Using Kalman filter algorithm for shortterm traffic flow prediction in a connected vehicle environment
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Abstract
We develop a Kalman filter for predicting traffic flow at urban arterials based on data obtained from connected vehicles. The proposed algorithm is computationally efficient and offers a realtime prediction since it invokes the connected vehicle data just before the prediction period. Moreover, it can predict the traffic flow for various penetration rates of connected vehicles (the ratio of the number of connected vehicles to the total number of vehicles). At first, the Kalman filter equations are calibrated using data derived from Vissim traffic simulator for different penetration rates, different fluctuating arrival rates of vehicles and various signal settings. Then the filter is evaluated for a variety of traffic scenarios generated in Vissim simulator. We evaluate the performance of the algorithm for different penetration rates under several traffic situations using some statistical measures. Although many of the previous prediction methods depend highly on data from fixed sensors (i.e., loop detectors and video cameras), which are associated with huge installation and maintenance costs, this study provides a lowcost mean for shortterm flow prediction only based on the connected vehicle data.
Keywords
Connected vehicle Flow prediction Kalman filter Vissim simulator1 Introduction
Traffic congestion takes a massive toll on cities’ economies. The congestion cost for Sydney and Melbourne is around $6.1 billion and $4.6 billion a year, respectively, and it is projected to increase twofold by 2030 [1]. To tackle the problem of traffic congestion, intelligent transportation systems (ITSs) are considered as an appropriate choice to provide a reliable transport network [2]. To this end, adaptive traffic control is perceived as a useful tool in the ITS toolbox designed to allocate a fair amount of green times to vehicles in signalized intersections. The successfulness of adaptive traffic control strategies depends highly on the accuracy of the input data. Hence, it is essential to enhance the efficiency of traffic prediction algorithms by providing more exact and realtime data. In addition to adaptive traffic signals, traffic prediction is also used in the advanced traveller information system (ATIS), emergency response system planning, variable message signs (VMSs) and realtime route guidance to assist drivers to select the best route among the existing alternatives [3, 4]. The data from various sources such as fixed sensors or floating sensors can be used as input for prediction algorithms.
With the emergence of the connected vehicle (CV) technology, traffic states can be predicted with much higher accuracy in comparison with point detectors such as loop detectors and video cameras since point detector sensors can only provide information about specific spots of a much larger network. Connected vehicles can transmit their information such as position, speed and acceleration/deceleration to other connected vehicles (V2V) and an installed infrastructure near the intersections (V2I) on a realtime basis. Moreover, the data from CVs can also be used to develop smart and intelligent control schemes for a network of signalized intersections. Some CV testbeds are underway around the world to test various applications of connected vehicle technology in the real world [5].
“Connected Vehicle” is no longer a distant idea or technology; it is currently becoming a new norm and reality. Recently, some car manufacturers have started to install an onboard unit (OBU) in their new products. The OBU is a small gadget for sending and transmitting signals that can be easily installed inside vehicles. The cost of OBU is no more than a few hundred dollars, and it is projected to decrease in the coming years [6]. The aim is to provide the possibility for vehicles and infrastructure to cooperate to enhance safety, mobility and environmental sustainability. However, there is evidence to support the idea that it takes a long time for new technology such as OBU to become available in all new vehicles and even longer for that technology to be in the majority of vehicles on the road [7]. Therefore, there is a need to develop traffic signal control strategies based on data from various penetration rates of connected vehicles, for which one prerequisite is to create accurate and versatile traffic prediction methods.
Moreover, traffic state estimation in a dense city with interlock intersections is a more complicated task compared to freeways and corridors. Nevertheless, a clear majority of research studies in traffic prediction methods are dedicated to traffic prediction in isolated freeways and arterials.
