Transportation Vehicle Arrival Time Prediction
5.6 Proposed Refined Kalman Filter (KF) Model-Based System
Fig. 5.8 Information relating to large spikes in Fig.5.7
5.6 Proposed Refined Kalman Filter (KF) Model-Based
A Kalman model implies the state of a system at a time kC1 developed from the previous state at time k. This can be expressed by the following state equation:
xkC1 D˛xkCßukCwk (5.1)
Here,xkC1 is the state vector containing the terms of interest at timekC1.uk is the vector with the control inputs at timek.˛is the state transition matrix which applies the effect of system state parameters at timek to the state at time kC1. ß is the control input matrix which applies the effects of the control input parameters (uk) to the state vector.wkis a vector containing process noise terms for each value in the state vector.
Measurements of the system are performed according to the formula
ykDxkCzk (5.2)
where yk is the vector of measurements, is the transformation matrix which maps the state vector parameters to the measurements, andzk is the vector which contains the measurement noise for each element in the measurement vector. This measurement formula is also referred to as output equation.
Consequently, Kalman filter (KF) model can be used to estimate the position of a vehicle by inputting the vehicle speed into the KF algorithm. The addition of state constraints to a KF model can significantly improve the filter’s estimation accuracy [11]. In this sense, the addition of information that is input linearly into the model may produce significant benefits. KF models can theoretically deliver the best and most up-to-date results when they have continuous access to dynamic information [12]. Many existing models therefore make use of dynamic information. However, the performance of these models can often suffer due to issues with accounting for scenarios that involve rapidly updating, real-world information. In these cases, constraints may be time varying or non-linear. Many road users rely on navigation systems to navigate and estimate the duration of their journeys. However, it is not always possible to handle some events solely by navigation systems’ information.
For instance, if there is an accident, and it is known that it will take 30 min or more to clear, a navigation system cannot detect or incorporate this information because it provides and predicts an arrival time based on GPS satellites. As discussed, social networks have the ability to provide plausible real-time information regarding road traffic, which could potentially improve the accuracy of arrival time prediction.
Intelligently/automatically selecting the best source of external traffic condition information from social networks for input into the KF model can produce improved traffic prediction results. This is achieved by comparing conventional GPS-based Traffic Management Systems (TMS) with new social media information sources. As this paper previously noted, the external/delay information can initially be ‘linearly’
added to determine total KF, based on arrival estimation times. For instance, if a KF model estimates the arrival time without external delay information to be 80 min, and there is delay information from social media of 20 min, then the estimated arrival time will beD80C20D100 min.
information from EdinburghTravelNews such as ‘10-min delays on Whitehouse Road towards Barnton’ can be leveraged. This can be input linearly into KF models.
Other social media data (see: Fig.5.6) can also be fed into KF models. Previously constructed KF models [12,13] focused on speed and position. Hence, a 30-min delay will impact vehicle speed and position. Due to this, the speed, calculated in Eq.5.3, will be set to 0 and a 30-min delay set in Eq.5.6. The key component of the previously configured KF model [19] is outlined below:
xD
Position Speed
(5.3) The system model is then set as follows:
xkC1DAxkCwk (5.4)
ykDHxkCvk (5.5)
AD 1 t
0 1
; HD 0ˇˇˇ1
(5.6) Similarly to the general Kalman filter equations above,xk represents the state variable,yk the measurement variable,A the state transition matrix, H the state measurement matrix, and vk the measurement noise, defined as any unusual disturbance during a journey. Further, Adescribes how the system changes over time.4t is the interval of the time measuring position. Equation 5.4predicts the state at the next time step for system modelA. Subscript k indicates that the KF model is executed in a recursive manner. The state measurement matrixHis used in Eq.5.6to predict the position based on measured velocity. Designing the optimum system model is an art and difficult to calculate precisely. The system model has to rely on the experience and capability of the Kalman filter. The additionalwkis introduced in Eq.5.4to show the noise taken into account and will give effects of the state variable.
To demonstrate the accuracy of prediction, and that the simulated system results correspond to real-world conditions, a simulation of urban mobility (SUMO) is used [14,15] to validate the KF models. SUMO is a road traffic simulation which is
capable of simulating real-world road traffic using digital maps and realistic traffic models. This is fully discussed in [12] and indicates that appropriately constructed KF models correspond to real-world data. In addition, the estimation arrival time in KF models is more accurate if more information is fed into the model.