An Interactive-Voting Based Map Matching Algorithm

Transcription

1 Eleventh International Conferene on Mobile Data Management An Interative-Voting Based Map Mathing Algorithm Jing Yuan* University of Siene and Tehnology of China Hefei, China Yu Zheng Mirosoft Researh Asia Beijing, China Chengyang Zhang University of North Texas Denton, TX. U.S.A. Xing Xie Mirosoft Researh Asia Beijing, China Guang-Zhong Sun University of Siene and Tehnology of China Hefei, China, Abstrat Mathing a raw GPS trajetory to roads on a digital map is often referred to as the Map Mathing problem. However, the ourrene of the low-sampling-rate trajetories (e.g. one point per minutes) has brought lots of hallenges to existing map mathing algorithms. To address this problem, we propose an Interative Voting-based Map Mathing (IVMM) algorithm based on the following three insights: ) The position ontext of a GPS point as well as the topologial information of road networks, ) the mutual influene between GPS points (i.e., the mathing result of a point referenes the positions of its neighbors; in turn, when mathing its neighbors, the position of this point will also be referened), and ) the strength of the mutual influene weighted by the distane between GPS points (i.e., the farther distane is the weaker influene exists). In this approah, we do not only onsider the spatial and temporal information of a GPS trajetory but also devise a voting-based strategy to model the weighted mutual influenes between GPS points. We evaluate our IVMM algorithm based on a userlabeled real trajetory dataset. As a result, the IVMM algorithm outperforms the related method (ST-Mathing algorithm). Keywords-map mathing; GPS trajetory; road network; voting I. INTRODUCTION The inreasing popularity of GPS-enabled devie has failitated users to trak moving objets, suh as vehiles and people. However, as the readings of a GPS sensor have positioning errors and sampling errors [4], the departure of the GPS traking data from the atual trajetory an hardly be avoided. To math an original GPS traking data to a digital map or a digital road network is often referred to as Map Mathing. The general purpose of a map mathing algorithm is to identify the true road segment on whih a user (or a vehile) is/was travelling. Map mathing is a key proedure in many loation-based appliations, suh as vehile navigation [8], fleet management [9], intelligent transport systems (ITS) [7], and many other loation based servies [, 7]. Sine most of the ivilian GPS devies and GPS modules in smart phones are low-end and low auray, a sophistiated and reliable map mathing algorithm is ruial for these loation-based servies. There has been an inreasing attention on the map mathing problem. The majority of the existing map mathing algorithms onsider the senario that the sampling rate is high (e.g. one sample per 0 seonds). However, in pratie, there exists a large quantity of low-sampling-rate GPS traking data (the sampling interval is more than minutes). For instane, to save the ommuniation and energy ost, the taxis usually report their GPS positions to the dispathing enter with lowsampling-rate data. Fig. presents the statistial distribution of the sampling intervals of the GPS trajetories generated by 0,000+ taxies in Beijing in a week. The average time interval of the data set is.7 minutes. Aording to the result in the hart, only 4% of the data is high-sampling-rate (the sampling interval is less than minute) data. More than 60% of the data is low-sampling-rate data. ~ minutes 8% 0~ minutes 4% ~0 minutes 58% ~6 minutes 86% 6~0 minutes 4% Figure. Distribution of the sampling interval This poses new hallenge for onventional map mathing algorithms sine as the sampling rate grows, the details between two sampling points are missing. For example, if the sampling interval is minutes, the distane between two sampling points may reah to 000 meters even if a vehile s speed is 40 km/h. In a dense urban road network, a taxi may pass through several road segments during this period. Another problem is that the mathed road segments of most exiting methods may be disonneted when the GPS sampling interval is large. Therefore, an adaptive and robust map mathing algorithm with aeptable aurate is needed for this senario. In this paper, we investigate the problem of map mathing low-sampling-rate GPS traking data. To math this kind of data to the real map, we need to make full use of the interior information of the traking data as well as the topology of the road networks. For instane, Fig. shows a real road map and the GPS traking data (the red points). We an easily manual *This work was done when Jing Yuan was an intern at Mirosoft Researh Asia /0 $ IEEE DOI 0.09/MDM

