The VQ and the fuzzy VQ

Since the relation between the VQ method（also hard clustering）and the fuzzy VQ method（also fuzzy c-means clustering）has been done well by Bezdek in［10］，we present here how to obtain the fuzzy VQ algorithm from the fuzzy EM algorithm in the GMM.Since the Gaussian density is of the form

The distance in（27）is now rewritten as

where d is the feature space dimension，|∑i|is the determinant of ∑ithere are onlyμiand ∑i；considered as variables，then we can assume that clusters have the same densities or are subject to the constraints wi=ai，where ai＞0 and fixed for each i，hence the term-log wi；in（31）is fixed and will be omitted in the optimisation procedure.Similarly，the termis fixed if it is subject to the constraints for fuzzy covariance matrices proposed by Gustafson and Kesse.Therefore maximising the fuzzy Q-functions in（24）is now equivalent to minimising the following function

where Mi=.The function Jm，in（32）is the fuzzy objective functions proposed by Gustafson and Kessel［11］.It means that the fuzzy VQ，also the fuzzy c-means algorithm can be derived from the fuzzy EM-based GMM algorithm using the above constraints.