Continuous HMM：
Consider the case that observations are continuous vectors.The most general representation of the model output probabilities is a mixture of Gaussians［3］
where ot，t=1，...，T，are the n-dimensional observation vectors being modelled，wjk，j=1，...，N，k=1，...，M are mixture coefficients，and N（ot，μjk，∑jk）is a Gaussian with mean vectorμjk and covariance matrix Σjk for the kth mixture component in state j.The following constraints are satisfied
Similarly，let u jkt=ujkt（O）be the membership function，denoting the degree to which the observation sequence O belongs to state j and mixture k at time f，satisfying
For the E step，it can be shown that the fuzzy reestimation formulas are of the form
where
For the M step，we obtain
where the prime denotes vector transposition.
The fuzzy EM-based HMMs are called the fuzzy HMMs（FHMMs）.They have been applied to speech and speaker recognition applications.［16］，［21］