2 math the trajetory to the blue dot line whih is the true path of this vehile. This proedure indiates our key insights: ) Position Context: The mathed positions of sampling points are effeted by other points. Point is mathed to E Yesler Way (the horizontal blue dot line) rather than th Ave street though point is muh loser to th Ave street. Why is that? This is beause when we are labeling point, we have onsidered, subonsiously, the positions of the other points nearby (e.g. point a, b, d, e) and their tendeny. We an definitely deide that is not loated at th Ave (the vertial road). In other words, we regard the neighboring points as point s referene points. The sampling points affet eah other. Figure. Illustation of our insights ) Mutual Influene: The sampling points have an interative influene on eah other. As stated above, when we are labeling point, we look upon its neighboring points as referene points. However, in the same time, point is also other points referene point when labeling its neighbors. Therefore, this referene relationship is mutual and interative. ) Weighted Influene: The farther away two sampling points are, the more limited they influene eah other. In Fig., when we are labeling point, as stated above, we refer to the positions of other points. However, the influene made by points d and b is obviously more important than point f whih is farther away from point. Whether point f mathed to the vertial road or the horizontal road hardly affets the position of point. This fat indiates that the influene made by the referene points varies aording to their distanes from the mathed point. Based on these insights, we present a novel Interative Voting-based Map Mathing algorithm (termed IVMM). We design a voting proess among all the sampling points to reflet their interative influene. For eah sampling point, we find out their andidate road segments, and for eah andidate, there exists an optimal path whih is passing through it. Every andidate will vote for their best path, and the global optimal path will be hosen aording to the voting result. In general, our ontributions an be summarized as: We study the interative influene of the GPS traking points and propose a novel voting-based algorithm IVMM for map mathing low-sampling-rate GPS traking data. Extensive experiments are onduted on real datasets. The data is olleted from real-world and labeled by real people. Therefore, the results are more reliable than syntheti data used in most existing work The evaluation results validate the advanement of our IVMM algorithm ompared to existing method for low-sampling rate data in terms of mathing auray. The outline of this paper is as follows. Setion II reviews the related work and ategories urrent map mathing algorithms. Setion III formulates the map mathing algorithm and gives an overview of IVMM algorithm. The detail of IVMM algorithm is introdued in Setion IV and analyzed in Setion V. The experiment results are presented in Setion VI. Setion VII onludes the paper and gives diretions for our future work. II. RELATED WORK This setion reviews the existing map mathing algorithms and ategories them. Approahes for these algorithms an be lassified by various riteria. A. Involved Information of Input Data. Aording to the information of input GPS traking data used, existing methods an be ategorized into four groups: geometri [], topologial [, 4], probabilisti [5] and other advaned tehniques [6, 7, 9]. Geometri map mathing algorithms utilize the geometri information of the spatial road network data by onsidering only the shape of the links without the onnetivity of the links []. A map mathing algorithm whih makes use of the the onnetivity and ontiguity information is referred to as a topologial map mathing algorithm [4]. Topologial methods use the topology of map features to onstrain the andidate mathes for a sampling point. For example, in Fig., though point A is loser to the vertial road, but A is sure to be mathed to Redmond Way sine there is no way to go from the vertial road to the Redmond Way rather than the vertial road. A probabilisti map mathing algorithm is developed in [5]. Advaned map mathing algorithms are referred to as those using more refined onepts suh as a Kalmam Filter [6], a fuzzy logi model [7] or the appliation of Hidden Markov Model [9, 9]. However, these algorithms still perform poor when the sampling rate is low. In [9], the evaluation shows the error rate is more than 50% when the sampling interval exeeds 5 minutes. Figure. Map mathing using topologial information 44

