The fuzzy EM algorithm
An infinite family of fuzzy Q-functions of observed data is proposed as follows
where
·μy（x）is the membership function，which denotes the degree of x belonging toy and satisfies
·m
1 is a weighting exponent on each fuzzy membershipμy（x）and is called the degree of fuzziness.
The task of the fuzzy EM algorithm is to maximise the fuzzy Q-functions in（3）on variables U and，e.g.，finding a pair of（
）such that Qm（
）≥Qm（U，A）.As shown in［15］，we obtain
where
Determining 
is performed using the same optimization method as in step 3 of the EM algorithm.The fuzzy EM algorithm is proposed as follows
The fuzzy EM algorithm：
1.Define y.fix m
1 and choose an initial estimate ʌ；
2.E-step：compute 
following（5）and then Qm（
，ʌ）based on the given 
A，
3.M-step：determine 
，for which Q（
，ʌ）is maximised；
4.Set 
=A and 
=U，repeat from step 2 until convergence.
For multiple-prototype classifier design，it should be noted that，the fuzzy Q-function is maximised.Therefore，the decision rule for the fuzzy EM-based classifier is the following
Assign x to model k*if
It should be noted from（5b）that，the fuzzy EM algorithm is reduced to the EM algorithm as m=1.