The fuzzy EM in the GMM

The fuzzy EM in the GMM

For multiple observed data X=｛x1，x 2，...，xT｝the family of fuzzy Q-function is［17］

Where

·A=｛wi，μi，∑i｝=，i=1，...，c，denotes the complete parameter set of the GMM，where wi，μi，and∑iare the mixture weight（prior probability），the mean vector and the covariance matrix of the ith Gaussian density N（xt，μi，Σi）respectively.

·uitdenotes ui（xt）and satisfies

Similarly，for the E step，we obtain

where

and for the M step

The fuzzy EM-based GMM is called the fuzzy GMM（FGMM）.It has been also applied to speech and speaker recognition applications.［17］-［19］