# A Progressive Extended Kalman Filter Method for Freeway Traffic State Estimation Integrating Multisource Data

Authors: Yingshun Liu; Shanglu He; Bin Ran; Yang Cheng
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

Variable techniques have been used to collect traffic data and estimate traffic conditions. In most cases, more than one technology is available. A legitimate need for research and application is how to use the heterogeneous data from multiple sources and provide reliable and consistent results. This paper aims to integrate the traffic features extracted from the wireless communication records and the measurements from the microwave sensors for the state estimation. A state-space model and a Progressive Extended Kalman Filter (PEKF) method are proposed. The results from the field test exhibit that the proposed method efficiently fuses the heterogeneous multisource data and adaptively tracks the variation of traffic conditions. The proposed method is satisfactory and promising for future development and implementation.


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## Body

## 1. Introduction

Various traffic detection methods have been used for the freeway traffic surveillance and ITS application, such as loop detectors, microwave sensors, video cameras, Bluetooth sensors, cellphone probes, and GPS probes. In most cases, more than one technology is available on a freeway segment. Thus, it is a challenge to fully unitize the multisource data for the construction of traffic state. The critical issues include the following:(i)
The heterogeneity in the multisource measurements(ii)
The difference in the spatial or temporal resolution(iii)
The inconsistency in the measurements when they should represent the same traffic stateAlthough the multisource data provide plentiful traffic information, they still have the limitation in the overall spatial or temporal coverage. A traditional method to construct the network-wide picture of traffic state with limited measurements is the model-based estimation [1–3]. One of the extensively used techniques for the estimation, known as online Bayesian method, is Kalman filter and its extended version, such as Extended Kalman Filter (EKF). Kalman filter [4] has been implemented to solve the traffic problems with linear relation [5]. Extended Kalman Filter fits the estimation with nonlinear traffic flow models [3, 6–9]. A series of works based on the EKF algorithm conducted by Wang and Papageorgiou [3, 6–8, 10] have been applied in freeway network with data from the fixed detector. The EKF algorithm also has efficient performance on the estimation using data from probe vehicles, such as GPS-based probe technique [9], as well as cellphone activity data-based probe technique [1] which uses the wireless communication records. However, these works used a single-source data.It has been proved that integrating multisource data can increase the accuracy, reduce the ambiguous, and improve the robust [11]. Bachmann et al. [12] provided a comparative assessment of the different data fusion techniques for traffic speed estimation, including distribution fusion techniques, Kalman filter techniques, ordered weighted averaging, fuzzy integral, and artificial neural network. The majority of recent works on traffic data fusion are data-driven approaches [11, 13–17], which required a set of training data for the calibration of method. Hence, a data-driven method might fail when there is no sufficient data for training and testing. Mannini et al. made a further application of fused results by using EKF to estimate the traffic speed [18]. On the other hand, some model-based works using the macroscopic traffic flow models were developed to fuse multisource data. Heilmann et al. applied the linear Kalman filter to combine data from local detector and electronic toll collection (ETC) system [19]. Deng et al. [20] applied Clark’s approximation method and Newell’s method to fuse data from three detectors, and their estimator is also Kalman filter-based for the linear measurement equations. Recently, Nantes et al. [2] devoted to the development of an incremental EKF estimator for the fusion of three heterogeneous data sources collected from the arterials, with the assumption that all measurements are independent. Comparatively, the model-based method is superior to the data-driven method by considering the relationship among the traffic variables.In the application of a model-based method, it is important to define the relation between the measurements and the traffic state variables. The newly developed data collection technique usually requires an update or adjustment to this relation model. The multisource data applied in this study are the traffic features that are extracted from wireless communication records and traffic measurements from microwave sensors. The traffic features from wireless communication records are not direct traffic state variables, which is a challenge for the application in the traffic state estimation. The wireless communication records are also known as the cellular communication records, including the records of handoff, normal location update, and data transition. The traffic features extracted from these wireless communication records are different from the traffic information from the handoff-based collection method [21]. Our recent work shows that the application of these traffic features in the traffic estimation is feasible [1]. This study aims to solve the integration of these traffic features with the data from the traditional fixed sensors for the traffic state estimation. Accordingly, the above three critical issues of traffic data fusion will be investigated in this study.In this paper, we develop a Progressive Extended Kalman Filter- (PEKF-) based estimator. There are several obvious difference between the work of Nantes et al. [2] and our work. First, we use different data sources, which leads to different measurement equations in state-space model. And it also results in different semantic, spatial, and temporal synchronization of data. Moreover, the assumption of their method is the dependence of each data source, while our proposed method is based on the foreknowledge of the precision of multiple collection approaches. Besides, our study focuses on the application on the freeway while they focus on the urban arterial roads.The organization of the rest of the paper is as follows. Section2 presents a brief analysis on these two-source heterogeneous data. The following Section 3 describes the link-based freeway traffic model. We built up a PEKF estimator in Section 4. Section 5 applied the field data to assess the performance of the proposed approach. Finally, several key conclusions were drawn in Section 6.


## 2. Analysis on the Characteristics of the Multisource Data

### 2.1. The Multisource Data

As mentioned above, the multisource data stem from the wireless communication records and the microwave sensors. Similar to other fixed sensors (i.e., inductive loop detectors), the microwave sensors collect the value of traffic flow and speed at the spot. However, the sparse placement of sensors on the freeway in China makes this technique only achieve the traffic information from a small part of the road network. To some extent, the probe-based techniques have the larger spatial coverage. The wireless communication records of cellphones provide a source to extract traffic state information. In some studies, they used parts of these records; for instance, some used the handoff records to obtain the traffic speeds [22–24]. To achieve more samples and to better mine the potential value of these wireless communication records, this study applied all the records of wireless communication activities from the cellular network to extract traffic information. Mainly, two types of activities are included in the records: (1) system signal, such as location area updates, handoffs, and billing records, and (2) user activities, such as phone calls, text messaging, and data services, such as web browsing, sending, and checking emails [1].The extraction procedure is named as the cellphone activity (CA) data-based method [1]. This method ambiguously locates the on-board cellphones on the freeway according to the geographic information of nearby cell towers. Two traffic features are extracted which are nominated as unique cellphone counts (UCC) and pseudo speed (PS). Compactly, the method includes the following steps. First, find out the cell towers whose signal covers the freeway. Second, project these cell towers on the freeway as the virtual sensor sites. Third, determine the moving direction of a on-board cellphone on the freeway by tracking the change of virtual sensor sites, that is, the cell towers that continuously record the activities of this cellphone. Fourth, the UCC of a freeway is calculated by counting the cellphones in the same direction and recorded by the cell towers on the same freeway link. Fifth, the PS of a cellphone ambiguously equals the distance of two adjacent virtual sensor sites divided by the time between two sequential records. Because of the ambiguous positioning of the on-board cellphones, UCC and PS could not directly equal the traffic state variables. It was found that these traffic features had a link-based characteristic, which meant that some freeway links could generate good data that closely related to the traffic state variables [1]. Besides, it was proven that the physical meaning of UCC changes depended on the traffic state. Accordingly, the relation models were set up as follows [1].( 1 ) The relation model of PS and traffic speed is as follows:(1)mivck=vik+ηivck,where vik (in km/h) denotes the traffic speed at Link i at the time interval k, mivck (in km/h) denotes the measurements of PS at Link i at the time interval k, and ηivck denotes the corresponding PS noise.( 2 ) The relation model of UUC and the traffic state variable (density or flow) is as follows:Since the relationship depended on the traffic state, a conditional function was developed [1]: (2)mick=θk·qik+ηicqkwhenmivck≥vcrφ·ρik+ηicρkwhenmivck<vcr,where qik (in veh/h) denotes the traffic flow at Link i during the time interval k.ρ i k (in veh/km/h) denotes the average traffic density of Link i during the time interval k, mick denotes the measurements of UCC during the time interval k, ηicqk and ηicρk denote the corresponding UCC noise, θ(k) is a linear coefficient which is a time-dependent variate, φ is a linear coefficient which is a constant, and vcr (in km/h) is the critical speed to identify the traffic state.The later six parameters will be discussed in Section5.