Based on a comprehensive review of the previous studies, developing shorttime prediction methods, which use the CV data, is still an open research area. Therefore, this paper presents a Kalman filter strategy to predict traffic flow on a realtime basis using data from CVs. The proposed method does not depend on the data from fixed sensors such as loop detectors. Moreover, it can work for various penetration rates of connected vehicles. The model is also applied to estimate the traffic flow for urban arterials where the signalized intersections can considerably affect the flow of vehicles approaching each intersection.
The paper is organized as follows. The next section is dedicated to the review of the existing traffic state prediction algorithms. In Sect. 3, the methodology of the flow prediction algorithm based on Kalman filter method is explained in detail. Section 4 presents the numerical results of the Kalman method for different penetration rates of connected vehicles as well as various traffic conditions. The conclusion and future research direction are presented in the last section.
2 Literature review
Regarding various prediction models used in the literature, the traffic prediction methods can be classified as parametric, nonparametric and a combination of both. Parametric models use the training data to adjust some finite and fixed set of parameters of the model and then use the model to estimate the traffic states for a set of different test data. For instance, a family of time series models [8, 9] such as linear regression model [10], autoregressive integrated moving average (ARIMA) [11] and Box–Jenkins time series model [12], Kalman filtering and particle filter models [13, 14, 15] are types of parametric models.
On the other hand, nonparametric models assume that the distribution of data cannot be easily defined by a set of fixed and finite parameters in the model. Some nonparametric models include neural networks [16, 17] and nonparametric regression models [18, 19]. Some prediction models are developed based on a combination of both parametric and nonparametric models, such as fuzzyneural network [20, 21], neural network and ARIMA model [22], autoregressive moving average with exogenous input (ARMAX) [23] and machine learning and neural network [24].
Although nonparametric methods show better accuracy in comparison with simple parametric methods such as time series, they require a high computational effort. Moreover, their accuracy is highly dependent on the quality and quantity of the training data [25].
The Kalman filter which uses a statespace model is a popular tool for shortterm traffic prediction thanks to its multivariate characteristic, in a sense; there exist several checkpoints to round up noisy data [26]. It can be used in both stationary and nonstationary traffic conditions (in other words, stable and highly volatile traffic circulation). In each step of the prediction, the state variables (which we aim to predict their quantity in our present work) will continuously be updated using the new and realtime traffic condition data collected from different sources [27, 28]. In this paper, a Kalman filter is developed to predict the traffic flow of each movement approaching an intersection using data gathered from connected vehicles in the previous step of the prediction time.
Prediction data can be obtained from stationary sensors such as loop detectors, radar and video cameras [3, 29] or nonstationary sensors such as GPS devices, mobile phones and probe vehicles [30].
Over the past decade, most research in traffic state prediction has mostly focused on data gathered from single point detectors such as inductive loop detectors [31]; however, the prediction of traffic states based on data from connected vehicles is still a new area of research and needs to be more investigated. The poor performance of previous traffic forecasting algorithms based on loop detectors is mostly due to the lack of widespread deployment of these sensors in the area of measurement [32]. Moreover, point detectors can only provide information about specific spots and therefore are not able to reflect the real situation of the traffic in the whole network [33]. These sensors are also associated with huge installation and maintenance costs [34]. With the emergence of connected vehicle technology, it is possible to collect information from connected vehicles in various parts of the network. This precise information can be obtained several times in a second. More accurate data can give rise to better accuracy of the prediction models [35]. One major drawback of this approach is the imperfect penetration rate of connected vehicles; recent studies have shown that even if US car manufacturers are mandated to install OBU on light vehicles, it takes approximately 25–30 years to have 95% of vehicles equipped with communication devices [36]. This implies the need to develop prediction methods based on a limited number of connected vehicles [37, 38].

