**6. Conclusions**

The research proposed a novel AKF model for estimating the number of vehicles on signalized approaches using only probe vehicle data. An AKF model was developed to provide real-time estimates of the statistical properties (mean and variance) for the state and measurement errors. The state equation is derived from the traffic flow continuity equation, while the measurement equation is constructed using the traffic hydrodynamic equation. Results show that the proposed AKF model outperforms the traditional KF model (improves the estimation accuracy by up to 29%), demonstrating the need to use real-time values of the statistical noise parameters in the KF model.

Two estimation models were presented, namely (a) the AKF and (b) the AKFNN. The AKF model uses only probe vehicle data assuming a fixed LMP value that is obtained from historical data, while the AKFNN uses a fusion of probe and single-loop detector data with real-time estimates of the LMP values (*ρin* and *ρout*). In this paper, a robust NN model was developed to provide accurate real-time estimates of the *ρout* values. The selected features of the NN model are *At* (observed from the single-loop detector), *Ap*, *us*, *S*1, and *S*2 (observed from probe vehicles).

The AKF and the NN models were combined to develop the novel AKFNN approach. Results demonstrate that the AKFNN approach significantly improves the vehicle count estimation accuracy since the *ρin* and *ρout* values are estimated better. Subsequently, the paper compared the AKF with the AKFNN models, showing that the AKFNN model outperforms the AKF model, enhancing the estimation accuracy by up to 26%.

Finally, the study investigated the impact of the initial conditions (*Ni*, *mi*, and *Pi*) on the AKF performance. Results show that the AKF model is very sensitive to the initial conditions. For instance, starting the simulation with an *Ni* value of 8 instead of 0 improves the estimation accuracy by 10%. In addition, starting the simulation with an *mi* value of 11 instead of 2 enhances the estimation accuracy by up to 10%. For the *Pi* parameter, an improvement of 7% could occur if the simulation starts with an initial value of 150 instead of 75 veh2. The study also tested the accuracy of the AKFNN estimation by allowing the *Pi* parameter to be tuned (Tuned AKFNN approach), showing that more improvement could be achieved. Specifically, the Tuned AKFNN improves the accuracy by up to 27%.

In conclusion, both models (AKF and AKFNN) produce high estimation accuracy when compared with the state-of-the-art KF model. Proposed future work entails testing traffic signal performance using the estimates of the total number of vehicles as inputs to the traffic signal controller.

**Author Contributions:** The work described in this article is the collaborative development of all authors, conceptualization, M.A.A., H.M.A., and H.A.R.; methodology, M.A.A., H.M.A., and H.A.R.; software, M.A.A., H.M.A., and H.A.R.; validation, M.A.A., H.M.A., and H.A.R.; formal analysis, M.A.A., H.M.A., and H.A.R.; investigation, M.A.A., H.M.A., and H.A.R.; writing—review and editing, M.A.A., H.M.A., and H.A.R.

**Funding:** This research effort was sponsored by the University Mobility and Equity Center (UMEC).

**Conflicts of Interest:** The authors declare that there are no conflicts of interest regarding the publication of this article.