### 2.2. Characteristics of the Heterogeneous Data

In this study, it is essential to identify the differences between the heterogeneous multisource data to guide the fusion research. Obviously, the multisource data in this study differ in semantic, temporal, and spatial coverage. The microwave sensors provide the spot measurements while the traffic features of CA data-based method are the link-average values. Besides, they exhibit different accuracy. To explore their different in the precision, we use the field data collected from the Jiangsu freeway. The comparison between the speed measurements from these two collection methods is shown in Figure1. It can be inferred from the figure that under free-flow condition (the speed is over 50 km/h), the PSs from the wireless communication records are not stable which fluctuate over a large range. Figure 1(a) shows that the PSs are all below 50 km/h from 11:00 to 24:00 on Link 1 and from 16:40 to 24:00 on Link 19, but the spot speeds from the microwave sensors show that the traffic switches between the free-flow state and the low-speed state. The incident/accident reports from the freeway operation center indicated that the traffic speeds were slow during that time period. It proves that the PSs are more accurate under the low-speed state compared with the speeds from microwave sensors. The possible cause for the instability of the microwave measurements is that the speed is collected on the spot which could not reflect the average speed of a link. There will be stop-and-go phenomenon especially under the low-speed or congested state.Figure 1
Speed measurements from multisensors: (a) on Link 1; (b) on Link 19.
(a) (b)Based on the comparison, we make the following inferences to promote the construction of the estimator. First, measurements from the microwave sensors are more stable and accurate than the traffic features from wireless communication records under the free-flow condition. Second, under the low-speed or congested condition, PS is regarded as the accurate speed measurements. Obviously, it is important to identify the traffic condition when both data sources are available but conflict. According to the inferences, PS is more reliable under the low-speed or congested condition. To avoid the impact of the biased PS under the free-flow condition, we set the following rules to identify the traffic condition when speeds from two-source conflict.The condition is low-speed or congested if one of the following conditions is met:(i)
mivck<vcrandmivck-1<vcrbutmivmk≥vcr.(ii)
mivck<vcrandmivck+1<vcrbutmivmk≥vcr.(iii)
mivmk<vcrbutmivck≥vcr.Otherwise, it is the free-flow condition, wheremivmk (in km/h) denotes the speed measurement (the time mean speed) from the microwave sensors on Link i at the time interval k.


## 2.1. The Multisource Data

As mentioned above, the multisource data stem from the wireless communication records and the microwave sensors. Similar to other fixed sensors (i.e., inductive loop detectors), the microwave sensors collect the value of traffic flow and speed at the spot. However, the sparse placement of sensors on the freeway in China makes this technique only achieve the traffic information from a small part of the road network. To some extent, the probe-based techniques have the larger spatial coverage. The wireless communication records of cellphones provide a source to extract traffic state information. In some studies, they used parts of these records; for instance, some used the handoff records to obtain the traffic speeds [22–24]. To achieve more samples and to better mine the potential value of these wireless communication records, this study applied all the records of wireless communication activities from the cellular network to extract traffic information. Mainly, two types of activities are included in the records: (1) system signal, such as location area updates, handoffs, and billing records, and (2) user activities, such as phone calls, text messaging, and data services, such as web browsing, sending, and checking emails [1].The extraction procedure is named as the cellphone activity (CA) data-based method [1]. This method ambiguously locates the on-board cellphones on the freeway according to the geographic information of nearby cell towers. Two traffic features are extracted which are nominated as unique cellphone counts (UCC) and pseudo speed (PS). Compactly, the method includes the following steps. First, find out the cell towers whose signal covers the freeway. Second, project these cell towers on the freeway as the virtual sensor sites. Third, determine the moving direction of a on-board cellphone on the freeway by tracking the change of virtual sensor sites, that is, the cell towers that continuously record the activities of this cellphone. Fourth, the UCC of a freeway is calculated by counting the cellphones in the same direction and recorded by the cell towers on the same freeway link. Fifth, the PS of a cellphone ambiguously equals the distance of two adjacent virtual sensor sites divided by the time between two sequential records. Because of the ambiguous positioning of the on-board cellphones, UCC and PS could not directly equal the traffic state variables. It was found that these traffic features had a link-based characteristic, which meant that some freeway links could generate good data that closely related to the traffic state variables [1]. Besides, it was proven that the physical meaning of UCC changes depended on the traffic state. Accordingly, the relation models were set up as follows [1].( 1 ) The relation model of PS and traffic speed is as follows:(1)mivck=vik+ηivck,where vik (in km/h) denotes the traffic speed at Link i at the time interval k, mivck (in km/h) denotes the measurements of PS at Link i at the time interval k, and ηivck denotes the corresponding PS noise.( 2 ) The relation model of UUC and the traffic state variable (density or flow) is as follows:Since the relationship depended on the traffic state, a conditional function was developed [1]: (2)mick=θk·qik+ηicqkwhenmivck≥vcrφ·ρik+ηicρkwhenmivck<vcr,where qik (in veh/h) denotes the traffic flow at Link i during the time interval k.ρ i k (in veh/km/h) denotes the average traffic density of Link i during the time interval k, mick denotes the measurements of UCC during the time interval k, ηicqk and ηicρk denote the corresponding UCC noise, θ(k) is a linear coefficient which is a time-dependent variate, φ is a linear coefficient which is a constant, and vcr (in km/h) is the critical speed to identify the traffic state.The later six parameters will be discussed in Section5.


