6, L-fuzzy sets and codes, Fuzzy Sets and Systems 53 (1993) 217-222.
379
Bulletin Editorial The interruption in publication of the Bulletin was caused by an administrative or postal error rather than by any dearth in or paucity of the material sent in. Thank you all for continuing to send me material. I hope you will be patient in waiting for your article to appear. It almost certainly will in due course. We now resume regular publication with some interesting and readable research reports, compelling conference announcements and bubbly book reviews. My particular thanks to Dr. Toth for his three contributions. lan Graham February 1994
Research Activities in the Area of Fuzzy Sets at the Scientific Analysis Group, Metcalfe House, Delhi, India We are a small group of mathematicians specialized basically in albegraic and number theoretic concepts, and are deeply engaged in applicative research on various kinds of problems. One of our main objectives has been to handle uncertainties in various decision support systems where the knowledge may be imprecise but may not be random to fit in the probabilistic model. Proper representation of 'the imprecision' is very much required in such cases and the latest developments Fuzzy Sets and Evidence Theory provides some means to achieve that. In addition to probability theory, Fuzzy Sets and Fuzzy Logic have now proved their applicability in various kinds of problems and have developed now in different dimensions. Be it the generalizations from crisp sets to fuzzy sets, from probability to fuzzy characteristic values or from Boolean logic to fuzzy logic, it has now become an area of interest for scientists of different backgrounds-mathematicians, engineers, computer scientists and so on. With such latest developments of this newly developed discipline, we have developed our interest in the theoretical as well as applicative research on this topic. On the theoretical research side, we have been pursuing the generalizations of various algebraic structures such as groups, fields, linear spaces to fuzzy theoretic concepts and have come up with some interesting results, some of which have already appeared in Fuzzy Sets and Systems and others have been submitted to Fuzzy Sets and Systems. On the applications side, we are working on handling uncertainties through fuzzy sets
in designing knowledge based systems for various kinds of real problems. Our interest lies in fuzzy pattern recognition and clustering, fuzzy isodata and C-means theory of Bezdek, since most of our problems need classifications under a supervised/unsupervised learning environment. More advanced concepts of AI such as hypothesis formation and discovery systems are also under study. We have developed some dissimilarity measures for learning systems which depend upon the setting of parameters on the basis of learning through known examples. In the future we are looking for some real 'hybrid systems' where all kinds of knowledge, precise or imprecise, could be tackled with whatever logic mathematical/fuzzy/Boolean is applicable. We also anticipate some more interesting theoretical results on fuzzy algebraic structures with more thrust on examples which are not so much appearing as they should have been at the beginning of generalizations in this direction. Persons interested in the above areas of Fuzzy Sets are welcome to interact with us at the address given.
References [1] Some A.I. applications in cryptology, Proc. DEFCAP-88 (Restricted) 2 (Sect. III) (1988) 10-20. [2] I.J. Kumar, P.K. Saxena and Pratibha Yadav, Fuzzy normal subgroups and fuzzy quotients, FuzzySets and Systems 46 (1992) 121-132. [3] P.K. Saxena, Fuzzy subgroups as union of two fuzzy subgroups, FuzzySets and Systems57 (1993) 209-218. 14] On some conventions in fuzzy fields and fuzzy linear spaces, submitted. [5] A Rule-Based clear text recognizer for English language, Proc. ICAPRDT-93,28-31 December (1993) 619-627. [6] A dissimilarity measure for learning systems based on fuzzy sets, (submitted). Dr. P.K. Saxena SAG, DRDO Metcalfe House Delhi- 110054, India
Research on Fuzzy Clustering at ChungYuan Christian University, Taiwan The works of Bellman, Kalaba and Zadeh [11 and Ruspini [2] opened the door for research on fuzzy clustering. Now fuzzy clustering has been widely studied and applied in a variety of substantive areas. In the literature of fuzzy clustering, the fuzzy c-means (FCM)
0165-0114/94/$07.00 (~) 1994---Elsevier Science B.V. All rights reserved
Bulletin
380
clustering algorithms defined by Dunn [3] and generated by Bezdek [4] are well-known and powerful methods in cluster analysis. My research interests focuses on this area. Yang and Yu [5, 6] investigated the stochastically asymptotic properties of these FCM's. Their existence and strong consistency were established. We also created their numerical convergence properties in Yang [7]. A class of fuzzy classification maximum likelihood procedures was proposed by Yang [8] which also gave a generalized type of FCM, called the penalized FCM clustering procedure. Yang and Su [9] study and compare the accuracy and efficiency of the parameter estimation for normal mixtures based on EM, FCM and penalized FCM algorithms. The fuzzy k nearest neighbor rule (k-NNR) was also widely studied. Many researchers constructed these algorithms and gave its applications. Recently, Yang and Chen [10, 11] gave its strong consistency and the rate of convergence. These results give well theoretical foundation for the fuzzy k-NNR. Finally, the researchers who are interested in fuzzy clustering may refer to the survey of Yang [12].
