Influence scores for traffic forecasts must pass through conservation constraints: classic rankings shift, and removing the top ten stations raises constrained-model MAE by 44.2%
Synopsis
When a traffic forecasting model enforces junction flow conservation at its output, a training sample's influence on predictions no longer travels only through the loss but also through the constraint projection. Classical influence functions assume unconstrained learning and cannot separate perturbations the physics keeps from those the projection removes. This work computes influence inside the feasible set, ranks station sensitivity on hourly PeMS District 3 data covering 318 freeway links, and shows that withholding the ten highest-ranked stations raises the constrained model's MAE by 44.2%.
Interpretation
The influence of training data on traffic forecasts can now be measured inside the feasible set that satisfies flow conservation, not only on the unconstrained neural-network output. Classical influence functions are built for unconstrained learning, where a sample changes parameters only through the empirical loss; junction flow conservation couples links, so a perturbation can propagate only along physically feasible directions. On hourly PeMS District 3 data covering 318 monitored freeway links from January to May 2025, the unconstrained output breaks junction flow conservation by 121.95 vehicles per hour on average and by as much as 536.143, while the hard-constrained output reduces this deviation to near zero and lowers MAE from 267.209 to 263.659.
Influence scores are computed from derivatives that pass through the projection layer, so they reflect how sensitive the physically feasible prediction is to each training sample. The construction places a hard equality projection layer after a spatiotemporal graph neural network, mapping raw predictions onto the conservation-constrained set, and then takes second-order sensitivity of the projected loss. Of 312 candidate conservation equations, 62 pass residual checks and form a full-row-rank constraint matrix; the model trains under these constraints, and influence is estimated with a damped, scaled iterative solver for inverse-Hessian-vector products.
The projection splits each sample's gradient into a conservation-consistent part and a part removed by projection, and a purity indicator between zero and one reports the shares. On the training set the mean purity is 0.923 and the median 0.931, meaning about nine-tenths of the loss-gradient energy lies in the conservation-consistent subspace. Purity varies by period: the morning peak is highest at 0.948, the evening peak is 0.938, and the overnight period from 22:00 to 05:00 is lowest at 0.891, 0.057 below the morning peak.
Perspective
The demonstrated scope is hourly flow forecasting on 318 monitored links of the Sacramento network in PeMS District 3, with 62 junction flow-conservation equalities that pass residual checks and next-hour flow as the target. The intended readers are transportation-modeling and machine-learning researchers interested in constrained learning and data attribution.
The first-order influence approximation relies on standard assumptions that the loss is twice continuously differentiable, the Hessian is nonsingular, and the perturbation is infinitesimal; the conservation structure is treated as fixed rather than changing with time or incidents. Extending to larger networks remains challenging as the number of parameters, stations, and constraints grows.