## 2.2. Characteristics of the Heterogeneous Data

In this study, it is essential to identify the differences between the heterogeneous multisource data to guide the fusion research. Obviously, the multisource data in this study differ in semantic, temporal, and spatial coverage. The microwave sensors provide the spot measurements while the traffic features of CA data-based method are the link-average values. Besides, they exhibit different accuracy. To explore their different in the precision, we use the field data collected from the Jiangsu freeway. The comparison between the speed measurements from these two collection methods is shown in Figure1. It can be inferred from the figure that under free-flow condition (the speed is over 50 km/h), the PSs from the wireless communication records are not stable which fluctuate over a large range. Figure 1(a) shows that the PSs are all below 50 km/h from 11:00 to 24:00 on Link 1 and from 16:40 to 24:00 on Link 19, but the spot speeds from the microwave sensors show that the traffic switches between the free-flow state and the low-speed state. The incident/accident reports from the freeway operation center indicated that the traffic speeds were slow during that time period. It proves that the PSs are more accurate under the low-speed state compared with the speeds from microwave sensors. The possible cause for the instability of the microwave measurements is that the speed is collected on the spot which could not reflect the average speed of a link. There will be stop-and-go phenomenon especially under the low-speed or congested state.Figure 1
Speed measurements from multisensors: (a) on Link 1; (b) on Link 19.
(a) (b)Based on the comparison, we make the following inferences to promote the construction of the estimator. First, measurements from the microwave sensors are more stable and accurate than the traffic features from wireless communication records under the free-flow condition. Second, under the low-speed or congested condition, PS is regarded as the accurate speed measurements. Obviously, it is important to identify the traffic condition when both data sources are available but conflict. According to the inferences, PS is more reliable under the low-speed or congested condition. To avoid the impact of the biased PS under the free-flow condition, we set the following rules to identify the traffic condition when speeds from two-source conflict.The condition is low-speed or congested if one of the following conditions is met:(i)
mivck<vcrandmivck-1<vcrbutmivmk≥vcr.(ii)
mivck<vcrandmivck+1<vcrbutmivmk≥vcr.(iii)
mivmk<vcrbutmivck≥vcr.Otherwise, it is the free-flow condition, wheremivmk (in km/h) denotes the speed measurement (the time mean speed) from the microwave sensors on Link i at the time interval k.


## 3. Freeway Traffic Flow Modelling with Multisource Data

The models include a macroscopic traffic flow model and a measurement model. The macroscopic traffic flow model describes the relationship between traffic variables including speed, density, and flow. And the measurement model shows the relationship between the heterogeneous measurements and traffic variables.


### 3.1. Macroscopic Freeway Traffic Flow Model

This study utilized one of the validated macroscopic traffic flow models, that is, the METANET model [3]. The macroscopic model is discretized in space and time. As shown in Figure 2, a freeway stretch in a travel direction is divided into N links with length Li,  i=1,…,N. Time is discretized in an equal time step T.Figure 2
A freeway stretch, link division, sensor location, and cellular signal coverage [1].For the spatial synchronization of multisource measurements, the division of the freeway stretch follows some rules. First, if there is a microwave sensor on a link, it is better to locate in the middle of the link (or within the link), which is different from some works that put the fixed detector at the boundary of a link [3, 6]. Second, as an advantage of CA data-based method, any length can be used as long as one or more cell towers vertically projected to a link and the signal from this or these cell towers could cover this link. Other rules including the geometric difference (e.g., lane drops, on/off-ramps) could be applied, and it is optimal to have at most one on- or off- ramp within a link [3]. In this study, we applied a uniform length of 1 kilometer as a link length shown in the following test section.Based on the divided freeway link and time step, the applied second-order macroscopic traffic flow model for Linki  (i is the number of the link) includes the following four equations.( 1 ) Conservation equation is(3)ρik+1=ρik+TLiqi-1k-qik+iik-oik.( 2 ) Dynamic speed equation is (4)vik+1=vik+Tτveρik-vik+TLivikvi-1k-vik-νTτLiρi+1k-ρikρik+κ+ξivk.( 3 ) Stationary speed equation is(5)veρik=vfexp⁡12ρikρcr2.( 4 ) Flow equation is(6)qik=ρik·vik+ξiqk,where T (in hour) is the discretized time step, Li (in km) is the length of Link i, ρi(k) (in veh/km/lane) is the traffic density of the Link i at the time interval k, qi(k) (in veh/h) is the traffic flow of the Link i at the time interval k, ii(k) (in veh/h) is the ramp inflow of the Link i at the time interval k, oi(k) (in veh/h) is the ramp outflow of the Link i at the time interval k, vi(k) (in km/h) is the space mean speed of the Link i at the time interval k, τ, ν, δ, and κ are model parameters, and all links share the same value of these parameters, ξiv(k) is zero-mean Gaussian white noise in the speed equation, veρi(k) (in km/h) is the average corresponding speed according to density, vf (in km/h) is the free low speed, ρcr (in veh/km/h) is the critical density, and ξiqk denotes zero-mean Gaussian white noise in the flow equation.


### 3.2. Measurement Model and Date Synchronization

According to the characteristics of the multisource measurements, a set of measurement equations are established as follows.


#### 3.2.1. Cellphone Density Measurement Equation

The conditional function (2) shows the relationship between UCC and the traffic state variables (flow and density) under different traffic conditions. Based on the relationship among flow, density, and speed, this relationship model could be transformed as follows:(7)miρck=mickθk·mivckmivck≥vcrmickφmivck<vcr,where miρck (in veh/km/h) denotes the transferred density from UCC at Link i at the time intervalk.Afterwards, the cellphone density measurement model is set up as(8)miρck=ρik+ηiρck,where ηiρc(k) is the measurement error of UCC.


#### 3.2.2. Cellphone Speed Measurement Equation

The speed measurement equation could directly apply (1) in Section 2.


#### 3.2.3. Speed Measurement Equation of Microwave Sensor

Speed measurements from the microwave sensors are average spot speeds, that is, the time mean speed. It is known that there is the difference between the time mean speed and space mean speed. In the model-based estimation, it is the space mean speed required. Rakha and Zhang [25] formulated the conversion between these two kinds of speeds. This formulation works when the variance of time mean speeds is known. However, it is difficult to achieve this variance some times. According to the analysis in Section 2, it is found that the speed measurements from the microwave sensors are more stable and reliable than the PS under the free-flow condition, because the PS are oscillating under the low-speed condition. Therefore, this study proposed the following conditional function using the two-source speeds as follows.Ifmivmk conflicts with mivck under the low-speed or congested condition, then (9)mivmk=mivck=vik+ηivck.Otherwise,(10)mivmk=vik+ηivmk,where ηivm(k) is the noise of speed measurements from the microwave sensor.


#### 3.2.4. Flow Measurement Equation of Microwave Sensor

Consider a microwave sensor installed in the middle Linki, as illustrated in Figure 2. For the flow measurement, we have(11)miqmk×60TI=qik+ηiqmk,where TI (in min) is the time interval to collect the flow measurements.m i q m k (in veh/TI) is the flow measurement from the microwave sensor located at Link i during the time interval k.η i q m ( k ) denotes the corresponding flow measurement noise.Considering the relation in flow equation (6), the traffic density can be calculated from the measurements of flow and speed. Accordingly, (11) can be transferred as follows, except for the measurement q0.(12)miqmk×60/TImivmk=ρik+ξiqk+ζiqmkmivmk.


