# Dynamic Tensor Modeling for Missing Data Completion in Electronic Toll Collection Gantry Systems
## 3 Materials and Methods
### ETC Gantry Data Completion Method Based on Improved Tensor Decomposition
#### 3.2.3 Calculation of the Incremental Approximate Decomposition of the Tensor Block  
The ETC gantry data exhibit a substantial spatial correlation. Given the minor variations within neighboring tensor blocks, these correlations can be harnessed to conserve \begin{table}
\begin{tabular}{c c} \hline \hline
**Vehicle Type** & **Number** \\ \hline Minibus & _1.0_ \\ LargeBus & _1.5_ \\ Minivan & _2.0_ \\ Medium-sized truck & _2.5_ \\ Large truck & _4.0_ \\ Extra-large truck & _5.0_ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical conversion factors for vehicle types. Figure 6: Maximum and minimum tensor window values based on varying criteria. computational resources. The sparse increment at time \(t\), denoted by \(\Delta\mathcal{X}_{t}=\mathcal{X}_{t+1}-\mathcal{X}_{t}\), facilitates a more efficient dynamic tensor decomposition. 1. Initialization of the Core Tensor and Constraint Terms: Utilizing Tucker decomposition, we acquire the initial tensor \(\mathcal{X}_{1}\), alongside the constraint term \(L_{(m)}|_{m=1}^{M}\in\mathbb{R}^{n_{m}\times n_{m}}\), within the M-dimensional tensor sequence \(\mathcal{X}_{t}\in\mathbb{R}^{l_{1}\times l_{2}\times\ldots\times l_{m}}\). 2. Initial Tensor Covariance Matrix Calculation: At time \(t=1\), the covariance matrix for the \(m\)th dimension of the initial tensor \(\mathcal{X}_{1}\) is defined as: \[C_{1}{}^{m}=\mathcal{X}_{1}^{(m)}\mathcal{X}_{1}^{(m)\mathrm{T}}+\mu_{(m)}L_{ (m)}\] (10) where \(\mathcal{X}_{1}^{(m)}\) is the unfolded matrix of \(\mathcal{X}_{1}\) along the \(m\)th dimension, and \(\mu_{(m)}\) is the weight for \(L_{(m)}|_{m=1}^{M}\in\mathbb{R}^{n_{m}\times n_{m}}\). 3. Factor Matrix and Core Tensor Calculation: The factor matrix \(U_{1}^{(m)}\in\mathbb{R}^{n_{m}\times r_{m}}\) comprises the leading \(r_{m}\) eigenvectors of the covariance matrix, \(C_{1}^{m}|_{m=1}^{M}\). The core tensor \(Y_{1}\in\mathbb{R}^{r_{1}\times\ldots\times r_{M}}\) is computed correspondingly. This procedure is outlined in Table 2.  
4. Update of the Factor Matrix \(U_{t+1}^{(m)}\) The revised factor matrix, \(U_{t+1}^{(m)}\), is realized through the calculation of the update of the eigenvalue, \(\lambda_{t+1;i}^{m}\), and the eigenvector, \(u_{t+1;i}^{m}\), as \(C_{t}{}^{m}=\mathcal{X}_{t}^{(m)}\mathcal{X}_{t}^{(m)T}+\mu_{(m)}L_{(m)}\), whose eigenvalue is \(\lambda_{t,i}^{m}\) with an eigenvector of \(u_{t,i}^{m}\), the size of the change in \(\mathcal{X}_{t}\) is \(\Delta\mathcal{X}_{t}\). \[\lambda_{t+1;i}^{m}=\lambda_{t,i}^{m}+\Delta\lambda_{t,i}^{m} \tag{11}\]  
\[u_{t+1,i}^{m}=u_{t,i}^{m}+\Delta u_{t,i}^{m} \tag{12}\]  
Bringing the expressions for \(\lambda_{t+1;i}^{m}\) and \(u_{t+1;i}^{m}\) into \(C_{t+1}{}^{m}\) and omitting t simplifies the equation. \[ \begin{array}{l}[(\mathcal{X}^{(m)}+\Delta\mathcal{X}^{(m)})(\mathcal{X}^{( m)}+\Delta\mathcal{X}^{(m)})^{T}+\mu_{(m)}L_{(m)}](u_{i}^{m}+\Delta u_{i}^{m}) \\ \qquad\qquad\qquad=(\lambda_{i}^{m}+\Delta_{i}^{m})(u_{i}^{m}+\Delta u_{i}^{m} )\end{array}  \tag{13}\]  
By focusing only on first-order variable terms, we simplify to:  
\[ \begin{array}{ll}\mathcal{X}^{(m)}\mathcal{X}^{(m)T}&\Delta u_{i}^{m}+\left( \mathcal{X}^{(m)}\Delta\mathcal{X}^{(m)T}+\Delta\mathcal{X}^{(m)}\mathcal{X}^ {(m)T}\right)u_{i}^{m}+\mu_{(m)}L_{(m)}\Delta u_{i}^{m}\\ &=\lambda_{i}^{m}\Delta u_{i}^{m}+\Delta\lambda_{i}^{m}u_{i}^{m}\end{array}  \tag{14}\]