Trajectory Pattern Mining via Clustering based on Similarity ...traffic data, that is vehicle trajectory, is selected as reliable data for discovering the underlying pattern via trajectory

MEI YEEN CHOONG et al: TRAJECTORY PATTERN MINING VIA CLUSTERING BASED ON SIMILARITY . . .

DOI 10.5013/IJSSST.a.17.34.19 19.1 ISSN: 1473-804x online, 1473-8031 print

Trajectory Pattern Mining via Clustering based on Similarity Function for Transportation Surveillance

Mei Yeen Choong, Renee Ka Yin Chin, Kiam Beng Yeo, Kenneth Tze Kin Teo Modelling, Simulation & Computing Laboratory

Faculty of Engineering, Universiti Malaysia Sabah Kota Kinabalu, Malaysia

E-mail: [email protected], [email protected]

Abstract — Recently, surveillance on moving vehicles for traffic flow monitoring has emerging in rapid rate. A comprehensive traffic data, that is vehicle trajectory, is selected as reliable data for discovering the underlying pattern via trajectory mining. As the task of monitoring moving vehicles via vehicle trajectory dataset can be tedious, researchers are keen to provide solutions that reducing the tedious task performed by the traffic operators. One of the solutions is to group the vehicle trajectory data according to the shape of the patterns. This grouping task is called as clustering. Each of the clusters formed represents a pattern. In this paper, the analysis of the implemented clustering algorithm on the trajectory data with similarity function is presented. Discussion on the issues concerning the trajectory clustering is also presented.

Keywords - vehicle trajectory; trajectory mining; trajectory clustering

I. INTRODUCTION

Surveillance plays an important role in intelligent transportation systems (ITS) in which government has invested heavily invested in it. It is indisputable that the rapid development of urbanization due to economic growth requires an efficient surveillance system to monitor traffic flow data. Despite that many researchers focused on traffic data in macroscopic scale, it is still crucial to analyze and monitor the microscopic data such as detecting the abnormal behaviors of traffic scene, which enables the determination of the origin of traffic congestion and conflicts [1]. Thus, the challenge of obtaining precise information about the traffic flow at region of interest such as roundabout is generally difficult.

In the past decades, many computer vision researchers have been working closely to study the pattern of moving traffic at road intersection by using vehicle trajectory (tracked point of vehicles) as input data. The traffic flow of a road intersection may contain statistical information which is of interest to researchers, such as: the traffic volume, left and right turning times, lane changes, as well as the U-turn event counts. With automatic identification of this behavioral information, modeling the traffic flow will be easier as the amount of manually inputted parameter is reduced [2]. However, describing the dynamics of traffic flow via conventional statistical methods still lacks of direct interpretation for human understanding. In the attempt to reduce manual works of monitoring for traffic flow, behavioral pattern learning from trajectory has been useful to understand surveillance activity.

This paper presents a study of the potential of trajectory modeling with clustering algorithm. The goal of the clustering algorithm is to group common similarities

exhibited in a large dataset of trajectory into designated clusters. In preliminary work of trajectory modelling, the retrieval of spatial-temporal trajectory dataset that usually comes in large amount has been introduced through various methods [3]. Although the trajectory query provides labeling task on the raw data for indexing the dynamics spatial-temporal queries, this information is insufficient to provide distinct statistics for dynamic traffic flow. With the clustered trajectory dataset, pattern that describes the maneuver behavior of travelling vehicles can be characterized. Thus, computing similarity measures for the clustering is a primary concerned by researchers [4]. This is due to the inference about the interactions between groups of trajectories are highly emphasized and it is necessitated to compute the measures for the pattern learning task [2].

As the data structure of the trajectories has to be considered when performing clustering, data that are stored in clumps enables clusters to be formed by finding the boundary of grouped elements in the dataset. However, in the case of trajectory dataset, boundary of grouped data is not easily obtained as the trajectories are more likely to intersect with each other in the dataset. Such condition requires a similarity function to compare the closeness of the trajectories in pairwise manner. The similarity function provides the measurement of distance between paired trajectories. The organization of this paper is as follows: Review of trajectory extraction methods is presented in Section II. Section III describes the similarity function for trajectory and Section IV explains clustering method for the trajectory dataset. The experimental study of trajectory clustering implementation is shown in section V. Lastly, conclusion is provided in section VI.



Figure 1. Illustration of trajectory modelling concept in surveillance

activity.