References [1] R. Bellman, R. Kalaba and L.A. Zadeh, Abstraction and pattern classification, J, Math. Anal. and Appl. 2 (1966) 581-586. I2] E.H. Ruspini, A new approach to clustering, Inform. and Control 15 (1969) 22-32. [3] J.C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact, well-separated clusters, J. Cybernetics 3 (1974) 32-57. [4] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms (Plenum Press, New York, 1981). [5] M.S. Yang and K.F. Yu, On stochastic convergence theorems for the fuzzy c-means clustering procedure, Int. J. of Genera/Systems 16 (1990) 397-411. [6] M.S. Yang and K.F. Yu, On existence and strong consistency of a class of fuzzy c-means clustering procedures, Cybernetics and Systems 23 (1992) 583-602. [7] M.S. Yang, Convergence properties of the generalized fuzzy c-means clustering algorithms, Computers & Math. Appl. 25(12) (1993) 3-11. i8] M.S. Yang, On a class of fuzzy classification maximum likelihood procedures, Fuzzy Sets and Systems 57(3) (1993) 365-375. [9J M.S. Yang and C.F. Su, On parameter estimation for the normal mixtures based on fuzzy clustering algorithms (submitted). [10] M.S. Yang and C.T. Chen, On strong consistency of the fuzzy generalized nearest neighbor rule, Fuzzy Sets and Systems 60 (1993) 273-281. [11] M.S. Young and C.T. Chen, On convergence rate of the fuzzy generalized nearest neighbor rule, Computers & Math. Appl. (to appear). [12] M.S. Yang, A survey of fuzzy clustering, Math. and Computer Modelling (to be published). Miin-Shen Yang Department o f Mathematics Chung-Yuan Christian University Chungli, Taiwan 32023, ROC
Fuzzy Sets in Novi Sad At the Institute of Mathematics, University of Novi Sad, fuzzy sets have been investigated for years. In Fuzzy Sets and Systems 43(3) (1991) 339-340, a survey article about the work on fuzzy sets at the Institute of Mathematics in Novi Sad was published. Recent activities of the research group investigating fuzzy sets from the algebraic point of view, will be presented here. As it is well known, fuzzy set in Zahedh's original definition was a mapping from a nonempty set to the real interval [0, 1]. Later on the codomain of the mapping was taken to be a complete lattice (Goguen, 1967) and a Boolean algebra (Brown, 1971). We have been working on some further generalizations. Our approach is based on a fuzzy set as a function the codomain of which is not only the lattice, but also a partially ordered set and, the most generally, a suitable relational system. From the algebraic point of view, there is an interest in considering the collection of cuts (level functions) of a fuzzy set. By the above mentioned approach, we can deal with some problems in representation of lattices and also in applications of fuzzy sets. It turns out that the results appear to be more readily applicable than those obtained by 'classical' (interval valued) fuzzy sets. (Roughly speaking, the difference between these new conceptions of fuzzy sets and the 'classical' one is comparable to the one between a colour and b / w picture). From the theoretic point of view, up to now, we have been investigating partially ordered, and relational valued fuzzy relations, algebras and congruences, and some other algebraic notions, and these results will appear in Fuzzy Sets and Systems and in Information Sciences. Moreover, we have applied these theoretic results on problems in coding theory, and more recently on problems which arose in biogeography. Namely, in some researches at the institute of Biology in Novi Sad a problem of classification of biogeographical areas appeared. Using partially order valued fuzzy relations, we gave a useful solution of the problem. By this approach, the character and the level of connections of fauna in different biogeographical territories could be seen very clearly and were easy to investigate. People working at the above mentioned problems are Branimir Se~,elja and Andreja Tepav~evi6. The doctoral thesis in which fuzzy sets were applied in biology was 'Taxonomic status and zoographic analysis of genus cheilosia meigen and of related genera (diptera:syrphidae) on Balkan peninsula', by Ante Vuji6.
References I1] B. ~e~elja and A. Tepav~evi6,On a construction of codes by P-fuzzy sets, Review of Research Fac. of Sci, Univ. of Novi Sad 20(2) (1990) 71-80. i2] B. ~e~,eljaand Tepav~;evi6,Relational valued fuzzy sets, Fuzzy Sets and Systems 52 (1992) 217-222. 13] B. ~e~,elja and A. Tepav~evi6, Fuzzy Boolean algebras, Automated Reasoning, IFIP Transactions A-19 (1992) 83-88. [4] B. ~e~elja, A. TepavP,evi6, and G. Vojvodi6, L-fuzzy sets and codes, Fuzzy Sets and Systems 53 (1993) 217-222.