## 3.1. Macroscopic Freeway Traffic Flow Model

This study utilized one of the validated macroscopic traffic flow models, that is, the METANET model [3]. The macroscopic model is discretized in space and time. As shown in Figure 2, a freeway stretch in a travel direction is divided into N links with length Li,  i=1,…,N. Time is discretized in an equal time step T.Figure 2
A freeway stretch, link division, sensor location, and cellular signal coverage [1].For the spatial synchronization of multisource measurements, the division of the freeway stretch follows some rules. First, if there is a microwave sensor on a link, it is better to locate in the middle of the link (or within the link), which is different from some works that put the fixed detector at the boundary of a link [3, 6]. Second, as an advantage of CA data-based method, any length can be used as long as one or more cell towers vertically projected to a link and the signal from this or these cell towers could cover this link. Other rules including the geometric difference (e.g., lane drops, on/off-ramps) could be applied, and it is optimal to have at most one on- or off- ramp within a link [3]. In this study, we applied a uniform length of 1 kilometer as a link length shown in the following test section.Based on the divided freeway link and time step, the applied second-order macroscopic traffic flow model for Linki  (i is the number of the link) includes the following four equations.( 1 ) Conservation equation is(3)ρik+1=ρik+TLiqi-1k-qik+iik-oik.( 2 ) Dynamic speed equation is (4)vik+1=vik+Tτveρik-vik+TLivikvi-1k-vik-νTτLiρi+1k-ρikρik+κ+ξivk.( 3 ) Stationary speed equation is(5)veρik=vfexp⁡12ρikρcr2.( 4 ) Flow equation is(6)qik=ρik·vik+ξiqk,where T (in hour) is the discretized time step, Li (in km) is the length of Link i, ρi(k) (in veh/km/lane) is the traffic density of the Link i at the time interval k, qi(k) (in veh/h) is the traffic flow of the Link i at the time interval k, ii(k) (in veh/h) is the ramp inflow of the Link i at the time interval k, oi(k) (in veh/h) is the ramp outflow of the Link i at the time interval k, vi(k) (in km/h) is the space mean speed of the Link i at the time interval k, τ, ν, δ, and κ are model parameters, and all links share the same value of these parameters, ξiv(k) is zero-mean Gaussian white noise in the speed equation, veρi(k) (in km/h) is the average corresponding speed according to density, vf (in km/h) is the free low speed, ρcr (in veh/km/h) is the critical density, and ξiqk denotes zero-mean Gaussian white noise in the flow equation.


## 3.2. Measurement Model and Date Synchronization

According to the characteristics of the multisource measurements, a set of measurement equations are established as follows.


### 3.2.1. Cellphone Density Measurement Equation

The conditional function (2) shows the relationship between UCC and the traffic state variables (flow and density) under different traffic conditions. Based on the relationship among flow, density, and speed, this relationship model could be transformed as follows:(7)miρck=mickθk·mivckmivck≥vcrmickφmivck<vcr,where miρck (in veh/km/h) denotes the transferred density from UCC at Link i at the time intervalk.Afterwards, the cellphone density measurement model is set up as(8)miρck=ρik+ηiρck,where ηiρc(k) is the measurement error of UCC.


### 3.2.2. Cellphone Speed Measurement Equation

The speed measurement equation could directly apply (1) in Section 2.


### 3.2.3. Speed Measurement Equation of Microwave Sensor

Speed measurements from the microwave sensors are average spot speeds, that is, the time mean speed. It is known that there is the difference between the time mean speed and space mean speed. In the model-based estimation, it is the space mean speed required. Rakha and Zhang [25] formulated the conversion between these two kinds of speeds. This formulation works when the variance of time mean speeds is known. However, it is difficult to achieve this variance some times. According to the analysis in Section 2, it is found that the speed measurements from the microwave sensors are more stable and reliable than the PS under the free-flow condition, because the PS are oscillating under the low-speed condition. Therefore, this study proposed the following conditional function using the two-source speeds as follows.Ifmivmk conflicts with mivck under the low-speed or congested condition, then (9)mivmk=mivck=vik+ηivck.Otherwise,(10)mivmk=vik+ηivmk,where ηivm(k) is the noise of speed measurements from the microwave sensor.


### 3.2.4. Flow Measurement Equation of Microwave Sensor

Consider a microwave sensor installed in the middle Linki, as illustrated in Figure 2. For the flow measurement, we have(11)miqmk×60TI=qik+ηiqmk,where TI (in min) is the time interval to collect the flow measurements.m i q m k (in veh/TI) is the flow measurement from the microwave sensor located at Link i during the time interval k.η i q m ( k ) denotes the corresponding flow measurement noise.Considering the relation in flow equation (6), the traffic density can be calculated from the measurements of flow and speed. Accordingly, (11) can be transferred as follows, except for the measurement q0.(12)miqmk×60/TImivmk=ρik+ξiqk+ζiqmkmivmk.


## 3.2.1. Cellphone Density Measurement Equation

The conditional function (2) shows the relationship between UCC and the traffic state variables (flow and density) under different traffic conditions. Based on the relationship among flow, density, and speed, this relationship model could be transformed as follows:(7)miρck=mickθk·mivckmivck≥vcrmickφmivck<vcr,where miρck (in veh/km/h) denotes the transferred density from UCC at Link i at the time intervalk.Afterwards, the cellphone density measurement model is set up as(8)miρck=ρik+ηiρck,where ηiρc(k) is the measurement error of UCC.


## 3.2.2. Cellphone Speed Measurement Equation

The speed measurement equation could directly apply (1) in Section 2.


## 3.2.3. Speed Measurement Equation of Microwave Sensor

Speed measurements from the microwave sensors are average spot speeds, that is, the time mean speed. It is known that there is the difference between the time mean speed and space mean speed. In the model-based estimation, it is the space mean speed required. Rakha and Zhang [25] formulated the conversion between these two kinds of speeds. This formulation works when the variance of time mean speeds is known. However, it is difficult to achieve this variance some times. According to the analysis in Section 2, it is found that the speed measurements from the microwave sensors are more stable and reliable than the PS under the free-flow condition, because the PS are oscillating under the low-speed condition. Therefore, this study proposed the following conditional function using the two-source speeds as follows.Ifmivmk conflicts with mivck under the low-speed or congested condition, then (9)mivmk=mivck=vik+ηivck.Otherwise,(10)mivmk=vik+ηivmk,where ηivm(k) is the noise of speed measurements from the microwave sensor.


## 3.2.4. Flow Measurement Equation of Microwave Sensor

Consider a microwave sensor installed in the middle Linki, as illustrated in Figure 2. For the flow measurement, we have(11)miqmk×60TI=qik+ηiqmk,where TI (in min) is the time interval to collect the flow measurements.m i q m k (in veh/TI) is the flow measurement from the microwave sensor located at Link i during the time interval k.η i q m ( k ) denotes the corresponding flow measurement noise.Considering the relation in flow equation (6), the traffic density can be calculated from the measurements of flow and speed. Accordingly, (11) can be transferred as follows, except for the measurement q0.(12)miqmk×60/TImivmk=ρik+ξiqk+ζiqmkmivmk.


## 4. Design of PEKF-Based Estimator

### 4.1. State-Space Model

Based on the traffic flow model and measurement model mentioned above, a state-space model is set up with the state vectorsx,u, ξ, and ηc, ηm defined as follows:x=ρ1v1ρ2v2⋯ρNvNT denotes the density-speed state vector.u=q0v0ρN+1i1⋯iNo1⋯oNT denotes the boundary state vector.ξ=ξ1qξ1v⋯ξNqξNvT denotes the vector of the process noise.ηc=η1ρcη1vc⋯ηNρcηNvcT denotes the vector of the measurement noise from CA data.ηm=η1qmη1vm⋯ηNqmηNvmT denotes the vector of the measurement noise from microwave sensors.Then the macroscopic flow model and measurement model are rewritten to a compact state-space form including a process function (13) and two measurement functions (14) and (15) as follows:(13)xk=fxk-1,uk,ξk-1(14)zck=hcxk,ηck(15)zmk=hmxk,ηmk,where the process function relates to (3), (4), (5), and (6). The measurement functions relate to (8), (1), (10), and (12). f, hc, and hm are nonlinear differential vector functions. The vector zc consists of all available measurements from CA data and zm contains the measurements from the microwave sensors. In the application, the measurements will be transferred via function (7) and (10). The random variables ξk, ηck, and ηmk represent the process and measurement noise, respectively. k is the number of time steps.