3 Our approah is based on both geometri and topologial information of the road network. The IVMM algorithm, different from existing topologial methods, also onsiders the temporal/speed information of the road network. B. Global Algorithms and Loal Algorithms Map mathing algorithms an also be ategorized into loal/inremental algorithms and global algorithms aording to the range of sampling points onsidered when mathing the trajetories. The loal/inremental algorithms follow a greedy strategy of sequentially extending the solution from an already mathed portion. These methods try to find a loal optimal point or edge based on distane and orientation similarity [, 8, 0]. The inremental algorithm desribed in [] using a onstant-depth reursive look-ahead strategy is also based on loally mathing geometries. Wenk et al. proposed an adaptive lipping method whih utilizes the Dijkstra algorithm to onstrut a shortest graph on loal free spae graph []. A segment-based mathing algorithm introdued in [] assigns onfidene values for different sampling points. This algorithm mathes high-onfidene segments first, and then mathes lowonfidene segments using previous mathed edges. In general, loal/inremental algorithms usually run fast when sampling rate is very high (e.g. -5 seonds a sample). Therefore, the loal/inremental algorithms are often adopted in the appliations with online requirement. However, the performane of these algorithms gets worse when the sampling rate is not high. As the sampling rate dereases, the problem of ar-skipping [] beomes prominent, ausing signifiant degrade of auray. The other group of algorithms alled global algorithms tries to find a trajetory whih is as loser as the sampling trak among all available trajetories in the road network. To measure the similarity between the mathed trajetory and the sampling trajetory, most algorithms employ the Fréhet distane or Weak Fréhet distane. The algorithm proposed in [5] applies parametri searh over all ritial values, then it solves the deision problem by finding a monotone path in the free spae from the lower left orner to the upper right orner. This work is extended in [] with average Fréhet distane to redue the effet of outliers. Paper [] also uses weak Fréhet distane that runs in ( log time with similar mathing quality. Paper [4] proposes an algorithm based on a weighted graph representation of the road network. This algorithm is based on a measurement similar to the Average Fréhet distane. Sine the global algorithms require the information of an overall traking trajetory, they are often used in offline situation. In ontrast to urrent global map mathing algorithms, we onsider the mutual influene of the sampling points themselves as well as the impat of remoteness on the positions of the sampling points. Based on a novel voting strategy, our algorithm is more robust and reliable than existing methods. C. Sampling Rate Aording to the sampling density of the traking data, we an ategorize existing map mathing algorithms to densesampling-rate approahes and low-sampling-rate approahes. As stated in [4], most global algorithms perform poor when GPS sampling intervals are larger than 0 seonds- the average orretness method is less than 60%. To the best of our knowledge, [] is the first and only paper to address the problem of map mathing for low-sampling-rate GPS trajetories. ST-Mathing [] is a map mathing algorithm for low-sampling-rate GPS traking data whih inorporates both the geometri topologi struture and the speed onstrains. This algorithm first retrieve a set of andidate points for eah sampling point, then define a similarity funtion with respet to every two sequential sampling points and their andidate points. Based on the summation of the similarity funtion during the whole path, the ST-Mathing algorithm finds a path with the largest summation to be the result path. The authors onduted extensive evaluation to investigate the performane of ST- Mathing algorithm. The experiment results validate ST- Mathing outperforms the methods based on Fréhet distane both in terms of mathing auray and running time. However, by experiments, we find that when the length of trajetory is long and the vehile passes through a multi-lane road, the mathing result given by ST-Mathing algorithm is still not reliable. For example, Fig. 5 is a sreenshot of the visualized result of ST-Mathing algorithm on a real GPS traking data. The red urve onnets the GPS sampling data and the blue line marked by green pushpins is the mathing result of ST-Mathing algorithm with respet to the sampling data. The mathed path still runs in a roundabout zigzag way whih obviously deviates from the real path. Figure 4. A sreenshot of ST-Mathing result The mismathing of ST-Mathing algorithm is aused by these reasons: The similarity funtion is built solely with respet to two adjaent andidate points, whereas the position of a sampling point is influened by all its neighboring points, both previous points and latter points. Thus the values of the edges in the andidate graph omputed by ST-Mathing algorithm do not reflet the true influene of the neighboring points enough. ST-Mathing algorithm uses a simple summation of all the values in the trajetory to evaluate the similarity of a andidate path with the sampling data. It neglets the impat of remoteness on the sampling point by other points. The reiproal influene mentioned in Setion I (our insight ) is not onsidered by ST-Mathing algorithm. 45

