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Information Theory Quantifies Interactions Among Car Drivers

By Jillian Kunze

Figure 1. In his study, Subhradeep Roy focused on the interactions of drivers with the cars to their immediate front and rear.
“While we drive in traffic, we constantly interact with other cars,” Subhradeep Roy of California State University, Northridge said during his minisymposium presentation at the 2021 SIAM Conference on Applications of Dynamical Systems, which is taking place virtually this week. “We look ahead of us, use our rearview mirrors to look behind us, and respond accordingly.” In his talk, Roy described how he examined the interactions that occur among drivers using information theoretic tools and data from previous real-world experiments. 

Understanding the behavior of humans as they drive has a number of useful applications: refining traffic simulation models to increase their predictability and examine the emergence of traffic conditions, improving road safety interventions, and informing autonomous vehicles about how drivers typically act. Roy specifically wanted to investigate the extent to which drivers respond to the cars in their immediate front and rear under different traffic conditions. Previous research has used eye-tracking devices to measure where drivers look while on the road, but people might not respond to all the information they see; analytical tools can quantify their responses.

A 2013 experiment conducted by Shin-Ichi Tadaki et al. provided useful data on vehicles traveling on a circular track. The study included 19 experimental sessions in which between 10 and 40 vehicles drove in circles on the one-lane track. The results showed that “stop-and-go” events—where traffic builds up into a jammed flow—primarily occurred when the number of vehicles was greater than 28. For 25 vehicles and under, there was generally a free flow of cars. A large amount of data is available from this study; the authors recorded trajectory data along the circumference of the track every 0.2 seconds for long periods of time, with a spatial resolution of 0.16 meters.

Figure 2. A still from an animation of the simulated circular motion of cars using the intelligent driver model and parameters from Tadaki et al.’s experiment. The right side of the circle shows jammed flow beginning to occur.
This circular track experiment was a good starting point for Roy’s analysis, since the large amount of data was beneficial for the data-hungry analytical tools. There was also no lateral interaction among cars, since there was only one lane; so, the only interactions to note were between the target car and the vehicles in its immediate front and rear (see Figure 1). Roy was particularly interested in the time series of three observables from this experimental data: the distance between a target car and the car immediately in front, the distance between the target car and the car immediately to its rear, and the distance the car travels in a particular time interval. He wanted to use this data to study the influence of the front and rear car on the target car. “Information theoretic tools are very useful to measure these directional influences,” Roy said.

The information theoretic tool of conditional transfer entropy—which deals with situations where there are more than two interacting units —was especially useful in this case, since there were three observables. “Things can be different in presence of a third variable,” Roy said. “You can see false coupling if you only look at two variables due to their common influencing.” Conditional transfer entropy adds additional conditional probabilities that account for this effect. Roy used the Java Information Dynamics Toolkit (JDIT) to implement this measure and determine the statistical significance. 

Roy first tested out this tool in a scenario where the ground truth was known. He used the intelligent driver model developed by Martin Treiber et al. in 2000—in which each vehicle only responds to the car directly in front—and set the parameters to match Tadaki et al.’s experiment. Roy’s output was similar to what had been seen in that experiment, as jammed flow began to occur when the simulation incorporated a sufficient number of cars (see Figure 2).

Figure 3. Results from performing a conditional transfer entropy analysis on real-world data from Tadaki et al.’s experiment. The blue lines represent the amount of influence of the car in front on the target car, and the yellow lines represent the amount of influence of the rear car. The two left plots are from free flow, and the two right plots are from jammed flow. \(N_V\) is the number of vehicles. The front car’s influence is always significant, but the rear car’s influence is only sometimes significant.

The next step was to perform a deeper analysis using simulated data, looking at both free flow (with up to 15 vehicles) and jammed flow (with up to 30 vehicles). In this scenario, the conditional transfer entropy analysis detected a significant amount of influence from only the front car. There was also not much variation in the front car’s influence, though it did become slightly larger for jammed flow with a large number of cars due to the increased interaction during stop-and-go events.

Roy next used conditional transfer entropy on the real-world data from Tadaki et al.’s experimental sessions; Figure 3 shows a sample of the results from this analysis. Roy found the influence of the front car to always be statistically significant, but the rear car’s influence to only be significant some of the time. “From this we see that on average, a driver responds to the front car but may or may not respond to the rear car,” Roy said. There was also as a lot of variation in the quantities that represented the front and rear cars’ influence, which seemed to be somewhat sensitive to the nature of the traffic — the influence of the front car was larger for jammed flow than for free flow, though the rear car’s influence stayed mostly the same.

In the future, Roy hopes to get more data using a virtual reality multi-driver driving simulator. This could enable a study of the interactions between drivers when there are vehicles of different sizes, multiple lanes, and different driving conditions. But Roy’s research—which was the first ever instance of applying information theoretic tools to quantify the interactions among car drivers—has already demonstrated how empirical data can reveal more about human driving behavior.

  Jillian Kunze is the associate editor of SIAM News