# CQFormer: Cyclic Quasi-Dynamic Transformers for Hourly Origin-Destination Estimation
## 5 Experiments
### Results and Discussion
#### 5.4.4 Result Analysis  
In this section, we will analyze the superiority of the proposed model against other baselines as shown in Section 5.4.1. Figures 7 and 8 further display the comparison results in an intuitive visual manner. In Figure 7, ten thousand pieces of data in the evaluation dataset are randomly selected, where the x-axis represents the true value of OD flow and the y-axis represents the estimated value of the model. The standard line at 45 degrees stands for a perfectly accurate estimation. The closer the state points are to the standard line, the better estimations they stand for. Figure 8 illustrates the dimensionality reduction using Principal Component Analysis (PCA), where the first two principal components are plotted [52; 53; 54; 55; 56]. Each density peak and its respective cluster in can be regarded as a type of traffic demand structure. The traffic demand structure is defined as the distribution pattern of traffic demand between different OD pairs and reflects the fundamental skeleton of the dynamic OD matrix [57; 58]. The daily travel activities of urban residents have multiple types of attributes, including work, leisure, shopping, and more. These travel attributes may lead to different traffic demand structures at different time and scenarios [59]. As shown in Figure 8a, the real road network used for conducting simulation contains multiple traffic demand structures. Diverse traffic demand structures may contain various travel patterns [60] and generate different traffic states (e.g., congested and un-congested state) and evolutionary patterns [61], and further affect the accuracy of OD estimation [62]. Figure 8b-f show the capability of each model to capture those traffic demand structures. More specifically, the more completely the model captures these structures, the more accurate the model's estimation is likely to be. According to Figures 7 and 8, the models taking multiple traffic structures into account tend to present higher estimation accuracies. Both c-SPSA and learning-based methods implicitly or explicitly include the information about multiple traffic demand structures. The Extended Kalman Filter captures the time-dependent traffic demand patterns by capturing the evolutionary characteristics of traffic demand and states over time [17; 63; 64]. It presents promising results when there are a homogeneous and significant traffic demand evolution pattern, which is suitable for the OD estimation problem in an arterial road or a network composed of several arterial roads. Nevertheless, in the presence of multiple traffic demand evolution patterns, the Kalman Filter might fail to track the trend of traffic demand in a real urban network, and the estimated OD matrices might cluster around the single prior traffic demand structure, as illustrated in Figure 8f. The explicit gradient-based estimation model, SPSA, introduces a prior OD matrix as the structural skeleton of the estimated OD matrices [40]. In order to capture the structural information of traffic demand more accurately, weight-SPSA (W-SPSA) and cluster-SPSA (c-SPSA) have been successively proposed [43; 65; 66]. Therein, c-SPSA provides a stable gradient strategy by clustering the traffic demand of different intensities with a specific set of hyperparameters for each cluster to accommodate OD estimation under multiple demand structures. However, due to the absence of correlation mappings from a traffic evolution pattern to the corresponding demand structure in model, the estimation results of c-SPSA are susceptible to the influence of the prior matrices and noises of DTA process, which can lead to the failure of capturing some non-significant demand structures. Neural networks excel in discovering multiple implicit patterns in the target distributions through feature extraction of the data and enable us to establish many-to-many mappings between known distributions (traffic counts) and target distributions (OD matrices).