### 4.2. The PEKF Estimator for Multisource Data

The state-space model contains the nonlinear equations, and, thus, this study set up the estimator based on the Extended Kalman Filter (EKF) technique. As the exploration in the previous sections, the relationship between the multisource measurements and the traffic state variables highly relies on the judgment of the traffic condition. Therefore, we make some adjustments to the traditional EKF estimator, and the updated estimator is nominated as the progressive EKF (PEKF) estimator. The operation diagram of PEKF estimator is shown in Figure3. The adjustments are mainly shown in the measurement update step. The purposes of these adjustments are to integrate the multisource data from the wireless communication records and microwave sensors in this study. As shown in Figure 3, the adjustments include the following contents. First, speeds from the microwave sensors (named as m-speed in the following description) and PSs are used to identify the traffic condition as shown in Section 2. Second, based on the identified traffic condition, the UCC are transformed to the corresponding traffic state variables via (7). Consequently, the speed measurement model can be decided via (10). Similar to the traditional EKF estimator, the partial derivatives of the process function and measurement function, transformed using the Jacobian matrix, are used to linearize the model [23, 26]. The proposed estimator includes the following steps.Figure 3
The operation of the Progressive Extended Kalman Filter estimator.Step  0(initialization). Set time interval k=1, and let E[x0]=x(0)^, and Ex(0)-x(0)^2=p(0), where the initial state x0 and covariance p(0) are determined by the historical data.Step  1(time update). State estimate extrapolation is as follows: x(k)^-=fxk-1^,uk,0Error covariance extrapolation is(16)Pk-=Ak·Pk-1·AkT+Wk·Qk-1·WkT,where P(k) is the error covariance at the time step k.Step  2(measurement update)Step  2.1. Identify the traffic state based on the rules in Section 2.2.Step  2.2. Transform UCC to traffic density according to the traffic state using the conditional (7).Step  2.3. Make a decision on the speed measurement model for the microwave sensor according to the traffic state by the conditional (10).Step  2.4. Update the input measurement matrix h.Step  2.5. Measurement update using (14) and (15) is as follows.Kalman gain calculation is(17)Kk=Pk-·HkT·Hk·Pk-·HkT+Vk·Rk·VkT-1State estimate update is as follows:x(k)^=x(k)^-+K(k)·zk-hxk^-,0, where z could be zc or zm and h could be hc or hm depending on which source data is using.Error covariance update is as follows:Pk=(I-K(k)·Hk)·Pk-.Step  3.Let k=k+1 and go back to Step 1 until the preset time periods end, where I is the identity matrix.A and W are the Jacobian matrix of partial derivatives of f with respect to x and ξ, respectively: (18)Ak=∂f∂xxk-1^,uk,0Wk=∂f∂ξxk-1^,uk,0.H and V are the Jacobian matrix of partial derivatives of h with respect tox and η. h is the updated measurement matrix stemming from the multiple sources. η is the corresponding noise matrix compounding ηc and ηm:(19)Hk=∂h∂xxk^,0Vk=∂h∂ηxk^,0.


## 4.1. State-Space Model

Based on the traffic flow model and measurement model mentioned above, a state-space model is set up with the state vectorsx,u, ξ, and ηc, ηm defined as follows:x=ρ1v1ρ2v2⋯ρNvNT denotes the density-speed state vector.u=q0v0ρN+1i1⋯iNo1⋯oNT denotes the boundary state vector.ξ=ξ1qξ1v⋯ξNqξNvT denotes the vector of the process noise.ηc=η1ρcη1vc⋯ηNρcηNvcT denotes the vector of the measurement noise from CA data.ηm=η1qmη1vm⋯ηNqmηNvmT denotes the vector of the measurement noise from microwave sensors.Then the macroscopic flow model and measurement model are rewritten to a compact state-space form including a process function (13) and two measurement functions (14) and (15) as follows:(13)xk=fxk-1,uk,ξk-1(14)zck=hcxk,ηck(15)zmk=hmxk,ηmk,where the process function relates to (3), (4), (5), and (6). The measurement functions relate to (8), (1), (10), and (12). f, hc, and hm are nonlinear differential vector functions. The vector zc consists of all available measurements from CA data and zm contains the measurements from the microwave sensors. In the application, the measurements will be transferred via function (7) and (10). The random variables ξk, ηck, and ηmk represent the process and measurement noise, respectively. k is the number of time steps.


## 4.2. The PEKF Estimator for Multisource Data

The state-space model contains the nonlinear equations, and, thus, this study set up the estimator based on the Extended Kalman Filter (EKF) technique. As the exploration in the previous sections, the relationship between the multisource measurements and the traffic state variables highly relies on the judgment of the traffic condition. Therefore, we make some adjustments to the traditional EKF estimator, and the updated estimator is nominated as the progressive EKF (PEKF) estimator. The operation diagram of PEKF estimator is shown in Figure3. The adjustments are mainly shown in the measurement update step. The purposes of these adjustments are to integrate the multisource data from the wireless communication records and microwave sensors in this study. As shown in Figure 3, the adjustments include the following contents. First, speeds from the microwave sensors (named as m-speed in the following description) and PSs are used to identify the traffic condition as shown in Section 2. Second, based on the identified traffic condition, the UCC are transformed to the corresponding traffic state variables via (7). Consequently, the speed measurement model can be decided via (10). Similar to the traditional EKF estimator, the partial derivatives of the process function and measurement function, transformed using the Jacobian matrix, are used to linearize the model [23, 26]. The proposed estimator includes the following steps.Figure 3
The operation of the Progressive Extended Kalman Filter estimator.Step  0(initialization). Set time interval k=1, and let E[x0]=x(0)^, and Ex(0)-x(0)^2=p(0), where the initial state x0 and covariance p(0) are determined by the historical data.Step  1(time update). State estimate extrapolation is as follows: x(k)^-=fxk-1^,uk,0Error covariance extrapolation is(16)Pk-=Ak·Pk-1·AkT+Wk·Qk-1·WkT,where P(k) is the error covariance at the time step k.Step  2(measurement update)Step  2.1. Identify the traffic state based on the rules in Section 2.2.Step  2.2. Transform UCC to traffic density according to the traffic state using the conditional (7).Step  2.3. Make a decision on the speed measurement model for the microwave sensor according to the traffic state by the conditional (10).Step  2.4. Update the input measurement matrix h.Step  2.5. Measurement update using (14) and (15) is as follows.Kalman gain calculation is(17)Kk=Pk-·HkT·Hk·Pk-·HkT+Vk·Rk·VkT-1State estimate update is as follows:x(k)^=x(k)^-+K(k)·zk-hxk^-,0, where z could be zc or zm and h could be hc or hm depending on which source data is using.Error covariance update is as follows:Pk=(I-K(k)·Hk)·Pk-.Step  3.Let k=k+1 and go back to Step 1 until the preset time periods end, where I is the identity matrix.A and W are the Jacobian matrix of partial derivatives of f with respect to x and ξ, respectively: (18)Ak=∂f∂xxk-1^,uk,0Wk=∂f∂ξxk-1^,uk,0.H and V are the Jacobian matrix of partial derivatives of h with respect tox and η. h is the updated measurement matrix stemming from the multiple sources. η is the corresponding noise matrix compounding ηc and ηm:(19)Hk=∂h∂xxk^,0Vk=∂h∂ηxk^,0.