To optimize the parameters of adaptive traffic signal controllers, it is of critical importance to predict the traffic flow for short intervals (i.e., realtime traffic prediction). Relatively long intervals cannot be used to accurately adjust the traffic controller parameters in accordance with the shortterm variation of the traffic condition. However, a vast majority of the literature on traffic forecasting is dedicated to longterm traffic prediction. Hence, it is necessary to develop new algorithms able to accurately predict the traffic state for a shorttime horizon toward the future.

Over the past decade, most research in traffic forecasting has emphasized the prediction of traffic flow in freeways and corridors; however, the urban traffic flow prediction is a more complex problem in the heart of cities with interlock signalized intersections [32].

A considerable amount of traffic forecasting literature has developed their methods based on data gathered from stationary sensors such as inductive loop detectors. With the emergence of CV technology, it is possible to have access to more accurate data from connected vehicles, so there is still an open area for research in the use of connected vehicle data to predict traffic states.

Since it is predicted that reaching a 100% penetration rate of connected vehicles is not in the scope of near future, it is of vital importance to develop prediction methods that can predict the traffic situation based on data from a limited number of CVs [39].
Based on the aforementioned knowledge gaps, this paper aims to develop a shortterm traffic prediction algorithm to predict the flow of vehicles in urban networks based on only data from a limited number of connected vehicles.
3 Methodology
To achieve this, we use a Kalman filter model. In other words, based on the realtime flow information of connected vehicles in each step of the prediction, the flow in the next time step is predicted. In this section, we first introduce the basic concepts of the Kalman filter algorithm followed by a calibration and evaluation process.
3.1 Kalman filter algorithm
The Kalman filter is a statespace model that was first introduced by Kalman [40]. It can be applied to model systems with multiinput and multioutput and can be used for both stationary and nonstationary situations. This feature of the Kalman filter makes it an appropriate choice for modeling the traffic states [3]. Kalman filter updates the prediction of state variables based on the observation in the previous step. Therefore, it only needs to store the previous estimate information. The Kalman filter has two distinct features: (1) It does not require any additional space to store the entire previously observed data. (2) It is computationally efficient since it does not need to utilize all the previous estimated/measured data in each step of the prediction process [41]. In this study, the Kalman filter is used to predict the flow of vehicles (i.e. state variables) on the basis of the realtime information received from connected vehicles (observation) in the last step of the prediction process. The prediction of the state variable is realized in a recursive procedure in which observations and previous states are used to calculate the flow state for the next step of the prediction process.

Step I: Initialization:
Set k = 0 and let \(E\left[ {\varvec{x}_{0} } \right] = \hat{\varvec{x}}_{0}\) and \(E\left[ {(\varvec{x}_{0}  \hat{\varvec{x}}_{0} )^{2} } \right] = \varvec{P}_{0}\), where \(\hat{\varvec{x}}_{k}\) and P_{k} are the estimates of the state and error covariance matrix at time k, respectively.

Step II: Extrapolation:
Extrapolation of state \(\hat{\varvec{x}}^{}_{{{k} + 1}} = {\varvec{\phi}}_{k} \hat{\varvec{x}}_{k}\), and extrapolation of the error covariance \(\hat{\varvec{P}}^{}_{{{k}}} = {\varvec{\phi}}_{k} { P}_{k} \varvec{ }{\varvec {\phi}}_{k}^{\text{T}} + \varvec{Q}_{k},\) where the superscript dash stands for prior estimation of the state or error covariance.
 Step III: Calculation of Kalman gain:$$\varvec{K}_{k} = \varvec{P}^{}_{{{k}}} \varvec{H}^{\text{T}} \left( {\varvec{HP}^{}_{{{k}}} \varvec{H}^{\text{T}} + \varvec{R}_{k} } \right)^{  1} . $$
 Step IV: State and error covariance update:$$\hat{\varvec{x}}_{k} = {\hat{\varvec{x}}}^{}_{{{k}}} + \varvec{K}_{k} \varvec{ }\left( {\varvec{z}_{k}  {\varvec{H}}{\hat{\varvec{x}}}^{}_{{{k}}} } \right),$$$$\varvec{P}_{k} = \left( {\varvec{I}  \varvec{K}_{k} \varvec{H}} \right)\varvec{P}_{k}^{  }. $$

