On the Uncertainty Identification for Linear Dynamic Systems Using Stochastic Embedding Approach with Gaussian Mixture Models

. 2021 Jun 1;21(11):3837. doi: 10.3390/s21113837

Algorithm 1 (EM algorithm with GMMs).

Input
$Y^{[1 : M]}$ , $U^{[1 : M]}$ , M, $κ$ , ${\hat{θ}}^{(0)}$ , ${\hat{γ}}^{(0)}$ and ${\hat{σ}}^{2^{(0)}}$ .
Output
$\hat{θ}$ , $\hat{γ}$ and ${\hat{σ}}^{2}$ .
1:
$m \leftarrow 0$
2:
procedureE-step
3:
Compute ${\hat{\bar{μ}}}_{i r}^{(m)}$ and ${\hat{\bar{Σ}}}_{i r}^{(m)}$ from Equations (24) and (25).
4:
Compute ${\hat{ζ}}_{i r}^{(m)}$ from Equation (23).
5:
Compute ${\hat{\tilde{μ}}}_{i r}^{(m)}$ and ${\hat{\tilde{Σ}}}_{i r}^{(m)}$ from Equations (31) and (32).
6:
end procedure
7:
procedureM-step
8:
Estimate ${\hat{γ}}^{(m + 1)}$ , ${\hat{σ}}^{2^{(m + 1)}}$ from Equations (26)–(29).
9:
Compute $B_{i r} (θ, θ^{(m)})$ from Equation (35) using ${\hat{γ}}^{(m + 1)}$ and ${\hat{θ}}^{(m)}$ .
10:
Estimate ${\hat{θ}}^{(m + 1)}$ by solving Equation (34).
11:
end procedure
12:
if stopping criterion is not satisfied then
13:
$m \leftarrow m + 1$ , return to 2
14:
else
15:
$\hat{θ} \leftarrow {\hat{θ}}^{(m + 1)}$ , $\hat{γ} \leftarrow {\hat{γ}}^{(m + 1)}$ , ${\hat{σ}}^{2} \leftarrow {\hat{σ}}^{2^{(m + 1)}}$
16:
end if
17:
End