4 If one point is mathed to a wrong road segment, the following points will all be measured based on this mismathed point, thus that the errors of the proess are umulative. III. PRELIMINARY In this setion, we first define the onepts used in this paper and then give an overview of the interative voting-based map mathing algorithm. A. Problem Definition Definition. GPS trajetory: A GPS trajetory is a point sequene linked by the time stamps of the GPS points. Eah GPS point is a triple (.lat,.lng,.t) whih are its latitude, longitude and timestamp respetively. Definition. Road network: A road network is a direted graph (,, where is a set of edges representing the road segments. is a set of verties representing the intersetions and terminal points of the road segments. Eah road segment is a direted edge that is assoiated with an id. eid, a typial travel speed. v, a length value. l, a starting point. start, an ending point. end as shown in Fig. 5. of the position ontext. After that, we build a stati sore matrix aording to the andidate graph. The mutual influene is modeled utilizing a weighted sore matrix whih is onstruted dynamially. Based on the weighted sore matrix, all the andidate points parallelly vote for their best mathing paths in the last phase. Then a global optimal path is eleted aording to the voting result. Figure 6. Overview of the IVMM algorithm IV. INTERACTIVE VOTING-BASED MAP MATCHING ALGORITHM In this setion, we introdue our Interative Voting-based Map Mathing Algorithm (IVMM) and give an example to show how IVMM works. A. Candidates Preparation Given a GPS trajetory, we retrieve a set of andidate road segments (CRS) for eah sampling point by a range query. The set CRS ontains all road segments within radius r (a fixed number) of with respet to Eulidian distane. Figure 5. Illustration of a path in a road network Definition. Path: Given two verties, in a road network, a path is a set of onneted road segments that start at and end at, i.e. :, where.,.,..,. Fig. 5 illustrate a typial path onneting several road segments. The map mathing problem then an be defined as: Given the road network G and a raw GPS trajetory T, find a path in G whih mathes T with its real path. B. Framwork Motivated by the insights we presented in setion I as well as the disadvantages of ST-Mathing we disussed in setion II, we develop the IVMM algorithm whih is aimed to make the best of the interative relationship among all the sampling points so as to find a global optimal path to math the trajetory, espeially for low-sampling-rate situation. The IVMM algorithm onsists of four phases: andidate preparation, sore matrix building, interative voting and path finding. Fig. 6 shows a framework of the IVMM algorithm. In the first phase, we perform a range query to selet the andidate road segments (CRS) and andidate points (CP) for eah sampling point, then in the seond phase, we onstrut a andidate graph by performing a spatial and temporal analysis Figure 7. Illustration of andidate road segements and andidate points The andidate points (CP) are seleted in the following way: if the projetion of the sampling point onto the road segment is between its endpoints, then hoose the geometry projetion as the andidate point; otherwise, hoose the endpoint whih is loser to the sampling point with regard to the Eulid distane as shown in Fig. 7. Thus we get a andidate point set CP i for eah. Fig. 8 gives an example of a trajetory. After the proess of andidates preparation, we obtain CRS and CP for eah depited in Fig. 8. For instane, CRS ={,, } and CP ={,, }. Let be the ardinality of CRS, thus we have CRS CP. e p e e e p e Figure 8. Candidate road segments and andidate points p e e e 4 p e 4 e 4 46

