## 4 Methodology
### CF Events Segmentation and Driving Mode Recognition
#### 4.1.2 Toeplitz Inverse Covariance-Based Clustering (TICC) Method
This optimization problem can be expressed as follows:  
\[\operatorname*{argmin}_{\Theta\in\mathcal{T},\mathcal{P}}\sum_{i=1}^{k}[ \|\lambda\circ\Theta_{i}\|_{1}+\sum_{X_{t}\in P_{i}}(-\ell\ell(X_{t},\Theta_{ i})+\beta 1\{X_{t-1}\
otin P_{i}\})]. \tag{2}\]  
where \(\mathcal{T}\) represents the set of symmetric block Toeplitz \(nw\times nw\) matrices and serves as an additional constraint for constructing MRFs to ensure the time-invariant property of each cluster. The first expression \(\|\lambda\circ\Theta_{i}\|_{1}\) denotes an additional sparsity constraint based on the Hadamard product of the inverse covariance matrix with the regularization parameter \(\lambda\in\mathbb{R}^{nw\times nw}\). The second component, \(\ell\ell(X_{t},\Theta_{i})\), specifies the core optimization problem of fitting cluster parameters given the assignment set \(P_{i}\) (log likelihood). The indictor function \(1\{X_{t-1}\
otin P_{i}\}\) is checks the temporal consistency, that is, whether adjacent data  
Figure 3: Exemplary MRF structures for two distinct clusters. points belong to the same cluster. If two observations are consecutive but assigned to different clusters \(X_{t-1}\
otin P_{t}\), there will be a penalty \(\beta\). There are two key input parameters for the TICC algorithm: the window size \(w\) and the number of the clusters \(K\). The TICC does not cluster each data point \(x_{t}\) individually, it instead clusters short subsequences from \(t-w\) to \(t\). The size of the Toeplitz \(nw\times nw\) matrices determines whether it is too short to contain the complete MRF structure of each cluster, and it is too long to choose a segment boundary or disobey the time-invariance as well. To effectively use this algorithm, we refer to [PERSON], [PERSON], [PERSON], and [PERSON] [36] and set the window size to 10 since they suggested that when the window size is bounded between 4 and 15, the clustering result of the car data would be reliable and robust. Another key parameter is \(K\). Since we do not have any proper precedent for CF mode extraction, we will test different values of \(K\) according to the behavior reproduction performance as well as the Bayesian information criterion (BIC). The TICC problem is a combinatorial optimization problem, which has two coupled non-convex problems to search the optimized cluster parameter \(\mathbf{\Theta}=\{\Theta_{1},\ldots,\Theta_{K}\}\) and the cluster assignments \(\mathbf{\mathrm{P}}=\{P_{1},\ldots,P_{K}\}\). Since it is difficult to solve global optimization, an expectation-maximization (EM)-like algorithm is applied, which constantly and alternately assign \(\mathbf{\mathrm{P}}\) with \(\mathbf{\Theta}\) determined and update \(\mathbf{\Theta}\) with \(\mathbf{\mathrm{P}}\) determined. For the solution process of the TICC, please refer to Appendix A. A brief outline of TICC clustering is shown in Algorithm 1. ```
1:initialization Cluster parameters \(\mathbf{\Theta}\) and assignments \(\mathbf{\mathrm{P}}\)
2:repeat
3:\(E\)-step: Assign points to clusters \(\rightarrow\)\(\mathbf{\mathrm{P}}\)
4:\(M\)-step: Update cluster parameters \(\rightarrow\)\(\mathbf{\Theta}\)
5:until Stationarity. return (\(\mathbf{\Theta}\), \(\mathbf{\mathrm{P}}\))
```  
**Algorithm 1** Brief outline of TICC clustering