## 3 Methodology  
To empirically measure the effectiveness of in-vehicle warning information, we undertook a microscopic analysis utilizing PVD collected at second intervals. Leveraging the big data allowed the identification of individual vehicle dynamics, enabling an investigation into the relative changes in speed and spacing before and after the provision of warning information within the speed-spacing plane. Conventionally, the relationship between speed and vehicle distance is illustrated in Figure 1a [34]. In fact, the relationship can be derived from the fundamental relationship in Figure 1b frequently employed in macroscopic traffic flow analysis [35]. For instance, traffic state A in Figure 1b is characterized by \(d_{A}\), \(q_{A}\), and \(V_{A}\) (the slope of a straight line from origin to state A as per the fundamental equation). Given that spacing is the reciprocal of density, point A' in Figure 1a corresponds to state A in Figure 1b. In both relationships, delineated by black and red solid lines in Figure 1, there are two distinct regimes: free flow and congestion. As depicted in Figure 1a, the relationship between spacing and speed is directly proportional in the congested regime. Conversely, density is inversely proportional to flow in fundamental relationships. However, density and flow are aggregated measurements in traffic flow theory, and it is not possible to reproduce them directly from the PVD. On the other hand, as both spacing and speed are extractable from the data, our focus lies in examining the speed-spacing relationship to accomplish the objective in this study. The speed-spacing relationship provides insight into how ordinary drivers establish spacing corresponding to a given speed, and conversely, how they determine speed based on spacing. Suppose that a target vehicle is at point A in Figure 2a and its preceding vehicle decelerates in a traffic jam. If the driver of the target vehicle is an ordinary driver, the driver slows down while decreasing its spacing and that driver theoretically follows the trajectory in speed-spacing plane moving from point A to point B in Figure 2a. However, if the driver is a conservative driver, it tends to maintain additional spacing while decelerating harder than the ordinary driver. As a result, the trajectory of the driver's speed-spacing relationship moves from point A to point C. In contrast, an aggressive driver is anticipated to maintain a relatively short spacing at a given speed and its trajectory moves from point A to point D in Figure 2a. That is, the internal area of a given speed-spacing relationship signifies the speed-spacing behavior of conservative drivers, whereas the external area represents that of aggressive drivers. The objective of the in-vehicle warning information provided by the C-ITS is to prevent accidents by inducing drivers to take proactive actions against potential hazards downstream. In other words, the warning information is a service that advises the average driver to be more conservative in response to the situation ahead. Such conservative driving behavior can be distinguished from the speed-spading trajectory of ordinary  
Figure 1: (**a**) Conventional speed–spacing relationship and (**b**) fundamental relationship between flow (q) and density (k). drivers. Figure 2b shows the hypothetical trajectories on the speed-spacing plane between an ordinary driver who has not been alerted and a conservative driver who has. If the driver responds to the in-vehicle warning, it is expected to slow down and create additional spacing as the conservative driver shown in Figure 2b. Therefore, the average speed decreases, while the spacing increases right after the warning. Conversely, for an ordinary driver who does not receive warning information, the speed-spacing dynamics continue to adhere to the usual pattern, resulting in spacing decreasing concurrently with deceleration, as per the black dotted line in Figure 2b.