5 B. Position Context Analysis In this phase, we perform the Spatial Temporal Analysis and onstrut a andidate graph (,. The distribution of the measurement error is assumed to satisfy the Gaussian distribution (, [8]. For a andidate point, its observation probability with respet to is formulated as: e. () where is the Eulid distane from andidate to sampling point. Figure 9. Candidate graph From andidate point to andidate point, the spatial analysis funtion is defined as below: ( where ( ( (,. () is the transition probability defined as: (. () (,(, where is the Eulidian distane from sampling point to sampling point, and (,(, is the length of the shortest path from andidate to. V' is the union of CP i and E' is a set of shortest paths between any two andidate points. The spatial analysis funtion is aimed to measure the similarity between the andidate paths with the shortest path with respet to two adjaent andidate points. This is based on the assumption that a driver is more likely to hoose a shorter route when driving. We also define a temporal analysis funtion onsidering the speed onstrains of the road segments as follows: ( (. (,(, (. (,(, where is the shortest path onneting with, and (,(, the average speed from to whih an be easily omputed. We adopt the definition of STfuntion for in [] whih is given as below: ( ( (4) ( (5) where is the spatial analysis funtion and is the temporal analysis funtion. (For the details of spatial/ temporal analysis, please refer to paper [].) After spatial and temporal analysis, a andidate graph is onstruted. The nodes of the graph are the set of andidate points for eah GPS observation, and the edges of the graph are set of shortest paths between any two neighboring andidate points. The nodes and edges are all assigned weight values based on the results of spatial/temporal analysis. C. Mutual Influene Modeling ) Stati Sore Matrix Building Based on the andidate graph generated in the previous phase, we an build a Stati Sore Matrix denoted as (, (,, ( where ( ( ( = ((. We provide the sore matrix of trajetory as an example Eah item in this sore matrix represents the probability of a andidate point like to be a orret projetion only onsidering the information of two onseutive points, e.g.,. But, this information does not reflet the interative mutual influene between the andidate points, as disussed in Setion II. ) Weighted Influene Modeling To model the weighted influene of the andidate points, we define a ( -dimension Distane Weight Matrix for eah sampling point. This diagonal matrix gives weights for the effet of all other points to assoiated with their distanes to. For,,, defined as below: (, (,, (, (, ( (6) and = (, (,, ( (7) where ( (,,,, and (, is the Eulidian distane between and. We term the f funtion as distane weight funtion, whih represents the influene of the point to the point based on the distane between them. We ll further disuss the f funtion later in Setion V. Regarding matrix M as a ( -order blok matrix, for eah,,,,, the Weighted Sore Matrix is defined as: (, (,, ( (8 For eah andidate point of sampling point, matrix represents the similarity of all the andidate road segments with the true path onsidering the influene of the remoteness. Note this matrix multipliation proess ensures that if two sampling points have the same distanes to the third sampling point, their 47

6 influenes on the third sampling point are also equal. That is beause matrix is an order matrix just without the element ( (. In fat, for,,,, (, ( ( ( ( ( if ( ( ( ( ( (9) otherwise To illustrate this proessing, let s follow with the previous example. If we set (,, and suppose, (in real implementation, the dist funtion is the atual Eulid distane, here just for onveniene of explanation), the weighted sore matrix are: For instane, matrix is the weighted sore matrix for ( sampling point. Compared with the stati matrix M, and ( are both halved sine ( is assoiated with the transmission from CP to CP, and ( is related to the transmission from CP to CP. Based on the assumed distane weight funtion, their impats on are the same. The stati sore matrix is onstruted in this phase; whereas in pratie, the distane weight matrix and weighted sore matrix an be generated dynamially in the next phase when needed so as to save the memory spae though we defined them here. D. Interative Voting Reall the insights presented in Setion I, the influene of the sampling points is mutual. When a sampling point refers to other points, it is referened by other points in the same time. Considering the interation of the andidate points, we design a novel and effiient voting algorithm for the andidate seletion. For eah sampling point, the weighted sore matrix is built. Then for eah andidate point, we tried to find an optimal path P whih passes aross utilizing the weighted sore matrix. After that, andidate point votes for all the andidate points on path P. Sine the entire voting proess is independent for eah sampling point, it an be implemented in parallel. We now present our algorithm in pseudo ode. Algorithm InterativeVoting Input: