Data: $\left(O_1,O_2,\cdots ,O_n\right)$ set of observations

Initialization

${\theta }_0$

$\left(a_{ij}\right)$ Matrix of transition

$p_{il}$ Weight of the l mixture component of state i,

${\mu }_{il}$ Mean vector of the l mixture component of state i,

${\sigma }_{il}$ Covariance matrix of the l mixture component of state i.

Forward $\left(O_1,O_2,\cdots ,O_n\right)$ ,

Backward $\left(O_1,O_2,\cdots ,O_n\right)$ .

While $\log \left(P\left(O_1,O_2,\cdots ,O_n\right)\right)\le p_0$ (fixed threshold)

Expectation ${\xi }_t\left(i,j\right)=P\left(s_t=i,s_{t+1}=j/0,\theta \right)$ ,

${\gamma }_t\left(i\right)=P\left(s_t=i/O,\theta \right)={\displaystyle \sum _j\xi \left(i,j\right)}$ ,

${\gamma }_{il}\left(t\right)=P\left(S_t=i,m=l/O_t,\theta \right)={\gamma }_i\left(t\right)\frac{p_{il}P\left(O_t/{\mu }_{il},{\sigma }_{il},\theta \right)}{P\left(O_t/S_t=i,\theta \right)}$ .

Maximization $a_{ij}=\frac{{\displaystyle \sum _{t=1}^T{\xi }_t\left(i,j\right)}}{{\displaystyle \sum _{t=1}^T{\gamma }_t\left(i\right)}}$ ,

$p_{il}=\frac{{\displaystyle \sum _{t=1}^T{\gamma }_{il}\left(t\right)O_t}}{{\displaystyle \sum _{t=1}^T{\gamma }_{il}\left(t\right)}}$ ,

${\sigma }_{il}=\frac{{\displaystyle \sum _{t=1}^T{\gamma }_{il}\left(t\right)}\left(O_t-{\mu }_{il}\right){\left(O_t-{\mu }_{il}\right)}^{\text{T}}}{{\displaystyle \sum _{t=1}^T{\gamma }_{il}\left(t\right)}}$ .

End While

$f_j\left(O_{t+1}\right)$ : density of probability of the state i,

${\gamma }_{il}\left(t\right)$ : probability that the component l of the state i generated the observation at time t,

${\gamma }_i\left(t\right)$ : probability that the state i generated the observation at time t.