A foundational assumption in traditional traffic safety analysis is that risk scales linearly with exposure. Under this assumption, doubling the traffic volume (Annual Average Daily Traffic, or AADT) on a road segment should inherently double the expected number of collisions.
However, when training our count-based network screening model, applying a strict linear exposure offset (base_margin = log_offset) resulted in systematic over-prediction on high-volume roads. To investigate this, we fitted a Poisson Generalized Linear Model (GLM) to the full network—crucially including zero-collision links—to extract the true exposure coefficient (\(\beta\)) for different road characteristics.
The Poisson GLM fits the following relationship:
\[E[Y] = \exp(\alpha + \beta \cdot \ln(\text{Exposure}))\]
Where: * \(E[Y]\) is the expected number of annual collisions. * \(\alpha\) is the baseline risk intercept (how dangerous the road is when nearly empty). * \(\beta\) is the exposure scaling factor. * \(\text{Exposure}\) is the annual vehicle-kilometers travelled on the link.
Unclassified: 1060014 links
Classified Unnumbered: 190921 links
Unknown: 442836 links
Not Classified: 224878 links
A Road: 155538 links
B Road: 89286 links
Motorway: 4084 links| road_classification | intercept | exposure_coef | |
|---|---|---|---|
| 0 | Unclassified | 0.677943 | 0.814647 |
| 1 | Classified Unnumbered | 0.531478 | 0.596444 |
| 2 | Unknown | -1.336228 | 0.613607 |
| 3 | Not Classified | -1.413974 | 0.633419 |
| 4 | A Road | 0.505879 | 0.456619 |
| 5 | B Road | 0.289329 | 0.544913 |
| 6 | Motorway | -0.330084 | 0.781436 |
The relationship between traffic volume and collision frequency is not universal; it varies significantly across different road types and forms of way.
While exposure dictates the frequency of collisions, the posted speed limit is a strong proxy for their severity. Mapping the proportion of collisions that result in a Killed or Seriously Injured (KSI) casualty against speed limits reveals a stark descriptive pattern.
These diagnostics reveal a known specification gap in our current v2 architecture. By using log(exposure) as a fixed Poisson offset, the model is effectively forced to assume β=1.
Because the empirical data shows β is actually between 0.4 and 0.8 depending on the road profile, the XGBoost model has to compensate for this misspecification by using other features to artificially absorb the exposure-curvature signal. This likely contributes to the residual variance observed on high-volume networks, and specifically the tendency to under-predict on Motorways.
Recognizing this sub-linear scaling provides a clear diagnostic baseline. It informs our v3 roadmap, where we can empirically test alternative exposure specifications—such as passing log(exposure) as a learned feature, or utilizing per-family fitted GLM offsets—to better capture these realities without forcing the trees to approximate smooth curves.