Step V: let k = k + 1 and go back to step II and continue the process until the end of the present time period.
3.2 Kalman filter calibration
In order to construct the state space and measurement equations in (1) and (2), the state transition matrix ϕ_{k}, the measurement mapping matrix H, the noise matrixes w_{k} and v_{k}, which are considered to be scalar in this study, are estimated from groundtruth data.
Note that, x_{k}, the traffic flow approaching an intersection (crossing the reference line) depends on different external factors such as the traffic signal control of the upstream intersections as well as their geometries and layouts. Therefore, the Kalman filter equations need to exclusively adjust for each layout. Although real traffic data is important to evaluate the prediction algorithms, because of the unavailability of CV data at this stage of the study, we use the Vissim data as the groundtruth data. However, since the authors of this article are recently involved in a connected vehicle testbed project in Melbourne, called Australian integrated multimodal ecosystem (AIMES) (https://eng.unimelb.edu.au/industry/aimes), testing the algorithm with the real data from a CV environment is one of the plans for this study. To this end, the data obtained from the Vissim traffic simulator are used as historical data to derive and calibrate the equations. To achieve this, two consecutive intersections are simulated in the Vissim traffic simulator. The flow information of all vehicles including connected vehicles pertaining to different signal settings and arrival patterns and various penetration rates of connected vehicles for 2 h of the simulation is collected. We deploy the MATLAB as a COM interface to control the Vissim simulator. Moreover, the training and test data used, respectively, for calibration and evaluation of the Kalman algorithm are collected through MATLAB. We also realize the Kalman filter algorithm in a MATLAB m.file. The aim is to predict traffic flow for short time intervals toward the future to be used later in traffic signal settings. Here, a time interval of 50 s is considered.
The state transition matrix ϕ_{k} and noise matrix w_{k} are calculated based on a linear regression model on the state variable vector x_{k} and previous state vector x_{k−1}. In order to determine the w_{k}, the variance of the error between the previous state vector x_{k−1} and state vector x_{k} need to be calculated.
In the measurement Eq. (2), the state variable in each time step is calculated based on the measured variable z_{k}. In this study, z_{k} indicates the flow of connected vehicles travelling from three approaches of the preceding intersection in each step of the prediction period. H maps the number of connected vehicles in the traffic flow to the total flow of vehicles. Note that H is calculated using a linear regression model based on the total number of vehicles in the flow as an independent variable. This coefficient is interpreted as the penetration rate of connected vehicles. Noise variance v_{k} is calculated based on the variance of the error between the measured variable z_{k} and state variable x_{k}. The calibration process is evaluated using three widely used metrics [42, 43], as discussed in the next section.
The summary of the case study information
Parameters  Value 

Distance of the reference line from the intersection (m)  55 
Number of lanes in each link  2 
Width of each lane (m)  3.5 
Saturation flow rate (veh/h)  3,600 
Free flow speed (km/h)  50 
Time step duration (s)  50 
Warmup period (s)  200 
4 Numerical results
In order to illustrate the performance of the proposed flow prediction method, the method is evaluated under various penetration rates of connected vehicles and different congestion levels.
In this study, we also evaluate the performance of the proposed Kalman filter technique to track the abrupt changes in the traffic condition. Here we assume that the data related to the sudden changes in traffic flow are not considered in the adjustment of the parameters of Kalman filter equations. The aim is to evaluate the performance of the Kalman filter when there is a considerable change in the flow of vehicles. In order to do so, at first, the Vissim traffic simulator will be run for an undersaturated traffic condition. Then two times it was suddenly converted to saturated traffic condition (the average flow of vehicles in this stage is two times more than the undersaturated condition) by changing the number of input vehicles at intersection B in Fig. 1. Each time after rising the flow, the traffic condition again turns to the undersaturated traffic condition. Therefore, we can make a fluctuation in the traffic and evaluate the ability of the Kalman filter to track the changes. The connected vehicle data from Vissim is used to predict the flow of vehicles based on the Kalman filter method. One may argue for expanding the reach of the proposed algorithm to include autonomous vehicles’ data. To this end, it is important to note that a driverless car basically is a car without a driver and hence is not necessarily a connected vehicle. An autonomous vehicle can be considered a connected vehicle only if it is equipped with onboard unit (OBU) and it can communicate with roadside units (RSUs) and other vehicles. Take for example the famous Google car which is an ad hoc autonomous vehicle and it is noted connected to anywhere (Google’s Autonomous Vehicle, 2017). In the situation where autonomous vehicles are equipped with communication devices, we can easily consider them as connected vehicles in our methodology.
Comparison of error indexes for different penetration rates of connected vehicles for normal traffic situation and abrupt changes in flow
Penetration rate (%)  Rsquared  RMSE  MAE  MAPE  