Candidate graph (,, stati sore matrix Output: The vote ounts of eah andidate point : for i = to n do in parallel : Compute and : for k = to do 4: P=FindSequene (G,, i, k) 5: Vote for eah andidate point in trajetory P For eah andidate of, the FindSequene proess is alled to find a path P whih passes through the andidate point and has the largest weighted sore summation. The FindSequene proess works in a dynami-programming paradigm. The pseudo-ode is as follows: Algorithm FindSequene Input: Candidate graph (,, matrix, i, k Output: The sequene : /* retrieve a path whih must get aross and has the largest weighted sore summation with respet to */ : Let, denote the highest sore omputed so far; : Let, denote the parent of urrent andidate; : for t = to do 4:, ( w ( ; 5: for s = to do 6: if then 7: for t = to do (, 8: 5: for to n do 6: for s = to do 7:, (, =max,,t,, 8:, (, argmax,,t,, 9: Initialize as an empty list; /* a list of mathed points in reverse order*/ 0: (,,,, ); :. (,,,, : for to do :.add(); 4:, ; 5:.add(); 6: return. (); The FindSequene proess retrieves the best path for eah andidate point (we all it a loal optimal path) as well as a fvalue whih is used in the next phase when two andidates have the same number of votes. Now ontinue with our example before. As stated in the seond phase, eah andidate point has an observation probability. Given the observation probability of CP as shown in Tab., after the Find Sequene proess, the best path for andidate and its fvalue are illustrated in Fig

7 p 's andidates p 's andidates p 's andidates p 4 's andidates Figure 0. FindSequene proess of andidate TABLE I. INITIAL OBSERVATION PROBABILITY After the interative voting, we pik the andidate point who has the most votes for eah sampling point ; if two andidate point have the same votes, ties are broken using the fvalue. Then we link the andidate points one by one, the global optimal path is obtained. TABLE II. VOTING RESULT Considering the above example, after Interative Voting, we ount the votes for eah andidate point. The result shows in Tab. II. Then the global optimal path is eleted whih is as lined out in Fig Figure. Mathing result of IVMM algorithm V. ALGORITHM ANALYSIS This setion first analyzes the omplexity of IVMM algorithm, and then disusses the seletion of the distane weight funtion. A. Algorithm Complexity The time omplexity of IVMM an be divided into four parts aording to the four phases of IVMM. Let denotes the number of sampling points on the given trajetory, and denotes the number of road segments in the road network. We further assume that the maximum number of andidates of one sampling point is. In the first phase, a range query is performed to get the andidate points. The time ost for range query is (4 by using ell index [4]. In the seond phase, the onstrution of the andidate graph takes a ( log time sine the CP N( ) CP Votes fvalue number of shortest paths in is ( and the shortest path an be omputed in( log time utilizing Dijkstra algorithm (n m). The weighted sore matrix an be built in ( time. The omplexity of FindSequene proedure is ( sine eah andidate is at most visited one. Therefore, the Interative Voting algorithm takes a ( time due to parallel omputation. Finally, the omplexity of the last phase is (. In total, the omplexity of IVMM is (. Note that even if the Interative Voting algorithm uses a round-robin way rather than a parallel omputation, the total omplexity of IVMM is still ( beause usually in a single trajetory nm for low-sampling-rate senario. Note that the omplexity of ST-Mathing algorithm and IVMM algorithm is the same. [] B. The Distane Weight Funtion In Setion IV, we introdue a distane weight funtion ( in the position ontext analysis phase of IVMM algorithm. Let x be the Eulidian distane of two sampling points, then ( denotes the impat of distane x on eah sampling point with respet to road network r. The hosen of f is an interesting open problem. However, the f funtion should satisfy the following properties: ) (0. ) ( 0. ) For 0, 0 ( In general, f should be a delining funtion sine the farther away, the less two sampling points impat on eah other. For example, an exponential delining funtion e is an available distane weight funtion. We believe that the impat of the distane on the position of the sampling points satisfies Gassian/Normal distribution sine if two sampling points are relatively lose, then their influene on another point is pretty muh the same, while if the distane of two sampling points exeeds a ertain limit, the influene of eah sampling point delines very fast. Therefore, in our work, the distane weight funtion is defined as: ( (0) where is a parameter with respet to the road network. In Setion VI, we investigate the impat of parameter β and the impat of different distane weight funtions. VI. EVALUATION In this setion, we ondut extensive experiments to evaluate our IVMM algorithm. We first desribe the settings, then introdue the evaluation approah, and lastly report the evaluation results. A. Setting ) Road Netwroks In our experiments, we use the road network of Beijing visualized in Fig.. The network graph ontains 58,64 verties and 0,74 road segments. The vertial length of the map is about 47.7 km and horizontal length is about 5.6 km. 49

