A map-matching algorithm with low-frequency floating car data based on matching path

Abstract

With the wide application and rapid development of Intelligent Transportation System (ITS), the floating car has been widely used in the collection of traffic information, which is also very important in the application of the wireless sensor networks. In addition to the high-frequency floating car, energy-saving low-frequency floating car has attracted great attention, but the low-frequency GPS data have a poor effect on map matching. Taking consideration of the distance, direction, speed, and topology of road and vehicle, we propose a global map matching algorithm with low-frequency floating car data based on the matching path. The proposed algorithm preprocesses the floating car data and road network data to determine the potential points and sections by constructing the error region. Then, we calculate the potential matching path graph with the analysis of time and space. Finally, we can obtain the matching result by parallel computing with section division methodology. The experiment results demonstrate that the proposed map-matching algorithm can improve the running time and matching accuracy compared with the existing methods.

1 Introduction

In the application of the wireless sensor networks, traffic information from the senor is very important to control the traffic congestion. With the rapid development of economy, traffic congestion are getting worse, and at the same time, more and more technology has been applied to the traffic management, where the Intelligent Transportation System (ITS) is a significant method [1]. ITS is a real-time, efficient, and comprehensive transportation management system which integrates advanced electronic technology and information technology. ITS makes a new way of interaction among three main traffic systems: car, road, and traffic, which can reduce traffic congestion to improve the traffic capacity [2, 3]. The implementation of ITS system depends on the accurate and real-time traffic information.

At present, most of the traffic information is collected by floating car [4], which can acquire the vehicle’s position, speed, direction, and other information directly with low monitoring cost and high efficiency. The floating car is an advanced method to obtain the traffic information. The floating car utilizes the vehicle GPS device to collect the vehicle position data information, then send the data information to the processing center with the wireless communication transmission technology. In the processing center, the map matching methodology is used to match the collected information and map information to obtain real-time traffic condition [5].

However, due to the complex city geographical features, when facing the viaduct, culverts, high-rise buildings, and other terrains, the positioning accuracy of floating car would be significantly reduced. In order to provide more accurate positioning data, it is necessary to process the corresponding correction to the data acquired by the floating car, where the map matching is the key technology to achieve this goal [6, 7].

Map matching is also called map-aided positioning technology, which is based on the pattern recognition theory. We set the positioning information obtained by the positioning device as the objects to be matched and set the geographic information in electronic map as the matching template. With the calculation of similarity between the objects to be matched and the matching template, we can choose the correct vehicle travel matching path according to such similarity [8]. The map-matching technology is normally implemented in the field of traffic control. At present, most of the map-matching algorithms mainly aim at the high-frequency GPS data.

In practical applications, due to energy consumption and economic consideration, the data acquired by floating cars are low-frequency GPS data whose sampling time interval is more than 2 min. The sampling time interval of high-frequency GPS data is less than 2 min; two collected adjacent positioning points are generally located in a section of the road. For the low-frequency GPS data, with the increase of sampling time interval, in the complex city road network condition, the vehicle may pass through multiple complex sections, the matching effect and the matching efficiency will be affected greatly. To solve this problem, considering the distance, direction, speed, connectivity, and other factors, we propose a map-matching algorithm with the low-frequency floating car data, which can improve the accuracy and efficiency of map matching to select the correct matching path rapidly.

The paper is organized as follows. In Section 2, we give the related work. The proposed map-matching algorithm with the low-frequency floating car data is presented in Section 3. Experiment simulation and result analysis are illustrated in Section 4 and conclusion remarks in Section 5.

2 Related work

Traditional map-matching algorithm is simple and fast, but with the construction and development of urban traffic, the urban road and the traffic condition are more and more complex, the traditional map-matching algorithm would produce inaccurate matching results. Therefore, many experts have an in-depth analysis and research about map-matching algorithm.

Bernstein and Kornhauser presented a geometric matching algorithm, mainly including the matching of point to point and point to the curve [9]. Geometric matching algorithm is the foundation to the map-matching algorithm. Taylor proposed road reduction filter (RPF) algorithm, aiming at the map matching with positioning device GPS, which can improve the matching of point to the curve and curve to curve and the matching accuracy with differential GPS correction [10]. Greenfield proposed a map-matching algorithm based on network topology, which has used weighted method to fitting out multiple matching factors, and then selected the matching section. The experiments have demonstrated the significance of such algorithm [11]. Alt used Fréchet distance to measure the similarity between the trajectory composed of positioning points and the road to be matched, where the similarity can be used as the standard to choose the matching path [12]. However, the complexity of the algorithm is a little high. Brakatsoulas improved such algorithm by replacing the Fréchet distance with the weak Fréchet distance, which can reduce the algorithm complexity [13].

