Fig. 2: Koopman modes demonstrating our method’s ability to uncover patterns hidden within traffic velocity data.

The on/off-ramp locations have been labeled with dark orange dotted lines. Modes 1–3 have a very localized structure near the post-off-ramp section, indicating that they correspond to pinned localized clusters (PLC). Furthermore, it is evident that the first three modes capture the general transition from high to low velocities that occurs during the onset of traffic. Mode 5 seems to be a spatial harmonic of the first three modes in that it has another peaked structure in the mid-ramp section of the highway. Modes 4 and 6 provide clear evidence for the pumping effect, where an apparent increase in amplitude followed by a decrease can be seen in these modes as they propagate past the off and on-ramps respectively. Overall, the Koopman modes uncover complex spatiotemporal wave structures that are hidden within traffic data. Furthermore, every mode oscillates with a single known frequency, according to the imaginary part of its corresponding eigenvalue. This can be contrasted to a Fourier analysis which would yield modes and frequencies specific to the positions along the highway. A complete list containing the periods of oscillation of the modes we discussed can be referenced in Supplementary Table 1.