## 4 Methodology
### CF Events Segmentation and Driving Mode Recognition
#### 4.1.2 Toeplitz Inverse Covariance-Based Clustering (TICC) Method  
Therefore, in this research, we adopted a novel unsupervised learning method for the subsequence clustering of multivariate time series to obtain proper driving modes in the CF process for the following calibration, which is called TICC. The discovery of driving modes involves the simultaneous segmentation and clustering of multivariate time series data, and TICC is an innovative method designed to address this problem. This issue is more challenging than conventional time series segmentation because several segments may belong to the same cluster [37; 38]. However, it is also more complex than the problem of subsequence clustering because individual data points cannot be clustered separately (since neighboring points are encouraged to be in the same cluster) [39; 40]. Distance-based methods, such as dynamic time warping, are often used to solve these types of problems [41]. However, they focus more on matching raw values rather than exploring the overall structural consistency of time subsequences by examining the correlations between multidimensional data. TICC was introduced in 2017 and has been widely used in engineering [42], finance [43], and medicine [44] due to its effectiveness. MRFs are  
Figure 2: Framework of the driving-mode-based modeling. used to describe the non-time-varying correlation structure within a window to identify and cluster different subsequences and represent stronger or more complex relationships than simple correlations. TICC alternates between assigning points to clusters, which it accomplishes through dynamic programming, and updating the cluster MRFs, which it does via alternating the direction method of multipliers (ADMM). In comparison with several advanced time series clustering methods, TICC has at least a 41% higher clustering accuracy [36]. The TICC method avoids clustering each observation, since for the CF process, an isolated observation may tell an instantaneous state of the CF pair. The TICC cluster short subsequence of size \(w\) (\(w<T\)), consisting of observations \(x_{t-w+1},\ldots,x_{t}\), was concatenated into an \(nw\)-dimensional subsequence called \(X_{t}\). Then, a new sequence was formed and named \(X\), from \(X_{1}\) to \(X_{T}\), on which the clustering process was applied. An \(nw\)-dimensional subsequence \(X_{t}\) allows more time-varying information to be analyzed and clustered in the driving process. Meanwhile, the adjacent subsequences are encouraged to be clustered into the same cluster, which is called temporal consistency [36]. Rather than choosing simple correlation-based models, the segmenting and clustering of the multivariate time series is based on Markov random fields (MRFs), which is a multilayer correlation network which encodes the structural relationships between data of different dimensions, as shown in Figure 3. It not only considers the interdependencies of all dimensions of \(x_{t}\); the dependency of neighbor points is also taken into account. The Toeplitz matrix \(\Theta_{i}\in\mathbb{R}^{nw\times nw}\) defines the MRFs edge structure using a Gaussian inverse covariance instead of covariance for the computational advantages as the result of a tendency of being sparse and preventing overfitting. One challenge associated with the TICC approach is addressing the assignment problem of allocating data points to one of the K clusters and determining the assignment sets \(\mathbf{P}=\{P_{1},\ldots,P_{K}\}\) with \(P_{i}\subset\{1,2,\ldots,T\}\). Additionally, the algorithm must update cluster parameters \(\mathbf{\Theta}=\{\Theta_{1},\ldots,\Theta_{K}\}\) based on previously calculated assignment mappings.