Peng Fei put forward a map-matching technology based on probability and statistics, which can establish a confidence region based on the received positioning points to determine the matching sets quickly to improve the matching efficiency [14]. Gao Jian introduced a dynamic filtering technology into direct and indirect modes, which can process sections with shape and sections without shape efficiently. Such algorithm can achieve good effect in the simulation experiments [15]. Sinn Kim presented an adaptive fuzzy neural network map-matching algorithm. The algorithm calculated C-measure value for each positioning point, where C-measure considered the factor of distance and direction to indicate the probability of a vehicle traveling on a road section [16]. According to the characteristics of the floating car, Wang Xiaomeng presented a map-matching algorithm based on hidden Markov chain. This algorithm has many improvements compared with a traditional model: introducing heading variable into the calculation of emission probability and using the path distance to build the road transfer matrix of the activities range within the floating car moving scope. With these improvements, such map-matching algorithm has both high matching accuracy and high matching efficiency [17].

The current map-matching algorithms mainly aimed at vehicle positioning data with high frequency. There is few study of low-frequency data map matching. In 2013, Yao Enjian presented a real-time map-matching algorithm with low-frequency floating car data, called the piecewise fuzzy matching algorithm. The algorithm divided low-frequency positioning data collected by floating car at a fixed time interval (5 min). Considering the connectivity between two periods, the last locating point in the previous period has been taken as the starting point of the next period. Then the matching path can be calculated for each time period and connected to obtain the matching path for the floating car track [18]. In 2015, Shen Jingwei proposed a map-matching algorithm based on improved activity on edge (AOE) network with low-frequency floating car. The vertices in the graph were all the positioning points of the positioning points, and the edge weight was the path length of the potential points. The shortest path from the start to the end of the AOE graph was the matching path of the floating car track [19]. The experimental results showed that the algorithm has obvious advantages both at running time and matching accuracy compared with the global matching algorithm based on weak Fréchet distance.

However, with the analysis, we can find that in the calculation of floating car matching path, edge weight calculation only considers the shortest path, making no use of other information which would affect the floating car track matching results, such as the direction of the floating car, the velocity of floating car, and the distance between the positioning points and the potential points. Our proposed map-matching algorithm will make full use of the information gathering by floating car to improve the matching effect highly.

3 Map-matching algorithm

The whole framework of the proposed map-matching algorithm is shown in Fig. 1, which consists of four modules: The first one is the GPS data and road network data preprocessing, which can sift out the redundant data and abnormal data collected by floating car and construct the topology structure and grid partition, illustrated in Section 3.1; the second one is the selection of potential road sections and potential points, which can filter out the potential road sections according to error region built by the theory of probability and statistics, the vehicle’s speed, and the angle of vehicle direction and road direction, illustrated in Section 3.2; the third one is the time-space analysis for positioning points, corresponding potential points, and potential matching segments to obtain the potential matching path map of floating vehicle trajectory, illustrated in Section 3.3; the fourth one is the results matching, where one matching path of potential matching path map is the matching path of the floating vehicle, illustrated in Section 3.4.

Fig. 1

The framework of map matching algorithm

Before illustrating these four modules in detail, we firstly define the map-matching problem.

Defining the trajectory T of GPS, T is a series of orderly GPS positioning pointsp₁, p₂, …, p _n , and the information contained in each GPS positioning point can be expressed as (p _i . lat, p _i . lng, p _i . time, p _i . v, p _i . β), corresponding to the longitude, latitude, time, the instantaneous velocity, and the moving direction angle of positioning point, respectively.

Defining the road network G(V, E), G(V, E) can be represented with a directed graph; V is a collection of starting point, ending point, intersection point, and shape point in a road network; and E is a collection of section in road network. Each section e is a directed edge, which can be expressed as a (e. id, e. v, e. l, e. start, e. end), corresponding to the serial number of section, the speed limit, the length of the road, the starting point, and the ending point of the section, respectively.

Defining the path P, the given two points v _i and v _j on the network, P is a collection of connected sections starting from v _i and ending in v _j .

Then the map-matching problem can be defined as follows: with the road network G and untreated GPS trajectory T, it is to solve the actual path of T in G.

3.1 Data preprocessing

3.1.1 GPS data preprocessing

As the urban road network condition is very complex, when floating car drives in a tunnel, under the overpass or meet with larger building block, it will cause the deviation of GPS data. In addition, the vehicle low-speed driving or parking, caused by the traffic congestion or vehicle failure, will produce a lot of redundant data. These abnormal data and redundant data will seriously affect the effective execution of map-matching algorithm. Therefore, before matching the GPS positioning points, we preprocess the GPS data as follows:

• 1. Delete the data according to the space limit;

• 2. Delete the data according to the speed limit threshold;

• 3. Delete the data according to the static data drift.

When the speed is slow or the vehicle stops, the positioning point theoretically should be in dense distribution along the road or gathered at a point. But because of the GPS positioning deviation, it will make the positioning point present as a random fluctuation phenome taking a point as the center, as shown in Fig. 2. Since these points are close to each other, they can be filtered out according to the distance. If the velocity of the floating car in current locating point is low (v < 3 m/s) and the distance between the current positioning point and the previous positioning point is less than the threshold, we can ignore the current positioning point and take the previous positioning point as the current positioning point.

