Confidence intervals for real-time freeway travel

In this paper we demonstrate two ways of assigning confidence intervals to real-time neural network based on freeway travel time predictions, which express the uncertainty in our model parameters. Both exploit a bootstrap mechanism, but of different sizes. The non-parameterized bootstrap approach estimates model variance by the variance of the ensemble prediction, while the approximate Bayesian approach utilizes a combination of the ensemble variance and the variance that can be calculated analytically through a Bayesian approach toward neural network training. Both methods yield plausible results, albeit that the Bayesian method requires less (computational) effort, since it suffices with a much smaller ensemble of neural networks. The Bayesian approach, however, does seem to overestimate variance in freeflow conditions. Both methods can be straight forwardly extended to calculate prediction intervals, given that we are able to model the noise inherent to the data.