II. TRAJECTORY EXTRACTION METHODS

In transportation surveillance, trajectory analysis serves as an important and basic research for behavior analysis. It is also becoming the main method used to understand vehicle motion behavior because it is relatively simple and apparent in its interpretation. As shown in Fig. 1, a trajectory is a path of a moving vehicle which contains a sequence of the vehicle’s tracked points over a period of time. These tracked points (illustrated as block dots) drawn from each action and surveillance activity can be defined with the set of sequential actions [5]. Each tracked point at a particular time instance can consist of the location of the vehicle, which can be either represented as the (x, y) coordinates or distance travelled. The (x, y) coordinates would be preferable as the collection of these coordinates points gives the vehicle’s maneuver (turning angle and direction). These tracked points are reliable data for traffic density analysis as they provide direct and unbiased measurement [6].

The vehicle tracking method and behavior pattern learning of the trajectory are not the primary focuses of this paper, but the approaches of the trajectory extraction are briefly explained here. Three approaches can be stated here, they are: sensor, global positioning system (GPS) [7] and vision based tracking.

A. Sensor

Sensor to detect traffic flow such as microwave radar, infrared sensors, ultrasonic sensors, passive acoustic array sensors, inductive loop detectors, magnetic sensor are commonly installed at fixed points or road sections. These sensors are static and generally used for detecting presence of vehicle and counting number of vehicles passing through certain location. However, discovering the trajectory pattern from these sensors solely is limited due to its difficulty in precisely obtaining the geographical information of the moving vehicles [8]. However, this vehicle trajectory acquisition method can be served as an important remedy when the modern positioning method is not available in service.

B. Global Positioning System (GPS)

Another method to probe vehicles for trajectory is through the collected geo-referenced coordinates by GPS receivers. These receivers require road map-matching with the targeted vehicle coordinates. Information of the vehicle speed can be computed quantitatively by measuring the spacing between two GPS tracked points of a specific region on a map [6].

Positioning error may arise and it is a main concern in extracting trajectory based on the GPS devices that provide time and location of tracked vehicles. In fact, the positioning error can be up to tens of meters, which is critical for safety-critical applications.

Many approaches on improving the vehicle positioning accuracy have been widely conducted. One of the prominent approaches is by measuring the inter-vehicle distances. Radio-based ranging methods such as Time of Arrival (ToA) and Angle of Arrival (AoA) are also used to alleviate the positioning error. Although the methods have demonstrated effectiveness in reducing the error, they still heavily rely on complicated techniques with high computational cost, and require additional sensors that are hard to be employed [9].

C. Video Based Vehicle Tracking

Generally this method uses image processing approaches to perform vehicle tracking with the source from video cameras. Well known methods such as Kalman filter, Markov Chain Monte Carlo (MCMC), particle filter [10, 11] are used to track vehicles via estimation of vehicle position based on color and other features that characterize vehicles. Although it is unavoidable that imperfect tracking due to occlusion, low illumination and other disturbances such as image noise resulted from camera vibration may occur, these tracking methods are constantly improved and are robust enough to handle these imperfections [12].

In general, this vehicle tracking process begins with some image segmentation method, such as foreground/background subtraction [13]. This segmentation excludes unwanted parts that do not belong to the vehicle. Throughout this process, each of the targeted vehicles in the continuous captured image frames is assigned with a particle to perform prediction for future vehicle position. To accommodate the non-linearity of the perspective projection of vehicles, three dimensional bounding box models can be fitted to the target vehicles. Sampling rate of the tracker has to be chosen optimally to eliminate the need of interpolation for the tracked points.

III. TRAJECTORY SIMILARITY FUNCTION

Considering a trajectory dataset containing N trajectories, each of the trajectories contains a sequence of spatial-temporal tracked points of a vehicle as shown in (1) Equation (2) shows a tracked point that constitutes its spatial location ),( k

iki yx at time instant it .

kn

kkkNk ppppT ,...,, 321,...3,2,1 (1)

Sequence of Actions

Surveillance Activity

Trajectory



Figure 2. An example of three trajectories via video based vehicle

tracking.

),()( ki

ki

ki yxitp (2)

In general, trajectory dataset is displayed as y coordinate

versus x coordinate (spatial) or 3 dimensional (3D) data with x coordinate, y coordinate, and time, t. An example of trajectory representation in x-y coordinates is shown in Fig. 2. The three different trajectories (labelled with three colours) in the Fig. 2 are intersecting each other. If these trajectories were to be clustered based on the tracked points solely regardless of which trajectories they are from, it will be difficult to group the tracked points into valid clusters. A pairwise comparison of trajectories should be measured based on their closeness and spatiality. Thus, a pre-clustering task that is formulating a similarity function to calculate the distance between paired trajectories shall be performed. These similarities include the aspect of shape and spatial closeness.