Fig. 2

Random fluctuations phenomenon

3.1.2 Network data preprocessing

For road information, in addition to the storage of nodes, shape point, and road section, we also need to store the connection information between them, namely the topology information. To build a network topology relationship, we need to consider the relationship between nodes, relationship between node and section node, and link relationship between section and section. Figure 3 is a simple road network, including four paths: R1, R2, R3, and R4. These paths are divided by nodes into eight sections, sections A, B, C, D, E, F, G, and H, where the purple dot presents the road intersection node, and the blue dot presents the shape point in the section.

Fig. 3

Sample of road network

We can build the road topology according to the following steps:

• 1. Reading the road R1, finding the starting point, end point of R1, and the intersection points with other roads, numbering these nodes following the sequence starting from number 1, then generating the node link table. Taking into account the road direction, one-way street only takes a row of records in the table, while two-way street takes two rows of records.

• 2. Processing road R2~R4 as R1 in turn. If road nodes have serial number, we can use the existing serial number, if not, order number in sequence and add them to the node links table.

• 3. Sorting the node link table according to the starting point of the section to obtain a new node link table.

• 4. Merging all records with the same starting point number in the node link table to build the topological relations between nodes and nodes, and nodes and links.

• 5. For two-way road section, finding out the associated sections with similar starting point or end point from the node link table. For one-way road section, finding out the associated sections with the similar end point in the node link table. Then merging the same section to obtain the topological relations between sections.

After building the road network topology, we process the road network with grid partition to associate each grid with the sections in each grid area, which can help the rapid filtering out potential sections of positioning points. Due to the high accuracy of current GPS positioning, the actual location of the floating car should be in the road section within 100-m distance to the positioning point. Therefore, we choose D = 100 m as the length of grid side. Taking the length and width of electronic map as L and W, respectively, we divide the electronic map with the square whose side is D into the number of M*N grids; therefore, M = ⌈L/D⌉, and N = ⌈W/D⌉. We can number the gird in the electronic map from top to bottom, from left to right in sequence. Assuming that the latitude and longitude coordinates of the positioning point is (x, y), the latitude and longitude coordinates of the point which is located in the upper left in the electronic map is (x₀, y₀), then the grid id of the positioning point can be calculated like this:

$$ {\displaystyle \begin{array}{c} id=\left(n-1\right)\ast M+m\\ {}n=\left\lceil dis\left(\left({x}_0,{y}_0\right),\left(x,{y}_0\right)\right)/D\right\rceil \\ {}m=\left\lceil dis\left(\left({x}_0,{y}_0\right),\left({x}_0,y\right)\right)/D\right\rceil \end{array}} $$

(1)

where dis is the Euclidean distance between two points. Considering the positioning point on the edge of the grid, the positioning point’s potential road section is fixed within nine grids centered with the positioning point grid. After grid partition, we need to filter out the road sections in these nine grids, which can improve the efficiency of filtrating the potential road section greatly.

3.2 Selection of potential sections and potential points

3.2.1 Determination of error region

After grid partition, taking all road sections in nine grids centered to the positioning point grid as the potential road sections, which would lead to too many potential road sections. Taking advantage of the theory of probability and statistics to construct the error region of the positioning point can further reduce the number of the potential sections. Because of the existence of various errors, all possible locations obtained by the vehicle GPS receiver present normal distribution in an area. According to the theory of probability and statistics, we can build an error model with the data obtained by positioning sensor, to gain the confidence region of vehicle real position. According to the variance and covariance parameters of GPS receiver, the error confidence region can be defined as the ellipse form as below:

$$ a={\sigma}_0\sqrt{\frac{1}{2}\left({\sigma}_x^2+{\sigma}_y^2+\sqrt{{\left({\sigma}_x^2-{\sigma}_y^2\right)}^2+4{\sigma}_{xy}^2}\right)} $$

(2)

$$ b={\sigma}_0\sqrt{\frac{1}{2}\left({\sigma}_x^2+{\sigma}_y^2+\sqrt{{\left({\sigma}_x^2-{\sigma}_y^2\right)}^2+4{\sigma}_{xy}^2}\right)} $$

(3)

$$ \phi =\frac{\pi }{2}-\frac{1}{2}\arctan \left(\frac{2{\sigma}_{xy}}{\sigma_x^2-{\sigma}_y^2}\right) $$

(4)

In the above three formulas, a is the long semi-axis of the error ellipse, b is the short semi-axis error of the error ellipse, and ϕ is the angle between the long semi-axis and the north direction; σ _x and σ _y are the standard deviation of positioning point in the direction of north and east, which can be available in the output message of the GPS receiver; σ _xy is the covariance; and σ₀ is called the extended factor. When the shape is kept constant by adjusting the value of σ₀, the error ellipse can be magnified and reduced to get different confidence level. When σ₀ = 1, the confidence level is 39%; when σ₀ = 2.15, the confidence level is 95%; and when σ₀ = 3.30, the confidence level is 99% [20]. The center of the error ellipse is set as the positioning point.

