# Dynamic Tensor Modeling for Missing Data Completion in Electronic Toll Collection Gantry Systems
###### Abstract  
The deployment of Electronic Toll Collection (ETC) gantry systems marks a transformative advancement in the journey toward an interconnected and intelligent highway traffic infrastructure. The integration of these systems signifies a leap forward in streamlining toll collection and minimizing environmental impact through decreased idle times. To solve the problems of missing sensor data in an ETC gantry system with large volumes and insufficient traffic detection among ETC gantries, this study constructs a high-order tensor model based on the analysis of the high-dimensional, sparse, large-volume, and heterogeneous characteristics of ETC gantry data. In addition, a missing data completion method for the ETC gantry data is proposed based on an improved dynamic tensor flow model. This study approximates the decomposition of neighboring tensor blocks in the high-order tensor model of the ETC gantry data based on tensor Tucker decomposition and the Laplacian matrix. This method captures the correlations among space, time, and user information in the ETC gantry data. Case studies demonstrate that our method enhances ETC gantry data quality across various rates of missing data while also reducing computational complexity. For instance, at a less than 5% missing data rate, our approach reduced the RMSE for time vehicle distance by 0.0051, for traffic volume by 0.0056, and for interval speed by 0.0049 compared to the MATRIX method. These improvements not only indicate a potential for more precise traffic data analysis but also add value to the application of ETC systems and contribute to theoretical and practical advancements in the field. t 14 November 202319 December 202319 December 2023223  
These factors often lead to common issues within ETC gantry system, such as unceroded license plates, misidentifications, and delayed data uploads [9]. Unrecorded license plates can result from various factors, like poor visibility, suboptimal camera angles, or technical glitches in recognition software, significantly impacting the accuracy and reliability of the data, which, in turn, affect toll collection and traffic monitoring processes. Misidentification of license plates often occurs because of limitations in the optical character recognition technology, especially under challenging conditions, such as low light, adverse weather, or high vehicle speeds. Such errors can lead to incorrect toll charges, causing inconvenience to users and operational complexities for the authorities. Additionally, delayed data uploads pose a significant challenge in systems designed for real-time or near-real-time processing. Delays in data transmission can disrupt the efficient functioning of traffic management systems, especially in situations in which immediate data availability is critical for decision making, like in traffic congestion management or emergency response scenarios. Another noteworthy challenge is the significant variation in distances between ETC gantries, which creates substantial obstacles for seamless traffic monitoring. This irregular spacing can lead to inconsistencies in data collection, with shorter distances potentially resulting in redundant data capture and longer distances risking the omission of important traffic pattern changes. The complexities and challenges inherent in ETC gantry systems have become pivotal areas for research and innovation. Addressing these challenges necessitates the development of sophisticated solutions that can effectively manage these complexities and enhance the overall efficacy of ETC systems. Our study aims to contribute to this field by introducing a novel approach that not only addresses the technical challenges but also significantly improves the operational aspects of ETC systems, ensuring more accurate and reliable traffic management. We introduce high-order tensor modeling methods for ETC hub data, as well as ETC hub data completion methods based on improved tensor decomposition techniques.