Fig. 3: Koopman modes demonstrating the ability to uncover growing or decaying patterns.

The on/off-ramp locations have been labeled with dark orange dotted lines. Mode 7 propagates across the entire highway without disturbances to its amplitude and thus corresponds to a highway wide traffic jam also known as a moving localized cluster (MLC). Mode 9 provides further evidence for the pumping effect, where an apparent increase in amplitude followed by a decrease can be seen as the mode propagates past the off and on-ramps respectively. However, mode 8 seems to display a decrease in amplitude followed by another decrease when propagating past the off and on-ramp respectively. This phenomenon to the author’s knowledge has not been reported by other empirical studies. Modes 10 and 16 demonstrate our method’s ability to uncover growing or decaying patterns. Specifically, mode 16 appears to contain its amplitude almost entirely during the first 10–15 minutes and concentrated in the pre-on-ramp location of the highway. This indicates that mode 16 corresponds to the stop and go waves (SGW) present during the exact same region of the spatiotemporal data in Fig. 1a. The exact growth or decay rate of the mode is dictated by the real part of its corresponding eigenvalue. This again is a distinguishing feature of our methodology from a Fourier analysis in that Fourier modes do not capture growing or decaying features. Lastly, mode 13 also demonstrates a double-peaked structure resembling a spatial harmonic feature of Fig. 2a–c. A complete list containing the periods of oscillation of the modes we discussed can be referenced in Supplementary Table 1.