3.2.2 Determination of potential sections and potential points

After getting the error region of the positioning point, the potential road section of the positioning point can be quickly screened out by judging whether the sections in the positioning point grid and the surrounding grid are located in the error region of the positioning point. The process of determination of potential sections and potential points is illustrated in Fig. 4. The potential points corresponding to the positioning points are the projection points on the corresponding potential sections. We select the projection points according to the following method: if the projection point of the GPS positioning point in the potential section is located in the potential section, select the projection point as the potential point of GPS positioning point and if the projection point of GPS positioning point in the potential section is located in the potential section’s extension line, select the nearest node of the potential section to GPS positioning point as the potential point. Taking into account the angle between the direction of the floating car and the direction of the potential section, the potential section can be selected twice, in order to reduce the number of invalid candidates. According to the research results of Ochieng et al. [21], when the speed v of the floating car is less than 3 m/s, the direction angle θ will become unstable. Therefore, the rules for potential section to be screened out twice are as follows:

• 1. When v ≥ 3m/s, the direction angle was limited to θ < 30^°.

• 2. When v < 3m/s, the direction angle was limited to θ < 120^°.

Fig. 4

Determining of potential sections and points

When two nodes n₁(x₁, y₁), n₂(x₂, y₂) on a potential section are known, the direction angle а of the potential section can be calculated according to formula (5):

$$ a=\left\{\begin{array}{lll}\left(\frac{\pi }{2}-\arctan \left(\frac{y_2-{y}_1}{x_2-{x}_1}\right)\right)\ast \frac{180}{\pi }&, & {x}_2-{x}_1>0\\ {}\left(\frac{3\pi }{2}-\arctan \left(\frac{y_2-{y}_1}{x_2-{x}_1}\right)\right)\ast \frac{180}{\pi }&, & {x}_2-{x}_1<0\\ {}180&, & {x}_2-{x}_1=0\kern0.5em {y}_2-{y}_1<0\\ {}0&, & {x}_2-{x}_1=0\kern0.5em {y}_2-{y}_1\ge 0\end{array}\right. $$

(5)

The direction angle β of the floating car can be obtained directly from the information of the positioning point. It can be concluded that the angle θ between the direction of the vehicle and the potential section can be calculated as:

$$ \theta =\left\{\begin{array}{ll}\left|\alpha -\beta \right|&, \left|\alpha -\beta \right|\le 180\\ {}360-\left|\alpha -\beta \right|&, \left|\alpha -\beta \right|>180\end{array}\right. $$

(6)

The screening process is shown in Fig. 5, and in the error region, we can obtain three potential sections L1, L2, and L3, where the floating car speed of positioning point P is 6 m/s.

Fig. 5

Selection of potential points

When we process the secondary screening according to the direction angle, since the angle between the direction of road L3 and the positioning point P is larger than 30°, L3 will be rejected from the potential sections. The final potential sections of point P are L1 and L2, and the corresponding matching points are ${c}_1^1$ and ${c}_1^2$, respectively. With the secondary screening, the invalid potential sections are discarded, which can reduce the number of potential points corresponding to the positioning point.

3.3 Analysis of time and space

The analysis of time and space is the core module of the proposed map-matching algorithm. By considering the time and space factors in the map matching, we can calculate the matching path map of the floating vehicle trajectory from the potential sections. Time and space analysis is based on the following three rules:

• 1. If there is no direct connection between two positioning points, the actual route of the floating car tends to choose the shortest path between the two positioning points. Taking the taxi as the research object of the floating car, under the urban road network, it will choose the shortest route.

• 2. The actual route of a floating car tends to be straight rather than circuitous.

• 3. The speed of a floating car obeys the speed limit of the road network.

3.3.1 Analysis of space

In the space analysis, we calculate the observation probability of each potential point and the transition probability between potential point pairs based on distance, direction, and topology connectivity. Observation probability represents the possibility that potential points would be matching points based on the distance between the positioning point and the potential point, the angle between the direction of the vehicle, and the road direction of the potential point. The transition probability represents the possibility of the path between potential point pairs as the actual path of two adjacent positioning points.

• 1. Observation probability

The observation probability is defined as a probability that positioning point P _i matches to the potential point ${c}_i^j$. Our proposed calculation of observation probability not only considers the distance between the positioning point and the potential point, but also takes into account the angle between the direction of the floating car and the direction of the potential road section, which can improve the accuracy of the observation probability.

