IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. PAMI-3, NO. 2, MARCH 1981 Theoretical remarks and application," in Proc. 4th Conf. Inform. Theory, Prague, 1965, pp. 635-660. [201 Y. T. Chien and K. S. Fu, "Selection and ordering of feature observations in a pattern recognition system," Inform. Contr., vol. 12, pp. 395-414, 1968. [21] J. T. Tou and R. P. Heydorn, "Some approaches to optimum fea- ture extraction," in Computer and Information Sciences II, J. T. Tou, Ed. New York: Academic, 1967, pp. 41-122. [22] T. Y. Young, "The reliability of linear feature extractors," IEEE Trans. Comput., vol. C-20, pp. 967-971, 1971. [23] T. M. Cover and P. E. Hart, "Nearest neighbor pattern classifica- tion,"IEEE Trans. Inform. Theory, vol. IT-13, pp. 21-27, 1967. [24] K. Kugunaga and D. L. Kessell, "Estimation of classification error," IEEE Trans. Comput., vol. C-20, pp. 1521-15 27, 1971. Tzay Y. Young (S'58-M'63-SM'80) received the B.S. degree from the National Taiwan Uni- versity, Taipei, Taiwan, China, in 1955, the M.S. degree from the University of Vermont, Bur- lington, in 1959, and the Dr.Eng. degree from the Johns Hopkins University, Baltimore, MD, in 1962, all in electrical engineering. From 1962 to 1963 he was a Research Asso- i ciate at Carlyle Barton Laboratory of the Johns Hopkins University, and from 1963 to 1964 he was a member of the Technical staff of Bell Telephone Laboratories, Murray Hill, NJ. He was on the faculty of Carnegie-Mellon University from 1964 to 1974, and was on leave at NASA Goddard Space Flight Center from 1972 to 1973. Since 1974 he has been a Professor of Electrical Engineering at the University of Miami, Coral Gables, FL. He is coauthor (with T. W. Calvert) of Classification, Estimation, and Pattern Recognition, published by American-Elsevier. He was an Associate Editor of the IEEE TRANS- ACTIONS ON COMPUTERS from 1974 to 1976, and is currently a member of Editorial Committee of IEEE TRANSACTIONS PATTERN ANALYSIS AND MACHINE INTELLIGENCE. Philip S. Liu (S'70-M'75) was born in Wai Chow, China, on November 19, 1945. He re- ceived the B.S. degree in electrical engineering from the University of Wisconsin, Madison, in 1970, and the M.S. and Ph.D. degrees in electri- cal engineering from Purdue University, West Lafayette, IN, in 1972 and 1975, respectively. 000--i--0X_ <--; He joined the faculty of the University of Miami, Coral Gables, FL, in 1975, and he is currently Associate Professor of Electrical Engi- neering. His current research interests include database systems and computer architecture. Dr. Liu is a member of the Association for Computing Machinery and Eta Kappa Nu. Romulo J. Rondon was born in Caripito, Vene- zuela, on November 24, 1950. He received the degree from the University of Carabobo in electrical engineering in 1974 and the M.S. i%0S +? t00i ledegree from the University of Miami, Coral Gables, FL, in 1978. He held research assistantships during his \ j g 4- t i0-; -raduate work at the University of Miami where > i m he was engaged in a research in data compres- sion and pattern recognition. He is currently with the Engineering Department of the Ford Motor Company of Venezuela. Mr. Rond6n is a member of Eta Kappa Nu. An Approximate Solution to Normal Mixture Identification with Application to Unsupervised Pattern Classification JACK-GERARD POSTAIRE, MEMBER, IEEE, AND CHRISTIAN P. A. VASSEUR Abstract-In this paper, an approach to unsupervised pattern classifi- cation is discussed. The classification scheme is based on an approxima- fton of the probability densities of each class under the assumption that the input patterns are of a normal mixture. The proposed technique for identifying the mixture does not require prior information. The description of the mixture in terms of convex- ity allows to determine, from a totally unlabeled set of samples, the number of components and, for each of them, approximate values of the mean vector, the covariance matrix, and the a priori probability. Manuscript received November 26, 1979; revised April 28, 1980. J.-G. Postaire is with the Laboratoire d'Electronique et d'Etude des Systemes Automatiques, Faculte des Sciences, Rabat, Morocco. C. P. A. Vasseur is with the Centre d'Automatique, Universite de Lille, Villeneuve d'Ascq, Cedex, France. Discriminant functions can then be constructed. Computer simula- tions show that the procedure yields decision rules whose performances remain close to the optimum Bayes minimum error-rate, while involving only a small amount of computation. Index Terms-Convexity, minimum error-rate classifilcation, normal mixture identification, unsupervised classification. SUMMARY IN THIS WORK, we explore a new approach to the Gaussian multicategory classification problem, in an unsupervised environment. All the statistics of each class, as well as the number of classes, are unknown. An approximate solution to 0162-8828/81/0300-0163$00.75 © 1981 IEEE 163