8 Figure. Road network of Beijing ) GPS Traking Data In our experiments, all GPS traking data is real data olleted from the Geolife [5, 6] System. Distintive from existing approahes, we use real human labeled true path data as the ground truth in our experiments. The raw GPS data set ontains 6 trajetories with varying number of points and average speed as Fig. (a) and Fig. (b) present. The trajetories over about 64 kilometers over more than 0 hours. Fig. () plots all the original trajetories on the map. Counts ~50 50~00 00~00 00~450 Number of Sampling Points a) Statistis on number of points b) Statistis on average vehile speed Counts ~0 0~0 0~0 0~40 40~50 50~60 60~ Average Vehile Speed (km/h) investigate the impat of different hoie of parameter and some other distane weight funtions on the mathing result. Platform: The algorithms are implemented in Java, on an Intel. GHZ PC with GB memory on Windows 7 operating system. B. Evaluation Approah Sine the ST-Mathing algorithm is the only map mathing algorithm whih performs well for low-sampling-rate data [], we ompared our IVMM algorithm with ST-Mathing both in terms of mathing quality and effiieny. To evaluate the effiieny of our algorithm, we ompared the running time of both algorithms. To evaluate the mathing quality, we use the orret mathing perentage (CMP) alulated by the following equation: CMP C N 00%. () We also investigate the impat of different distane weight funtions on the mathing quality. C. Results ) Vitualized results Fig. 4 and Fig. 5 present the sreenshots of the visualized mathing results of IVMM algorithm ompared with ST- Mathing algorithm on the same GPS traking data. The red line in Fig. 4 represents the GPS traking data and the blue line marked with red pushpins shows the mathing results. The left piture of Fig. 4 is the result of ST-Mathing algorithm. It runs in a roundabout way whih obviously deviates from the true path. The right piture is the mathing result of IVMM algorithm. It mathes all ground truth road segments in this portion. ) Sreenshot of all the GPS trajetories in the dataset Figure. Distribution of the GPS trajetories ) Parameter Seletion The sampling rate: This work fous on the low-samplingrate senario, thus the sampling interval ranges from 0 seonds to 0.5 minutes in our experiments. Parameters for IVMM algorithm: In our experiments, we set k=5 as the maximum number of andidates for eah sampling point. The radius of the range query is set to 00 meters. For observation probability estimation, we use a normal distribution (Equation ()) with 5 meters and 0 meters. The distane weight funtion: We use the distane weight funtion defined as equation (0) for the estimation. The parameter is set to 7 km as the default set. We also Figure 4. A sreenshot of mahing result (ST-Mahing V.S. IVMM) Fig. 5 is another sreenshot of the visualized mathing result. The original GPS traking data is the same with the example presented in Fig. 4. In both of the two pitures below, the red line represents the original traking data and blue line marked with green pushpins is the mathing result. On the left, ST-Mathing mismathes several road segments suh that the result is a zigzag way whereas on the right our IVMM straightens the mismathed road segments and mathes the ground truth path quite well. Compared with ST-Mathing algorithm, the result of our IVMM algorithm is more reliable and adaptive. 50

9 be mathed is orrespondingly redued suh that time ost of both algorithms dereases fast. Figure 5. Another sreenshot of mathing result (ST-Mathing V.S. IVMM) ) Corret Mathing Perentage Fig. 6 presents the mathing auray omparison result based on the orret mathing perentage (CMP). It is learly that IVMM algorithm signifiantly outperforms ST-Mathing algorithm for all sampling intervals. Reall the statisti result of real taxi GPS data in Beijing (Fig. ), the sampling interval of most low-sampling-rate GSP trajetories are ~6 minutes (about 86% among all low-sampling-rate data). When sampling interval ranges from.5 minutes to 6.5 minutes, the auray of IVMM is always about 70% with a 0% improvement of ST-Mathing algorithm. When the sampling interval