# Dynamic Tensor Modeling for Missing Data Completion in Electronic Toll Collection Gantry Systems
## 4 Results
### ETC Gantry Plate Data Preprocessing
For instance, a vehicle with a license plate ending in 375 recorded passage times at two gantries, G005032001000310010 and G005032001000410010, approximately 2 min apart. With the gantries about 6 km apart, this would calculate to a speed of 180 km/h, which is inconsistent with road traffic rules. (3) ETC gantry video detectors are used for vehicle plate data collection. These video detectors rely on optical principles, and factors such as dust, shadows, weather, and lighting can affect the detection results; hence, natural elements like rainy weather and nighttime conditions impact the accuracy of gantry detection. (4) The minimum distance between gantries in the same direction on the G50 Nanjing section is set at 400 m, with two ETC gantries spaced 500 m apart, and three road segments have relatively small distances between them. Figure 9 below shows the proportion of duplicate vehicle detection data for January among ETC gantry segments, with blue, red, and green representing the three closely spaced segments, and grey representing other segments. Duplicate detection issues are more significant in the three closely spaced segments. For tensor model construction, PyTorch 1.11 was the framework of choice. The sub-tensor block models were formulated using data fields from each ETC gantry within the G50 Nanjing section, encompassing aspects such as token recognition, transactions, foundational information, and atypical state indicators. These subtensor components were methodically categorized per dimension into basic and expandable attributes. The expandable attributes were then amalgamated and preserved based on the foundational attributes of each dimension to establish a three-dimensional, high-order tensor block model for the ETC gantry system. The method's efficacy was gauged by its accuracy and computational complexity. We contrasted this approach with a matrix decomposition-based completion model (MATRIX) and a prediction model founded on Long Short-Term Memory (LSTM) neural networks. The MATRIX model relies on the low-rank nature of matrices and known elements to decompose the target matrix into two lower-rank matrices, thereby reconstructing the missing entries [33]. Conversely, the LSTM-based model capitalizes on the predictive capabilities of neural networks for ETC gantry data restoration. Preliminary processing of the ETC gantry data revealed notable instances of missing information for January, which included discrepancies arising from incorrect data removal and delays in data transmission. These instances are encapsulated in Table 6.  
Figure 9: Proportion of duplicate detection data within ETC gantry segments. In each missing data situation, two days of missing data were selected as the original data input, and the average time headway, \(h_{i}{}^{ij}\); traffic volume, \(q_{i}\); and mean interval speed, \(\overline{v_{s}}{}^{(ij+n)}\) were the traffic data to be completed. After the data are calibrated using the transaction data sheet, the data can be considered accurate and set to the true value. The Root Mean Square Error (RMSE) was selected as the evaluation index for this data supplementation, and the equation is as follows:  
\[\text{RMSE}=\sqrt{\frac{\sum_{i=1}^{n}(\hat{x_{i}}-x_{i})^{2}}{n}} \tag{29}\]  
where \(\hat{x_{i}}\) is the value of complementary traffic data, \(x_{i}\) is the true value of traffic data, and \(n\) is the number of complementary traffic data points.