# Dynamic Tensor Modeling for Missing Data Completion in Electronic Toll Collection Gantry Systems
## 4 Results  
### ETC Gantry Plate Data Preprocessing  
Our empirical analysis centered on a segment of the Shanghai-Chongqing Expressway, specifically the G50 Nauxun to Jingze Interchange (G50 Nanjing section). Over the 30 days from 1-30 January 2022, we collated 24 sets of ETC gantry data via the toll system on this stretch. These datasets comprise both static information (including vehicle, toll station, and pass card details, alongside toll system specifics) and dynamic records (detailing travel, toll transactions, and system validations). This study focused on the investigation and analysis of ETC gantry plate recognition data for a specific road section. Three main types of anomalies in ETC gantry plate recognition data were identified: incorrect plate recognition data, missing plate recognition data, and duplicate plate recognition data. The causes and forms of these anomalies are outlined in Table 5: \begin{table}
\begin{tabular}{c l} \hline \hline Input: \(\mathcal{X}\in\mathbb{R}^{1\times 1\times 2\times b}\), Dynamic Tensor Flow \(W_{T}\), \\ Iteration error \(\varepsilon_{0}=10^{-6}\) \\ Output: \(\mathcal{X}_{NEW}\) \\ \hline
**Begin** & \\
**1** & Calculate \(P_{\Theta}X\); \\  & Apply \(W_{T}\) approximate tensor decomposition to obtain the initial \\
**2** & solution \(G_{0}\), \(U_{0}^{(1)}\), \(U_{0}^{(2)}\), \(U_{0}^{(3)}\); \# initialization of the objective function \\  & solution \\
**3** & Calculate \(\varepsilon=\frac{\|\mathcal{X}-\mathcal{X}_{NEW}\|_{\varepsilon}}{\| \mathcal{X}\|_{\varepsilon}}\); \# Error calculation \\
**4** & While \(\varepsilon>\varepsilon_{0}\), do \# Set iteration conditions \\
**5** & \(G_{k}=max\left|0,G_{k-1}-\frac{1}{L_{G}}\
abla_{G}f\left(G_{k-1},\widetilde{U} \right)-\frac{\lambda_{G}}{L_{G}}\right|\); \\
**6** & \(U_{k}^{(i)}=max\left|0,U_{k-1}^{(i)}-\frac{1}{L_{U^{(i)}}}\
abla_{U^{(i)}}f \left(\widetilde{G},\widetilde{U}^{(j\
eq i)}U^{(i)}\right)-\frac{\lambda_{G} }{L_{G}}\right|\); \\
**7** & end for \\
**8** & \(\mathcal{X}_{NEW}=G_{k}\times_{1}U_{k}^{(1)}\times_{2}U_{k}^{(2)}\times_{3}U_{ k}^{(3)}\) \\
**9** & Return \(\mathcal{X}_{NEW}=G_{k}\times_{1}U_{k}^{(1)}\times_{2}U_{k}^{(2)}\times_{3}U_{ k}^{(3)}\) \\
**End** & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computational process for ETC gantry data completion via tensor decomposition. Directly applying abnormal ETC gantry plate recognition data in the computation of traffic parameters can severely disrupt model results and analyses. The preprocessing of abnormal plate recognition data in this section includes the identification of ETC gantry abnormal plate recognition data based on classification concepts, preliminary repair of abnormal plate recognition data, and evaluation of the integrity of ETC gantry plate recognition data. Initially, by establishing a normal plate recognition data model using standards and norms, one can predetermine the characteristics of normal data within the data table. This includes correct data formats/rules and distribution models for each attribute. The ETC gantry plate recognition data table is then checked against these standards, including data format, degree of fit within distribution models, repetition of vehicle passage information through each gantry, and vehicle travel direction. Differences between the tested data and the normal data characteristics are used to identify anomalies. In the second step, the prematch of abnormal ETC gantry plate recognition data involves the correction of erroneous, duplicate, and missing data through deletion, expansion, completion, and transformation, thus enhancing the data usability. As indicated above, different processing methods can be selected based on the cause and form of the anomalies. For incorrect plate recognition data, identified errors are deleted from the standard data table and then reclassified as missing data, moving into the repair process for missing plate recognition data.