exeed 6.5 minutes, the performanes of IVMM and ST-Mathing get loser but still have a 5% margin. The result validates that IVMM algorithm represents the interative influene of eah sampling points effetively and is more robust ompared with ST-Mathing algorithm. Figure 6. Mathing aruray (ST-Mathing V.S. IVMM) ) Running Time The time omplexity of IVMM algorithm is the same with ST-Mathing as analyzed in Setion V. We onduted experiments to evaluate the running time of both algorithms. The bar hart in Fig. 7 gives the result. Sampling Interval (minute) Corret Mathing Perentage (%) Sampling Interval (minute) Running Time(s) ST-Mathing IVMM(β=7km) IVMM ST-Mathing Figure 7. Running time (ST-Mathing V.S. IVMM) In this evaluation, the sampling interval ranges from 0.5 minutes to 0.5 minutes. The result demonstrates that IVMM algorithm is as fast as ST-Mathing algorithm for both lowsampling-rate and high-sampling-rate data. Fig. 7 also implies that as the sampling interval inreases, the number of points to 4) Impat of Distane Weight Funtions In Setion V, we disuss the hoie of distane weight funtions. Fig. 8 plots some different distane funtions we evaluate on the same data set. The x-axis is the distane of two sampling points; y-axis is the value of distane weight funtion whih represents the influene on the positions of two sampling points. Figure 8. Different distane weight funtions Fig. 9 presents the mathing performane of different weight funtions. The exponential funtion and linear funtion perform worse than the Gaussian funtion in terms of mathing auray. The performane of IVMM algorithm without distane weight funtion (denoted as IVMM (none) in Fig. 9) is also worse than approahes with a Gaussian distane weight funtion for all sampling intervals. This validates again that the influene of sampling points is related to their distane. Corret mathing perentage (%) Distane Weight Value f(x) Distane of Sampling points (km) Sampling Interval (minute) β=km β=7km β=0km exponential Figure 9. Impat of different distane weight funtions We also estimate the impat of on the distane weight funtion as shown in Fig. 0. The auray rate when is lower than when 4, 7 and 0, but still better than ST-Mathing algorithm in Fig. 6. The reason for this is that when, the value of distane weight funtion dereases too fast as showed in Fig. 8. VII. CONCLUSION In this paper, we investigate the problem of map mathing for low-sampling-rate GPS trajetories. A novel algorithm termed IVMM is proposed and analyzed. This algorithm employs a voting-based approah to reflet the mutual influene of the sampling points. We define a distane weight funtion to evaluate the impat of distane to the mathing positions. Extensive experiments are onduted with real GPS traking data. Both for low-sampling-rate data and high-sampling-rate data, the orret mathing perentage of IVMM algorithm is linear IVMM(β=0) IVMM (exponential) IVMM (none) IVMM (linear) 5

10 higher than ST-Mathing algorithm. In partiular, when the sampling interval ranges from to 6 minutes, the auray rate of IVMM algorithm always has a more than 0% improvement over the ST-Mathing algorithm. The results demonstrate that IVMM algorithm signifiantly outperforms ST-Mathing algorithm whih is so far the only approah aimed at lowsampling-rate GPS data in terms of mathing quality. Corret Mathing Perentage (%) Sampling Interval (minute) Figure 0. Impat of parameter with IVMM IVMM(β=km) IVMM(β=4km) IVMM(β=7km) IVMM(β=0km) The distane weight funtion whih plays an important role in IVMM algorithm is intriguing. We note that the weighted influene of the sampling points is not only related with the distane but also with the topology of the road networks. Typially, the influene of two sampling points in a dense road network is muh less than that of a sparse one. That is beause the more the number of possible paths traversing two points is, the less weighted influene they might brought on eah other. We will do more researh on this topi in our future work. ACKNOWLEDGEMEN This work is partially supported by Mirosoft Researh Asia Internet Servies Theme Researh Program. REFERENCES [] C.Y. Zhang, Y. Zheng and X. Xie, Map-Mathing for Low-Sampling- Rate GPS trajetories, in Proeedings of ACM SIGSPATIAL Conferene on Geographial Information Systems (ACM GIS 009). [] I. Raushert, P. Agrawal, R. Sharma, S. Fuhrmann, I. Brewer and A. MaEahren, Designing a Human-entered, Multimodal GIS Interfae to Support Emergeny Management, in Proeedings of the 0th ACM international symposium on Advanes in geographi information systems, MLean, Virginia, USA, 00. [] Bing Map Searh. Mirosoft Corporation [4] D. Pfoser and C. S. 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