Generally, the distance between positioning point P _i and potential point ${c}_i^j$ obeys the normal distribution with a mean value of 0. The angle ${\theta}_i^j$ between the floating car’s direction on positioning point P _i and the direction of the potential road L _j obeys the exponential distribution. Therefore, the probability function of distance ${d}_i^j$ is expressed as formula (7), and the probability function of direction angle ${\theta}_i^j$ is expressed as formula (8).

$$ {f}_d\left({c}_i^j\right)=\frac{1}{\sqrt{2\pi}\sigma }{e}^{-\frac{{\left({d}_i^j\right)}^2}{2{\sigma}^2}} $$

(7)

$$ {f}_{\theta}\left({c}_i^j\right)=\lambda {e}^{-{\lambda \theta}_i^j} $$

(8)

In formula (7), σ is the standard deviation of the normal distribution that distance ${d}_i^j$ is satisfied with. In formula (8), λ is the parameter with the exponential distribution that direction angle ${\theta}_i^j$ is satisfied with. Taking into account the distance and direction angle, the observation probability can be calculated by the weighted method:

$$ N\left({c}_i^j\right)={w}_d{f}_d\left({c}_i^j\right)+{w}_{\theta }{f}_{\theta}\left({c}_i^j\right) $$

(9)

In formula (9), w _d and w _θ are the weighting factors of the distance ${d}_i^j$ and direction angle ${\theta}_i^j$, respectively, meeting the condition of w _d  + w _θ  = 1. When the instantaneous speed v of the positioning point is less than 3 m/s, the direction of the vehicle becomes unstable at this time. The rules for determining w _d and w _θ are as follows:

• (a) When v < =3 m/s, ignoring the direction angle, take w _d  = 1, w _θ  = 0.

• (b) When v > 3 m/s, in order to determine the weight w _d and w _θ of the observation probability, we select different w _d and w _θ to process map matching. Comparing the accuracy of the map matching with different values, the higher the matching accuracy is, the better the value reflects the geometric relationship. Therefore, we choose the value of w _d and w _θ as the value when the matching accuracy is the maximum.

If the observation probability does not take into account the topological connectivity among the road sections of the potential points corresponding to the adjacent positioning points, mis-matching will occur. Figure 6 shows an example of mis-matching; if matching points are selected only by the observation probability value of potential matching point, the value of $N\left({c}_i^1\right)$ is the maximum, then the matching point of p _i is ${c}_i^1$. However, if considering the previous and next point p _i-1 and p _i + 1 , then we find that the correct matching point is ${c}_i^2$ of p _i . Because if the matching point of p _i is ${c}_i^1$, the vehicle must arrive on p _i , starting from p _i - 1 , then return to p _i + 1 , which is contrary to the previous rule 2.

• 2. Transition probability calculation

Fig. 6

An example of mis-matching

For the GPS location of two adjacent points p _i - 1 and p _i , their corresponding potential points are ${c}_{i-1}^s$ and ${c}_i^t$, respectively, and the transition probability $V\left({c}_{i-1}^s\to {c}_i^t\right)$ from ${c}_{i-1}^s$ to ${c}_i^t$ is defined as the coincidence degree between the real path and the shortest path. Consider the following conditions:

• (a) When two potential points ${c}_{i-1}^s$ and ${c}_i^t$ are located on the same section or adjacent sections, the topological relation is determined, then $V\left({c}_{i-1}^s\to {c}_i^t\right)=1$.

• (b) Transfer probability is mainly to analyze the topology from ${c}_{i-1}^s$ to ${c}_i^t$, then we should take the distance from ${c}_{i-1}^s$ to ${c}_i^t$ as the standard of measurement. Therefore, transfer probability $V\left({c}_{i-1}^s\to {c}_i^t\right)$ is always in the range (0, 1].

Based on the above analysis, the transition probability is calculated as formula (10):

$$ V\left({c}_{i-1}^s\to {c}_i^t\right)\left\{\begin{array}{l}1,\mathrm{in}\kern0.17em \mathrm{the}\kern0.17em \mathrm{same}\kern0.17em \mathrm{or}\kern0.17em \mathrm{adjacent}\kern0.17em \mathrm{sections}\\ {}\frac{d_{\left(i-1,s\right)\to \left(i,t\right)}}{w_{\left(i-1,s\right)\to \left(i,t\right)}}, others\end{array}\right. $$

(10)

In formula (10), d_{(i − 1, s) → (i, t)} is the Euclidean distance between ${c}_{i-1}^s$ and ${c}_i^t$, and w_{(i − 1, s) → (i, t)} is the shortest path from ${c}_{i-1}^s$ to ${c}_i^t$.

• 3. Space analysis function

With the observation probability and transition probability, the space analysis function can be defined as follows:

$$ {F}_s\left({c}_{i-1}^s\to {c}_i^t\right)=N\left({c}_i^t\right)\ast V\left({c}_{i-1}^s\to {c}_i^t\right) $$

(11)

Formula (11) shows the possibility of a vehicle from p _i - 1 moving to p _i . Therefore, the distance between the positioning point and the potential point, the instantaneous velocity of the positioning point, and the topology of the potential points are all utilized. Based on the space analysis, for any two adjacent positioning points p _i - 1 and p _i , the potential points ${c}_{i-1}^s$ and ${c}_i^t$ construct a series of weighted potential paths.

