*Article* **Developing a Neural–Kalman Filtering Approach for Estimating Traffic Stream Density Using Probe Vehicle Data**

#### **Mohammad A. Aljamal 1, Hossam M. Abdelghaffar 2,3 and Hesham A. Rakha 1,***∗*


Received: 20 August 2019; Accepted: 2 October 2019; Published: 7 October 2019

**Abstract:** This paper presents a novel model for estimating the number of vehicles along signalized approaches. The proposed estimation algorithm utilizes the adaptive Kalman filter (AKF) to produce reliable traffic vehicle count estimates, considering real-time estimates of the system noise characteristics. The AKF utilizes only real-time probe vehicle data. The AKF is demonstrated to outperform the traditional Kalman filter, reducing the prediction error by up to 29%. In addition, the paper introduces a novel approach that combines the AKF with a neural network (AKFNN) to enhance the vehicle count estimates, where the neural network is employed to estimate the probe vehicles' market penetration rate. Results indicate that the accuracy of vehicle count estimates is significantly improved using the AKFNN approach (by up to 26%) over the AKF. Moreover, the paper investigates the sensitivity of the proposed AKF model to the initial conditions, such as the initial estimate of vehicle counts, initial mean estimate of the state system, and the initial covariance of the state estimate. The results demonstrate that the AKF is sensitive to the initial conditions. More accurate estimates could be achieved if the initial conditions are appropriately selected. In conclusion, the proposed AKF is more accurate than the traditional Kalman filter. Finally, the AKFNN approach is more accurate than the AKF and the traditional Kalman filter since the AKFNN uses more accurate values of the probe vehicle market penetration rate.

**Keywords:** real-time estimation; probe vehicle; traffic density; neural network; level of market penetration rate