## 5. Test and Results

### 5.1. A Test Freeway Segment and Data Collection

#### 5.1.1. Freeway Test Bed

The test bed is a 19 km, four-lane northbound segment of Ning-Hu freeway connecting Wuxi and Suzhou, China, which is part of the busiest freeway in China, as shown in Figure4. Two microwave sensors (M1 and M19) are located near the ends of the segment. According to the freeway-division rules, the segment is divided into 19 links with uniform lengths of 1 km each. The links are labelled Link 1 to Link 19 from north to south opposite to the traveling direction. Microwave sensors are within Link 1 and Link 19, respectively. The segment is almost geometrically homogeneous without any lane drops.Figure 4
The test bed, part of Ning-Hu freeway, and a northbound segment.Note. The background photo is from Google Earth.


#### 5.1.2. Data Collection

The multisource data for the field test was provided by the Jiangsu Freeway Operation Center and was collected on September 30, 2014. Both collection methods (i.e., the microwave sensors and the CA data-based method) integrated their measurements by a time interval of ten minutes starting from 00:00, September 30, 2014. The spatial coverage of the microwave sensor is limited to Link 1 and Link 19, while the cellular signal covers all links.In addition to these two data sources, there is a traffic incident/accident report which could be used in the following evaluation of the estimation results. The report indicates that an incident occurred on Link 4 at about 11:12 AM. The traffic was also quite heavy. These two traffic phenomena together lead to the traffic moving slowly from Link 1 to Link 10 as well as the nearby affected links. The report also shows that the traffic moved slowly from 10:00 to the late evening of that day due to the heavy traffic. The cause is the China National Day Holiday which begins from October 1 and the free tolling policy on freeway that works during this holiday, and, thus, it attracts tremendous traffic. Therefore, the traffic flow climbs high when it is closing to the midnight of the test day. Consequently, the high traffic flow triggered the speed drop and the large density.


### 5.2. Model Parameters

#### 5.2.1. Parameters about Time

The time stepT of the estimator is 10 seconds to meet the requirement that T<L/vf=1km/100(km/h)=36seconds. The time interval for measurement update is 10 minutes. The estimator works when the measurements are available. It means that the time update and measurement update will work 60 steps (10 min/10 sec) using the same measurements.


#### 5.2.2. Parameters about Links and Boundaries

Although there are 19 links, the measurements from the microwave sensors are only available on two links (Link 1 and Link 19). Our recent work found that the traffic features (UCC and PS) extracted from the wireless communication records on Link 6, Link 10, Link 14, and Link 18 had the closest relation with the traffic variables [1]. This study uses the UCC and PS from these links for the sake of high accuracy. Besides, the PSs on Link 1 and Link 19 are also used under the low-speed or congested condition, because these PSs are more reliable than the microwave speeds under this condition as discussed in Section 2. To avoid the impact of inaccurate boundary inputs, Link 1 and Link 19 are regarded as the boundary links, and, thus, the measurements on these links are used as the boundary variables.


#### 5.2.3. Parameters in the Macroscopic Traffic Flow Model

The empirical parameters are set as Table1. And the models of all links shared the same values. This study uses the value suggested in the work of Wang et al. [6] as the process noise.Table 1
Parameter settings.
Parameter Value τ 0.02 h κ 40 veh/km v f 100 km/h ν 24 km2/h ρ c r 120 veh/km


#### 5.2.4. Parameters in the Measurement Model

The coefficients in the measurement model for the UCC were calibrated using the historical data as follows [1]. The constant parameter φ approximately equals 8.5 for all links (Link 6, Link 10, Link 14, and Link 18). The time-varying parameter θ(k) is valued as follows: from the time 00:00 to 05:00, it approximately equals 0.3; from 05:00 to 07:30, it gradually increases from 0.3 to 1.0; from 07:30 to the rest of a day, it approximately equals a constant 1.0 under the free-flow condition. Another critical parameter in the measurement update step is the critical speed for the identification of traffic condition. It is valued empirically as 50 km/h.The measurement noise should be the standard deviation of measurements from cellphone activity data and microwave sensors. However, it is difficult to value the noise when there is no sufficient ground-truth data. Some existing studies indicate that the Extended Kalman Filter was found to have a slight sensitivity to the standard deviation-based measurement noise, which guaranteed the estimator’s performance even with poor foreknowledge of model noise [3, 7]. Therefore, the deviation was set as a small value; that is, the density deviation was 1 veh/km and the speed deviation was 1 km/h for both sources.


### 5.3. Estimation Results

Figure5 shows the estimation results via the proposed PEKF-based method. The gradual changes in the color represent the variations in the traffic state. The blue represents the low-density and high-speed condition, while the red indicates the high-density and low-speed condition. As shown in Figure 5, both the estimated speeds and densities are changing among the free-flow, dense, and congested conditions. For instance, the speeds of the entire freeway segment during 00:00 to 10:00 AM are rarely red as shown in Figure 5(b). With the color getting red, it means that the traffic slows down. When the color is dark red, the traffic speed is quite low which indicates a congestion. In Figure 5(a), the density is lower than 40 veh/km from 00:00 to 7:00 showing in the blue. It is reasonable that there is little traffic on the freeway at the midnight of a working day and the traffic increases in the day time.Figure 5
The estimation results. (a) The density estimates. (b) The speed estimates.
(a) (b)On the other hand, the incident/accident report data are used to qualitatively evaluate the accuracy of results for lacking the “ground-truth” speeds and density. Figure5(b) shows that the speed drops firstly on Link 1 and Link 2 at about 11:00 AM. Consequently, this phenomenon happens on Link 3 and Link 4 quickly. And speeds on Link 5 and Link 6 also decrease around that time. The variation of speed indicates that the estimation results exhibit the influence of traffic incidents. However, the reported incident happened on Link 4 at about 11:00, but the estimation results on Link 4 have a small delay. Since the measurements are available only on Link 1 and Link 6 and the distance between two links is 6 km long, it is understandable for such a small delay. In Figure 5(a), the density on Link 1, Link 2, and Link 3 decreases when the incident happens. Since Link 1, Link 2, and Link 3 are downstream links of Link 4, this decline phenomenon reflects the propagation of a shock wave. Another incident record is about the heavy traffic later that day which induces the low speed. From Figure 5(b), we can see the large area of red later that day which validates the correctness of the estimation results. And the changing color in Figure 5(a) shows the variation of density which reflects the oscillation of traffic under the low-speed traffic condition. Roughly, the estimation results efficiently integrated the measurements from the multiple sources and exhibit the dynamic and real-time traffic state.