3.3.2 Analysis of time

In most cases, the proposed algorithm can identify the actual path from the potential paths. However, there are still some situations where we cannot effectively calculate the actual path only by the space analysis.

As shown in Fig. 7, the thick line is a main road, the thin line next is an auxiliary road, we can get ${F}_s\left({c}_{i-1}^1\to {c}_i^1\right)={F}_s\left({c}_{i-1}^2\to {c}_i^2\right)$ with the space analysis. However, if the average speed from p _i - 1 to p _i is 80 km/h, considering the road speed limit, we can select the path from ${c}_{i-1}^1$ to ${c}_i^1$ as the matching path.

Fig. 7

Example of time analysis

Given two GPS positioning points p _i-1 and p _i , the corresponding potential points are ${c}_{i-1}^s$ and ${c}_i^t$, respectively, the shortest path from ${c}_{i-1}^s$ to ${c}_i^t$ can be expressed as a series of sections e₁, e₂, …, e _k , the average vehicle speed in the shortest path can be expressed as:

$$ {\overline{v}}_{i-1\to i}=\frac{\sum \limits_{u=1}^k{l}_u}{\Delta {t}_{i-1\to i}} $$

(12)

In formula (12), l _u is the length of the road section e _u , Δt_{i − 1 → i} = p _i . t − p_i − 1. t, and Δt_{i − 1 → i} is the time interval of p _i and p _i - 1 . Note that each section e _i has a specific speed limit which can be used to measure the correlation between the average speed of the vehicle and the path speed limit. For the vector $\left({\overline{v}}_{i-1\to i},{\overline{v}}_{i-1\to i},\dots, {\overline{v}}_{i-1\to i}\right)$ and (e₁. v, e₂. v, …, e _k . v), the time analysis function can be defined as formula (13):

$$ {F}_t\left({c}_{i-1}^s\to {c}_i^t\right)=\frac{\sum \limits_{u=1}^k\left({e}_u.v\ast {\overline{v}}_{i-1\to i}\right)}{\sqrt{\sum \limits_{u=1}^k{\left({e}_u.v\right)}^2}\ast \sqrt{\sum \limits_{u=1}^k{\left({\overline{v}}_{i-1\to i}\right)}^2}} $$

(13)

3.4 Result matching

With the analysis of time and space, we can obtain the initial trajectory. However, if obtaining the accurate matching path results, we should process the result matching. We propose a segment partition method to accelerate the process of obtaining matching path.

3.4.1 Acquisition of matching path

With the analysis of time and space, it can generate a potential map G _T (V _T , E _T ) for the floating car trajectory T : p₁ → p₂ → … → p _n . V _T is a set of potential points corresponding to all the GPS positioning points on the T. E _T is a collection of edges, where each edge represents the shortest path between two adjacent GPS positioning points corresponding to the potential points. G _T (V _T , E _T ) is shown as Fig. 8, where each node is related to $N\left({c}_i^j\right)$ and the weight of each edge is related to $V\left({c}_{i-1}^s\to {c}_i^t\right)$ and ${F}_s\left({c}_{i-1}^s\to {c}_i^t\right)$.

Fig. 8

Potential graph G _T (V _T , E _T )

Combining formula (11) and formula (13), the space-time function can be defined as below:

$$ F\left({c}_{i-1}^s\to {c}_i^t\right)={F}_s\left({c}_{i-1}^s\to {c}_i^t\right)\ast {F}_t\left({c}_{i-1}^s\to {c}_i^t\right) $$

(14)

The potential matching path of trajectory T can be expressed as ${P}_c:{c}_1^{q_1}\to {c}_2^{q_2}\to \dots, \to {c}_n^{q_n}$. From the above analysis, we can find that the weight of each edge in the graph can be calculated with formula (14). The bigger the sum of the edge weights on the potential matching path is, the higher the possibility of being the matching result is. The total weight of each potential matching path is represented as $F\left({p}_c\right)=\sum \limits_{i=2}^nF\left({c}_{i-1}^{s_{i-1}}\to {c}_i^{s_i}\right)$. We can search for a path from the potential matching paths to meet the maximum of F(P _c ), where the found path is the matching path of T. Formally, the best matching path P of trajectory T can be expressed as:

$$ P=\arg {\max}_{p_c}F\left({P}_c\right)\kern0.5em ,\kern0.5em \forall {P}_c\in {G}_T\left({V}_T,{E}_T\right) $$

(15)

3.4.2 Sections division

From the process that obtainss the matching path, our proposed map-matching algorithm is a global map-matching algorithm. Only after calculating the cumulative weight of all the potential points in the track T, we can obtain the best matching path. When the trajectory T contains a large number of positioning points or the need for online matching, the processing efficiency of the algorithm may be in difficulty in meeting the actual requirements. In order to improve the algorithm efficiency, our paper proposes a method of section dividing the potential matching path graph G _T (V _T , E _T ). After division, the matching path can be calculated in parallel for each section can calculate the matching path with the proposed methodology.

