How does noise change car-following?
Short answer: the colour of the noise matters, not only its size. All three rings below carry a random error on every driver's acceleration and differ only in how long that error stays correlated with itself. Correlated residuals concentrate their variance in the low frequencies a car-following chain amplifies, so they seed stop-and-go waves more efficiently than white noise of comparable magnitude. The autocorrelation plot is where the three models separate most sharply.
Three ring roads run side-by-side with identical IDM parameters and initial
conditions. The only difference is the driver-noise model on the acceleration
residual η(t):
white noise (B-IDM),
AR(p) (DR-IDM),
Gaussian process (MA-IDM).
The autocorrelation of η(t), the fundamental diagram, and summary
metrics show how the temporal structure of the residual differs — slowly
decaying memory under AR(p) and GP noise, no correlation at all under white
noise. Each ring runs at its own paper-calibrated noise scale, so the
three do not share a common marginal variance; read the comparison as
"each model as calibrated", not as a controlled equal-variance experiment.
Three rings, three noise models identical IDM parameters and initial conditions
Residual structure and its macroscopic effect noise trace, autocorrelation, fundamental diagram, metrics
Acceleration noise η(t) — one tagged car
White noise looks like hash; AR is jagged but persistent; GP is smooth.
Autocorrelation ACF(τ) of η(t)
White noise: spike at τ=0, then within the ±2/√N band (grey dashed, 95% CI under i.i.d. noise). AR / GP: slow decay — the temporal correlation that distinguishes them from white noise. The window runs to 12 s so that the DR-IDM signature is visible: per Zhang, Wang & Sun (2024) the AR residual carries positive correlation over the first few seconds and, for the mid orders the paper recommends (p ≈ 4–6), turns negative between about 5 and 10 s. Low orders (p = 1, 2) stay positive across the whole window. Each curve is drawn only out to a quarter of its available sample length, so it extends as the run accumulates data.
Fundamental diagram — flow vs density
Each dot is one instantaneous sample of (density, flow) taken on a quarter of the ring, with flow computed as density × space-mean speed. Correlated noise widens the scatter. The dots are not time-ordered, so this panel shows the spread of flow–density states, not a hysteresis trajectory.
Summary metrics
| Metric | White (B-IDM) | AR(p) (DR-IDM) | GP (MA-IDM) |
|---|---|---|---|
| Empirical std of η(t) (m/s²) | – | – | – |
| Lag-1 autocorrelation of η | – | – | – |
| Effective correlation time (s) | – | – | – |
| Std. of avg ring speed (m/s) | – | – | – |
| % time with min speed < 3 m/s | – | – | – |
Let the sim run ~60 s for stable estimates. The η statistics use a rolling window of the most recent samples, ring-speed statistics a longer one, and the jam percentage accumulates from reset. "Effective correlation time" is the sum of the ACF over its leading run of positive values, stopping at the first value below 0.05 and capped at 10 s — so it deliberately ignores the negative lobe and is censored rather than a full integrated correlation time.