# The Car-Following Model and Its Applications in the V2X Environment: A Historical Review
## 3 Research on Car-Following Behavior in the V2X Environment
### Applications of the Car-Following Models
#### 3.3.1 Calibrating the Car-Following Model  
In the V2X environment, connected vehicles are usually equipped with multi-sensors. The data detected by these sensors along with the data obtained from the V2X technologies form a typical heterogeneous data set with multiple sources. How to utilize this kind of data set to calibrate the car-following model is a key problem to be solved in the research field of modeling car-following behavior in the V2X environment. Aiming at this, [PERSON] and [PERSON] proposed an improved calibration algorithm based on the maximum likelihood estimation method and a cross-comparison method of calibration results with the indexes, including the standard errors [113]. The time consumption and accuracy of the calibration process are directly determined by the performance of the calibration algorithm itself. Improvement of the calibration algorithm will contribute significantly to the development of the whole process of calibrating car-following models. Aiming at this, [PERSON] and [PERSON] constructed a general calibration algorithm for parameters in the car-following model based on the extended maximum likelihood estimation algorithm [114]. With [PERSON]'s algorithm, the joint estimation of multiple parameters in the model can be carried out, and statistical analysis results such as standard error can be obtained. Considering that the error may accumulate in the calibrating process with previous calibration algorithms, [PERSON] et al. pro posed an improved error control method based on the correlation between the parameters and the model stability [115]. Utilizing the genetic algorithm and a field data set, [PERSON] et al. verified their method with the Mean Absolute Error (MAE) index in calibrating typical car-following models, such as the GM, OV, Gipps, FRESIM, and ID model. It was proved by [PERSON] et al. that the objective function in the calibrating process is Lipschitz continuous and around the global optimal solution [116]. Based on this, they proposed a global optimization rapid calibration algorithm by combining the global direct search and the local gradient search and verified this algorithm utilizing the NGSIM data set based on the comparison with the Nelder-Mead algorithm, sequential quadratic programming algorithm, genetic algorithm, and simultaneous perturbation stochastic approximation algorithm. [PERSON] and [PERSON] derived the gradient of the optimization problem using the adjoint method and, based on this, proposed an improved quasi-newton algorithm for rapidly calibrating parameters in the car-following model [117]. The results confirm that the calibrating process with this algorithm is significantly faster than that with the genetic algorithm. As mentioned in the previous sections, there is heterogeneity in the car-following behavior with the consideration of various kinds of information, drivers, and traffic conditions. This heterogeneity makes the calibration difficult to \"once for all\", and re-calibration needs to be carried out when these conditions change, which will greatly increase the workload. Considering this, [PERSON] et al. proposed an online calibration algorithm for parameters in the models for micro traffic simulation based on the dynamic multi-step prediction of traffic measures [118]. [PERSON] et al. proposed an online calibration algorithm for parameters in the car-following models based on the Kalman filtering method and Takagi-Sugeno Fuzzy Reasoning System [119]. Model to be Calibrated  
The parameters in parts of the car-following models are not all independent of each other, which means that there is a kind of relevance. Based on this, [PERSON] et al. discussed the relevance of parameters in the car-following models and evaluated the impacts of this relevance on the calibration results [120]. Later, [PERSON] et al. proposed a method to simplify the parameters in car-following models based on the sensitivity analysis and Monte Carlo framework and verified this method employing the FVD model [121].