In the actual matching process, we find that the positioning point corresponding to the potential point is unique sometimes by using the error region, direction angle, and other factors to filtering out the potential points. For example, as shown in Fig. 9, where P ₃ has only one potential point ${c}_3^1$. In this case, the matching path of G _T (V _T , E _T ) definitely pass ${c}_3^1$. Thus, the map can be divided by a series of single potential point. And the topological relationship between the potential points in the adjacent segment boundary is determined. After division, the potential matching path graph in Fig. 8 will be divided into three sections as shown in Fig. 9. The division of the potential matching path graph not only can reduce the execution time of the algorithm, but also does not affect the accuracy of the map-matching algorithm.

Fig. 9

Division graph with a single potential point

3.5 Time complexity analysis of the proposed algorithm

This section firstly analyzes the time complexity of the map-matching algorithm, and then optimizes the algorithm according to the analysis results. N is used to indicate the number of positioning points in the trajectory T, M represents the number of road sections, and K represents the maximum number of potential points of each positioning point.

The establishment of potential matching path graph G _T (V _T , E _T ) requires the observation probability of each potential point. The weight E _T of each edge needs to calculate the transition probability and process time analysis. After the establishment of G _T (V _T , E _T ), it is necessary to search for the critical path. The following is a detailed analysis for each calculation step:

• 1. The time complexity of observation probability: The maximum of all positioning points is nk, and the complexity of calculating the probability of each potential point is O(1), so the time complexity is O (nk).

• 2. The time complexity of transfer probability and time analysis: The number of the shortest paths that need to be computed in the potential matching path graph is (n-1)k². The time complexity with Dijkstra algorithm to calculating the shortest path is O(v²), where v is the number of segment endpoints in the road network (only counting once for repeating), then v = m + 1, then the time complexity can be expressed as O((m + 1)²). Therefore, the time complexity of calculating the transition probability and process of time analysis is O(nk²m²).

• 3. The time complexity of matching path computation: Because each edge of the potential matching path graph should be visited once, and the number of edges is nk², then the time complexity of the matching path computation is O (nk²).

Based on the above analysis, the time complexity of our proposed algorithm is O(nk²m² + nk² + nk). Considering that the number of potential matching points for each GPS positioning point is a small positive integer, the time complexity of the matching algorithm can be approximated as O (nm²). The number m of urban road network is generally large, so the running time of the proposed algorithm would be a little big. In order to reduce the running time of our proposed algorithm, we can optimize the computation of shortest path.

Firstly, considering the calculation process of Dijkstra algorithm, s is used to indicate the starting point, t is the end point, the array path represents the distance from the starting point to other points, and the set S stores the set of points in shortest path. The optimization steps are shown as follows:

• 1. Setting the initial value of the set S as empty, path[s] = 0, and the value of other points in the path as positive infinity;

• 2. Searching for the point whose path value is the smallest one which is not in the set S, and then adding it to the set S, and use it to update the path value of the surrounding points;

• 3. If t is not in the set S, repeat step (2) until t is involved in the set S, and the process ends.

In the step (2), to search for the point whose path value is the smallest, normally the Dijkstra algorithm traverse the points not in the S, the complexity is O(v²). Here we can optimize the algorithm with the minimum heap algorithm, where the time complexity of the minimum heap algorithm is O(vlog v) to reduce the time complexity of the algorithm. Therefore, the total time complexity of the proposed map-matching algorithm is reduced from O(nm²) to O(nmlogm). Besides, with the section division, the processing of difference sections can be done in parallel, which is helpful to improve the computational efficiency of the proposed algorithm.

4 Experiments simulation and results analysis

4.1 Experimental data

For the road network data, due to the limitations of the actual conditions, the experiment uses the electronic map of Wuhan in 2016, as shown in Fig. 10. The network data is extracted with Coordinate Extractor, a plug-in unit of MapInfo software. The Network’s partition is divided by Grider Maker, which is also a plug-in unit of MapInfo software. The data is transferred and stored in the form of text files.

Fig. 10

Wuhan’s electronic map (2016)

For the floating car data, data are acquired from a Wuhan company’s vehicle in 2016, where the GPS sampling time interval is 1 s. We select 20 groups of the floating vehicle trajectory data.

For matching result visualization, the map-matching results are output to the text file. Then we use the desktop GIS system of MapInfo to display the matching results in the electronic map.