### 5.4. Evaluation

An evaluation of the proposed PEKF-based estimator is conducted by making a comparison between the results from multisource data and single-source data. We applied the estimation results from the previous work [1], which estimated the traffic state using the wireless communication data and EKF-based data assimilation method. Since the distance between the microwave sensors is extremely sparse, the data generated from these sensors is not qualified for the application of a model-based approach. Hence, the direct measurements from microwave sensors are used for the comparison. To make a precise comparison, the absolute difference (AD) is chosen as the performance measure. It can be calculated by the following equation:(20)ADk=xsk-xfk,where ADk is the absolute difference at the time interval k, xsk is the state estimates (density or speed) from single data source at the time interval k, xfk is the state estimates (density or speed) from multiple sources at the time interval k.The iteration process of the progressive EKF-based estimator shows that the measurements from microwave sensors are usually taken as the more precise value compared with the traffic features from the wireless communication records. ADs between estimates from multisources and wireless communication data and ADs between estimates from multisources and measurements from microwave sensors in different links are present in Figure6. Obviously, the progressive EKF-based method generated estimates which are more close to the more accurate measurements from microwave sensors. Generally, it will improve the total precision of estimation by fusing multisource data.Figure 6
The AD results. (a) The ADs of density on Link 1. (b) The ADs of speed on Link 1. (c) The ADs of density on Link 19. (d) The ADs of speed on Link 19.
(a) (b) (c) (d)


## 5.1. A Test Freeway Segment and Data Collection

### 5.1.1. Freeway Test Bed

The test bed is a 19 km, four-lane northbound segment of Ning-Hu freeway connecting Wuxi and Suzhou, China, which is part of the busiest freeway in China, as shown in Figure4. Two microwave sensors (M1 and M19) are located near the ends of the segment. According to the freeway-division rules, the segment is divided into 19 links with uniform lengths of 1 km each. The links are labelled Link 1 to Link 19 from north to south opposite to the traveling direction. Microwave sensors are within Link 1 and Link 19, respectively. The segment is almost geometrically homogeneous without any lane drops.Figure 4
The test bed, part of Ning-Hu freeway, and a northbound segment.Note. The background photo is from Google Earth.


### 5.1.2. Data Collection

The multisource data for the field test was provided by the Jiangsu Freeway Operation Center and was collected on September 30, 2014. Both collection methods (i.e., the microwave sensors and the CA data-based method) integrated their measurements by a time interval of ten minutes starting from 00:00, September 30, 2014. The spatial coverage of the microwave sensor is limited to Link 1 and Link 19, while the cellular signal covers all links.In addition to these two data sources, there is a traffic incident/accident report which could be used in the following evaluation of the estimation results. The report indicates that an incident occurred on Link 4 at about 11:12 AM. The traffic was also quite heavy. These two traffic phenomena together lead to the traffic moving slowly from Link 1 to Link 10 as well as the nearby affected links. The report also shows that the traffic moved slowly from 10:00 to the late evening of that day due to the heavy traffic. The cause is the China National Day Holiday which begins from October 1 and the free tolling policy on freeway that works during this holiday, and, thus, it attracts tremendous traffic. Therefore, the traffic flow climbs high when it is closing to the midnight of the test day. Consequently, the high traffic flow triggered the speed drop and the large density.


## 5.1.1. Freeway Test Bed

The test bed is a 19 km, four-lane northbound segment of Ning-Hu freeway connecting Wuxi and Suzhou, China, which is part of the busiest freeway in China, as shown in Figure4. Two microwave sensors (M1 and M19) are located near the ends of the segment. According to the freeway-division rules, the segment is divided into 19 links with uniform lengths of 1 km each. The links are labelled Link 1 to Link 19 from north to south opposite to the traveling direction. Microwave sensors are within Link 1 and Link 19, respectively. The segment is almost geometrically homogeneous without any lane drops.Figure 4
The test bed, part of Ning-Hu freeway, and a northbound segment.Note. The background photo is from Google Earth.


## 5.1.2. Data Collection

The multisource data for the field test was provided by the Jiangsu Freeway Operation Center and was collected on September 30, 2014. Both collection methods (i.e., the microwave sensors and the CA data-based method) integrated their measurements by a time interval of ten minutes starting from 00:00, September 30, 2014. The spatial coverage of the microwave sensor is limited to Link 1 and Link 19, while the cellular signal covers all links.In addition to these two data sources, there is a traffic incident/accident report which could be used in the following evaluation of the estimation results. The report indicates that an incident occurred on Link 4 at about 11:12 AM. The traffic was also quite heavy. These two traffic phenomena together lead to the traffic moving slowly from Link 1 to Link 10 as well as the nearby affected links. The report also shows that the traffic moved slowly from 10:00 to the late evening of that day due to the heavy traffic. The cause is the China National Day Holiday which begins from October 1 and the free tolling policy on freeway that works during this holiday, and, thus, it attracts tremendous traffic. Therefore, the traffic flow climbs high when it is closing to the midnight of the test day. Consequently, the high traffic flow triggered the speed drop and the large density.


## 5.2. Model Parameters

### 5.2.1. Parameters about Time

The time stepT of the estimator is 10 seconds to meet the requirement that T<L/vf=1km/100(km/h)=36seconds. The time interval for measurement update is 10 minutes. The estimator works when the measurements are available. It means that the time update and measurement update will work 60 steps (10 min/10 sec) using the same measurements.


### 5.2.2. Parameters about Links and Boundaries

Although there are 19 links, the measurements from the microwave sensors are only available on two links (Link 1 and Link 19). Our recent work found that the traffic features (UCC and PS) extracted from the wireless communication records on Link 6, Link 10, Link 14, and Link 18 had the closest relation with the traffic variables [1]. This study uses the UCC and PS from these links for the sake of high accuracy. Besides, the PSs on Link 1 and Link 19 are also used under the low-speed or congested condition, because these PSs are more reliable than the microwave speeds under this condition as discussed in Section 2. To avoid the impact of inaccurate boundary inputs, Link 1 and Link 19 are regarded as the boundary links, and, thus, the measurements on these links are used as the boundary variables.


### 5.2.3. Parameters in the Macroscopic Traffic Flow Model

The empirical parameters are set as Table1. And the models of all links shared the same values. This study uses the value suggested in the work of Wang et al. [6] as the process noise.Table 1
Parameter settings.
Parameter Value τ 0.02 h κ 40 veh/km v f 100 km/h ν 24 km2/h ρ c r 120 veh/km


### 5.2.4. Parameters in the Measurement Model

The coefficients in the measurement model for the UCC were calibrated using the historical data as follows [1]. The constant parameter φ approximately equals 8.5 for all links (Link 6, Link 10, Link 14, and Link 18). The time-varying parameter θ(k) is valued as follows: from the time 00:00 to 05:00, it approximately equals 0.3; from 05:00 to 07:30, it gradually increases from 0.3 to 1.0; from 07:30 to the rest of a day, it approximately equals a constant 1.0 under the free-flow condition. Another critical parameter in the measurement update step is the critical speed for the identification of traffic condition. It is valued empirically as 50 km/h.The measurement noise should be the standard deviation of measurements from cellphone activity data and microwave sensors. However, it is difficult to value the noise when there is no sufficient ground-truth data. Some existing studies indicate that the Extended Kalman Filter was found to have a slight sensitivity to the standard deviation-based measurement noise, which guaranteed the estimator’s performance even with poor foreknowledge of model noise [3, 7]. Therefore, the deviation was set as a small value; that is, the density deviation was 1 veh/km and the speed deviation was 1 km/h for both sources.


