Algorithm 1. The Execution Process of the EM Algorithm

Step 1: Initialize ${\Phi }^{(0)}=({\alpha }^{(0)},{\mu }^{(0)},{\Sigma }^{(0)})$ and set the iteration counter l = 0 and set ε > 0.

Step 2: For ${\Phi }^{(l)}=({\alpha }^{(l)},{\mu }^{(l)},{\Sigma }^{(l)})$, execute the following procedures:

(E-step): Calculate     ${\psi }_{ij}^l=\frac{{\alpha }_i^{(l)}{p}_i(z_j;{\mu }_i^{(l)},{\Sigma }_i^{(l)})}{p(z_j;{\Phi }^{(l)})},j=1,2,\cdots ,N,\hspace{1em}i=1,2,\cdots ,s$

${\Psi }_i^{(l)}={\displaystyle \sum _{j=1}^N{\psi }_{ij}^{(l)},i=1,2,\cdots ,s}$

(M-step): Update ${\Phi }^{(l)}=\left({\alpha }^{(l)},{\mu }^{(l)},{\mathsf{\Sigma }}^{(l)}\right)$ by computing ${\alpha }_i^{(l+1)}=\frac{{\Psi }_i^{(l)}}{N},i=1,2,\cdots ,s$ ${\mu }_i^{(l+1)}=\frac{1}{{\Psi }_i^{(l)}}{\displaystyle \sum _{j=1}^N{\psi }_{ij}^{(l)}{z}_j,i=1,2,\cdots ,s}$ ${\displaystyle {\sum }_i^{(l+1)}=}\frac{1}{{\Psi }_i^{(l)}}{{\displaystyle \sum _{j=1}^N{\psi }_{ij}^{(l)}(z_j-{\mu }_i^{(l+1)})(z_j-{\mu }_i^{(l+1)})}}^{\mathrm{T}},i=1,2,\cdots ,s$

Step 3: If the prescribed convergence condition $\max \left\{\frac{‖{\alpha }_i^{(l+1)}-{\alpha }_i^{(l)}‖}{‖{\alpha }_i^{(l)}‖},\frac{‖{\mu }_i^{(l+1)}-{\mu }_i^{(l)}‖}{‖{\mu }_i^{(l)}‖},\frac{‖{\Sigma }_i^{(l+1)}-{\Sigma }_i^{(l)}‖}{‖{\Sigma }_i^{(l)}‖};\right\}\le \epsilon ,\hspace{1em}\epsilon \ll 1$   is satisfied, then set $l^{\ast }=l+1$ and exit. The optimal ML estimate of Φ is obtained by   ${\Phi }^{\ast }={\Phi }^{(l^{\ast })}$. Else set $l=l+1$ and go back to Step 2.