Normal  Abrupt change  Normal  Abrupt change  Normal  Abrupt change  Normal  Abrupt change  
0*  − 0.2508  − 0.8350  4.9633  7.8604  4.2310  5.5667  0.4812  0.5083 
10  0.5205  0.3151  8.9841  4.7177  6.2202  3.4360  0.4049  0.4492 
20  0.6119  0.6506  6.9565  3.8485  5.1029  2.8743  0.2883  0.3798 
30  0.6234  0.6532  4.6346  3.6278  3.5055  2.4680  0.2647  0.3156 
40  0.6584  0.6537  3.8721  3.5603  2.7077  2.3279  0.2941  0.2969 
50  0.8491  0.7754  3.6984  2.7708  2.5327  1.9656  0.2091  0.2384 
60  0.8798  0.8359  2.9354  2.3818  2.0760  1.6189  0.1941  0.1950 
70  0.9177  0.8659  2.4256  2.1293  1.8234  1.3945  0.1973  0.1668 
80  0.9557  0.9059  1.6978  1.8384  1.2400  1.1960  0.1272  0.1450 
90  0.9529  0.9299  1.6723  1.5179  0.9799  0.8924  0.0838  0.0983 
100  0.9797  0.9736  0.3313  1.2770  0.0366  0.8714  0.0050  0.0050 
It is important to note that the Kalman filter is a frugal model, in the sense that, it does not need a high number of connected vehicles to perform well. As can be seen in Fig. 5, the penetration rate of at least 20% is a commensurate and healthy choice for the Kalman filter. This is a compelling result that proves the effectiveness of the proposed Kalman model to predict the flow for a low market penetration rate. For the penetration rate of 60% or more, there exists only a marginal improvement in the performance criteria. Therefore, for the reasonable and accurate performance of the proposed prediction algorithm, there is no need to have a near perfect market penetration rate (around 100%).
5 Conclusion
This paper presents a Kalman filter technique to predict traffic flows approaching an intersection based on the data of connected vehicles. At first, parameters of the Kalman equations are adjusted through the use of Vissim microscopic traffic simulator. Then, we evaluate the performance of the model for different penetration rates of connected vehicles under various traffic conditions. The results obtained from this study show the Kalman filter performs well when the penetration rate is more than 20%.
We also test the accuracy of the model to track abrupt changes in the traffic condition and show the effectiveness of the model based on several error indexes. It is apparent from the results that the proposed method has an acceptable accuracy to predict the traffic flow even in the presence of abrupt changes in traffic condition. Moreover, there is a positive correlation between the model’s accuracy and the penetration rates, in the sense that, as the penetration rate increases, the model predicts traffic flow with more resolution.
Future work should focus on the improvement of the algorithm tailored to situations in which the penetration rate is significantly low. For instance, we could also use the data from inductive loop detectors and Bluetooth data to predict the flow with more accuracy under low penetration rates. Moreover, to predict the flow with higher accuracy, we can consider the traffic signal states of upstream intersections as input variables in the state equation of the Kalman filter. Testing the accuracy of the algorithm against groundtruth data from the real world connected vehicle testbed is one of our focuses for future study. Moreover, the estimation algorithm can also be integrated into an adaptive traffic signal plan [44] to control the traffic based on the real situation of the traffic in the network.
Notes
Acknowledgements
This work has been sponsored by the Australian Integrated Multimodal EcoSystem (AIMES), https://industry.eng.unimelb.edu.au/aimes. The authors would like to thank many industry and government partners, in particular, Cisco, CUBIC, Cohda Wireless, VicRoads, Department of Transport, Traffic Accident Commissions, and PTV Group. The authors would like to thank Prof Zhai, the editor in chief and two anonymous reviewers for their constructive and insightful comments.
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