4.2 Experimental simulation results

In order to measure the matching effect of the proposed map-matching algorithm for trajectory data of different sampling frequencies, we do the sample on each GPS trajectory data with the interval of 30, 60, 90, 120, 150, and 180 s, respectively, as the data to match. Then we preprocess these data and select the initial potential road and the potential points. The data in Table 1 present the shortest path information among part of potential points of the vehicle 1. The first line indicates the number of the shortest path of the potential points for vehicle 1. The second line indicates the shortest path information of two potential points, where the first and the second number represent the ID of potential points, For example, 00 means the ID 0 potential point corresponding to the ID 0 positioning point, the third number indicates the number of sections in the shortest path, the following numbers are the ID of the corresponding sections, and the last number represents the length of the shortest path.

Table 1 Shortest path between some potential points

Then we process the analysis of the time and space and result matching to obtain the corresponding GPS matching path. Table 2 shows the matching results of the positioning points of vehicle 1. The first number represents the ID of the positioning point, the second number indicates the ID of matching section, and the third and the fourth numbers indicate the longitude and latitude corresponding to the positioning point.

Table 2 Matching results of some positioning points

Finally, we can import the matching result data into MapInfo, and the GPS trajectory and the matching results are presented in the electronic map, as shown in Fig. 11, where the black flag represents GPS positioning point of floating car, and blue pushpin represents the corresponding matching point.

Fig. 11

Matching result

4.3 Analysis of experimental results

The map-matching algorithm can be evaluated with two factors: running time and matching accuracy. The running time is the execution time of the algorithm in the actual environment. The matching accuracy can be measured with two aspects: one is the correct matching ratio of positioning point, called A _N ; the other is the length accuracy ratio A _L which is the ratio of the matching path length to the actual path length. And the calculation of A _N and A _L can be defined as follows:

$$ {A}_N=\frac{N_r}{N} $$

(16)

$$ {A}_L=\frac{L_r}{L} $$

(17)

where N _r is the number of correct matching points, N is the total number of positioning point, L _r is the matching path length, and L is the actual driving length. We compare our proposed algorithm with the global matching algorithm based on weak Fréchet distance and the piecewise fuzzy matching algorithm over the running time and matching accuracy.

4.3.1 Measurement with running time

The running time of the algorithm is an important factor to measure whether the algorithm is practical when processing map matching for massive floating car data. In order to evaluate the running time of the proposed algorithm, we compare the running time among the proposed algorithms, the global matching algorithm based on weak Fréchet distance [13], and the piecewise fuzzy matching algorithm [18]. The sampling time interval of the low-frequency floating car data is 120 s. Taking a different number of positioning points for matching, the running time results of three algorithms are shown in Fig. 12. The experimental results show that the running time of the proposed algorithm is very close to the piecewise fuzzy matching algorithm, and it is better than the matching algorithm based on weak Fréchet distance. When the number of matching points increase, the computation time of the matching algorithm based on weak Fréchet distance increases exponentially, but our proposed algorithm is almost stable. It indicates that the algorithm proposed in this paper is of good time efficiency.

Fig. 12

Running time of different algorithms

4.3.2 Matching accuracy

We take a sample of 20 sets of floating vehicle data with different sampling intervals and consider the average matching accuracy of these 20 sets of data as the final matching accuracy. We compare the matching accuracy among the proposed algorithms, the global matching algorithm based on weak Fréchet distance, and the piecewise fuzzy matching algorithm with the factor at different sampling intervals. We mainly compare the correct matching ratio of positioning point A _N , as shown in Fig. 13, and the length accuracy ratio A _L as shown in Fig. 14.

Fig. 13

Results of A _N at various intervals

Fig. 14

Results of A _L at various intervals

We can find that our proposed algorithm is superior to the other two algorithms in terms of accuracy. With the increase of the sampling time interval, the accuracy of three matching map-matching algorithms would decrease then. When the sampling interval is 180 s, A _N of the proposed algorithm is still as high as 83% and A _L is 81%. These results demonstrate that the proposed map-matching algorithm is able to make full use the low-frequency floating car data and obtain high matching accuracy with building the optimized map-matching model combined with the network data.

5 Conclusions

In this paper, based on the analysis of low-frequency floating car data and the research of existing map-matching algorithms, we propose an optimized map-matching algorithm based on matching path with low-frequency floating car data. The proposed algorithm takes full consideration of the factors such as vehicle distance, direction, speed, and road topology. Firstly, we preprocess the GPS data and road network data to determine the potential points and sections by constructing the error region. Then, we calculate the potential matching path graph with the analysis of time and space. Finally, we can obtain the matching result by parallel computing with sections division methodology. With the contrast experiment in running time and matching accuracy, we can find out when the sampling time of the floating car data is 120 s and the number of positioning point is 400, the running time of our proposed algorithm can be within 2 s, which is of high efficiency. Meanwhile, when the sampling interval is 180 s, the correct matching ratio of positioning point of the proposed algorithm is up to 83%, and the length matching accuracy ratio is up to 81%, which is more accurate than other algorithms.

With the construction of the city in the future, the road will be more and more three-dimensional. In the future, the map-matching algorithm will expand to the three-dimensional map matching.