## 5.2.1. Parameters about Time

The time stepT of the estimator is 10 seconds to meet the requirement that T<L/vf=1km/100(km/h)=36seconds. The time interval for measurement update is 10 minutes. The estimator works when the measurements are available. It means that the time update and measurement update will work 60 steps (10 min/10 sec) using the same measurements.


## 5.2.2. Parameters about Links and Boundaries

Although there are 19 links, the measurements from the microwave sensors are only available on two links (Link 1 and Link 19). Our recent work found that the traffic features (UCC and PS) extracted from the wireless communication records on Link 6, Link 10, Link 14, and Link 18 had the closest relation with the traffic variables [1]. This study uses the UCC and PS from these links for the sake of high accuracy. Besides, the PSs on Link 1 and Link 19 are also used under the low-speed or congested condition, because these PSs are more reliable than the microwave speeds under this condition as discussed in Section 2. To avoid the impact of inaccurate boundary inputs, Link 1 and Link 19 are regarded as the boundary links, and, thus, the measurements on these links are used as the boundary variables.


## 5.2.3. Parameters in the Macroscopic Traffic Flow Model

The empirical parameters are set as Table1. And the models of all links shared the same values. This study uses the value suggested in the work of Wang et al. [6] as the process noise.Table 1
Parameter settings.
Parameter Value τ 0.02 h κ 40 veh/km v f 100 km/h ν 24 km2/h ρ c r 120 veh/km


## 5.2.4. Parameters in the Measurement Model

The coefficients in the measurement model for the UCC were calibrated using the historical data as follows [1]. The constant parameter φ approximately equals 8.5 for all links (Link 6, Link 10, Link 14, and Link 18). The time-varying parameter θ(k) is valued as follows: from the time 00:00 to 05:00, it approximately equals 0.3; from 05:00 to 07:30, it gradually increases from 0.3 to 1.0; from 07:30 to the rest of a day, it approximately equals a constant 1.0 under the free-flow condition. Another critical parameter in the measurement update step is the critical speed for the identification of traffic condition. It is valued empirically as 50 km/h.The measurement noise should be the standard deviation of measurements from cellphone activity data and microwave sensors. However, it is difficult to value the noise when there is no sufficient ground-truth data. Some existing studies indicate that the Extended Kalman Filter was found to have a slight sensitivity to the standard deviation-based measurement noise, which guaranteed the estimator’s performance even with poor foreknowledge of model noise [3, 7]. Therefore, the deviation was set as a small value; that is, the density deviation was 1 veh/km and the speed deviation was 1 km/h for both sources.


## 5.3. Estimation Results

Figure5 shows the estimation results via the proposed PEKF-based method. The gradual changes in the color represent the variations in the traffic state. The blue represents the low-density and high-speed condition, while the red indicates the high-density and low-speed condition. As shown in Figure 5, both the estimated speeds and densities are changing among the free-flow, dense, and congested conditions. For instance, the speeds of the entire freeway segment during 00:00 to 10:00 AM are rarely red as shown in Figure 5(b). With the color getting red, it means that the traffic slows down. When the color is dark red, the traffic speed is quite low which indicates a congestion. In Figure 5(a), the density is lower than 40 veh/km from 00:00 to 7:00 showing in the blue. It is reasonable that there is little traffic on the freeway at the midnight of a working day and the traffic increases in the day time.Figure 5
The estimation results. (a) The density estimates. (b) The speed estimates.
(a) (b)On the other hand, the incident/accident report data are used to qualitatively evaluate the accuracy of results for lacking the “ground-truth” speeds and density. Figure5(b) shows that the speed drops firstly on Link 1 and Link 2 at about 11:00 AM. Consequently, this phenomenon happens on Link 3 and Link 4 quickly. And speeds on Link 5 and Link 6 also decrease around that time. The variation of speed indicates that the estimation results exhibit the influence of traffic incidents. However, the reported incident happened on Link 4 at about 11:00, but the estimation results on Link 4 have a small delay. Since the measurements are available only on Link 1 and Link 6 and the distance between two links is 6 km long, it is understandable for such a small delay. In Figure 5(a), the density on Link 1, Link 2, and Link 3 decreases when the incident happens. Since Link 1, Link 2, and Link 3 are downstream links of Link 4, this decline phenomenon reflects the propagation of a shock wave. Another incident record is about the heavy traffic later that day which induces the low speed. From Figure 5(b), we can see the large area of red later that day which validates the correctness of the estimation results. And the changing color in Figure 5(a) shows the variation of density which reflects the oscillation of traffic under the low-speed traffic condition. Roughly, the estimation results efficiently integrated the measurements from the multiple sources and exhibit the dynamic and real-time traffic state.


## 5.4. Evaluation

An evaluation of the proposed PEKF-based estimator is conducted by making a comparison between the results from multisource data and single-source data. We applied the estimation results from the previous work [1], which estimated the traffic state using the wireless communication data and EKF-based data assimilation method. Since the distance between the microwave sensors is extremely sparse, the data generated from these sensors is not qualified for the application of a model-based approach. Hence, the direct measurements from microwave sensors are used for the comparison. To make a precise comparison, the absolute difference (AD) is chosen as the performance measure. It can be calculated by the following equation:(20)ADk=xsk-xfk,where ADk is the absolute difference at the time interval k, xsk is the state estimates (density or speed) from single data source at the time interval k, xfk is the state estimates (density or speed) from multiple sources at the time interval k.The iteration process of the progressive EKF-based estimator shows that the measurements from microwave sensors are usually taken as the more precise value compared with the traffic features from the wireless communication records. ADs between estimates from multisources and wireless communication data and ADs between estimates from multisources and measurements from microwave sensors in different links are present in Figure6. Obviously, the progressive EKF-based method generated estimates which are more close to the more accurate measurements from microwave sensors. Generally, it will improve the total precision of estimation by fusing multisource data.Figure 6
The AD results. (a) The ADs of density on Link 1. (b) The ADs of speed on Link 1. (c) The ADs of density on Link 19. (d) The ADs of speed on Link 19.
(a) (b) (c) (d)


## 6. Conclusions

This paper proposed a Progressive Extended Kalman Filter (PEKF) method to estimate the freeway traffic state and integrate the heterogeneous data from the wireless communication data and the microwave sensors. The challenges in the application of these two data sources are the heterogeneities in the spatial coverage and the semantic of measurements. Via the characteristic analysis of the multisource data, we set up the relations between the measurements and traffic state variables and propose some rules to solve the conflicts between two data sources. The state-space model is constructed accordingly. The EFK technique is applied to establish the estimator, and it is improved to fit the multisource data and is nominated as the PEKF estimator. The test with the field data indicates that the proposed method successfully integrates the heterogeneous data, especially the new combination of measurements from the wireless communication records and microwave sensors. Moreover, the estimation results indicated that the PEKF is able to track the dynamic traffic conditions. The qualitative analysis between the estimates and traffic incident reports validates the accuracy of the proposed method roughly. The final comparisons indicates the advantages of the proposed methods; that is, it has larger coverage compared with the microwave sensors, and it is more accurate than the estimates from wireless communication data.This study makes an effort to use the limited, heterogeneous, and multisource data for the traffic state estimation, but there are still plenty of works before the field application. The most urgent task is to validate the method with the ground-truth data. Second, the test on other freeway links is required due to the spatial characteristic of the traffic features from the wireless communication data.



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*Source: 6745726-2018-05-06.xml*