## Appendix A Solving Process of the TICC
### Maximization Part: Updating Cluster Parameters  
With fixing cluster assignments \(P\), the next task is to update the inverse covariance matrices of the clusters. Since the temporal consistency \(1\{X_{t-1}\
otin P_{i}\}\) is mainly connected to the problem A.1, the optimization problem can be simplified as:  
\[\operatorname*{argmin}_{\Theta\in\mathcal{T}\mathcal{P}}\sum_{i=1}^{K}[\| \lambda\circ\Theta_{i}\|_{1}+\sum_{X_{t}\in P_{i}}-\ell\ell(X_{t},\Theta_{i})]\] (A2)  
In addition, since in this part there is no dependency on previous and cross-connected results of other clusters in [55], the updates can be conducted in parallel [36]. The negative log likelihood in terms of each \(\Theta_{i}\) can be expressed as:  
\[\sum_{X_{t}\in P_{i}}-\ell\ell(X_{t},\Theta_{i})=-|P_{i}|(\text{log det}\Theta_ {i}+\text{tr}(S_{i}\Theta_{i}))+C\] (A3)  
where \(|P_{i}|\) represent the number of the points assigned to cluster \(i\); det and tr are the determinant and the trace of a matrix, respectively; \(S_{i}\) is the empirical covariance of the random sample, which is open adopted to approximate the true covariance matrix \(\Sigma\) of a stochastic problem [56]; and \(C\) is a constant. Therefore, the update representation of this optimization problem is expressed as:  
\[ \begin{array}{rl}\text{minimize}&-\text{log det}\Theta_{i}+\text{tr}(S_{i} \Theta_{i})+\frac{1}{|P_{i}|}\|\lambda\circ\Theta_{i}\|_{1}\\ &\text{subject to}&\Theta_{i}\in\mathcal{T}\end{array} \] (A4)  
[PERSON], [PERSON], [PERSON], and [PERSON] [36] defined this optimization problem as the Toeplitz graphical lasso, and as a variation of the classical graphical lasso problem [57]. It is required to solve an individual Toeplitz graphical lasso for every cluster at each iteration. The alternating direction method of multipliers (ADMM) was applied to solve this problem as a well-performed algorithm in solving related large-scale convex problems [58]. In order to construct the ADMM-friendly form, a consensus variable \(Z\) is introduced to reconstruct the problem as:  
\[ \begin{array}{rl}\text{minimize}&-\text{log det}\Theta+\text{tr}(S\Theta) +\parallel\lambda\circ Z\parallel_{1}\\ &\text{subject to}&\Theta=Z,Z\in\mathcal{T}\end{array} \] (A5)  
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