A simple approach of measuring the similarity between paired trajectories is to find common trajectories having the closest pair of points. There are two prominent trajectory similarity functions that are popular for measuring the Euclidean distance between paired trajectories: Longest Common Subsequence (LCSS) and Dynamic Time Warping (DTW). One of the reason that these two similarity functions are still widely used by researchers is that their capability to cope with different lengths (number of tracked points) of trajectories. In order to have similarity function that is robust to noises and outliers [14], LCSS is preferable in this paper.

To explain the formulation of the LCSS similarity function, let TA and TB be the two trajectories with their length m and n respectively. Whereby, trajectory TA contains

tracked points with Am

AAA pppp ,...,, 321 . The LCSS, L(TA, TB)

between the TA and TB is defined in (3). dE is the Euclidean distance, ɛ is the threshold value for the paired trajectories to be matched and δ limits how far in time during the matching process. For example, parameter setting of high ɛ depicts the

closeness of the paired trajectories is less strict whereas high δ conveys big time gap apart is allowed at two different time instants for the paired trajectories that are close in space. The computed LCSS distance for all the paired trajectories is then transformed into DLCSS as shown in (4). With the computed DLCSS, a similarity matrix, S whereby each element, s(A,B) is constructed using the Gaussian kernel function as defined in (5). Hence, valid clusters can be identified from this similarity matrix. Factorisation of the similarity matrix can be performed via eigen decomposition. This is to provide the feasibility of finding valid cluster using the clustering algorithm which will be discussed in the next section.

otherwise ,),( ),,(max

and

,),( if ),,(1 0or 0 if , 0

),(

11

11

Bn

ABAm

Bn

AmE

Bn

AmBA

TTLTTL

nm

TTdTTLnm

TTL

(3)

),min(

),(1,

nm

TTLTTD

BABA

LCSS (4)

]1 ,0[,2

2

2

),(

BALCSS TTD

BA eTTS (5)

IV. TRAJECTORY CLUSTERING

As the last stage of the trajectory pattern mining is aimed to group the trajectories into specific clusters, the eigenvectors resulted from the eigen decomposition are needed to be partitioned into exact k clusters. In this paper, fuzzy c-means clustering (FCM) is used for the partitioning. Fuzzy c-means (FCM) clustering is an iterative centroid-based clustering and was firstly introduced by Dunn in 1973. It was later further improved by Bezdek in 1981 [15]. FCM clustering is very similar to k-means clustering, such that it partitions a set of observations into a pre-determined number of clusters. Each observation is assigned to the cluster with the nearest mean value. FCM clustering converges when the minimization of the objective function is met. Equation (6) shows the minimization of the objective function in FCM clustering, where uij is the membership of an observed data, c is the cluster centroid (mean) and q is the level of fuzziness [16]. FCM is a soft version of k-means clustering, which allows an observed element to belong to more than one cluster, rather than just single cluster in the case of k-means clustering. The steps of FCM clustering in general are shown in Algorithm I. The stopping criterion for the iteration of FCM clustering is determined by α.

N

i

C

jj

ki

ki

kqij qcyxTu

1 1

21 ,),(min (6)

60 80 100 120 140 160 180 200120

130

140

150

160

170

180

190

200

210

220

x

y



ALGORITHM I. FUZZY C-MEANS CLUSTERING FOR TRAJECTORY

1: MEMBERSHIP FUNCTIONS INITIALISATION

2: Determine the total number of vehicle tracked points, N and number of clusters, C

3: Initialise the membership matrix with U = [uij]

4: CENTROIDS COMPUTATION

5: FOR j = 1:C

6: centroid of cluster j,

N

i

qij

N

iii

qij

j

u

yxTuc

1

1

),(

7: END FOR

8: MEMBERSHIP UPDATE

9: Update U to compute Unew with

1

2

1 ),(

),(

qC

k kii

jii

ij cyxT

cyxTu

10: If UU new then stop, otherwise return to centroids

computation step

V. EXPERIMENT RESULTS AND DISCUSSIONS

To illustrate the collected trajectory superimposed with an image of monitored scene, a trajectory dataset from Massachusetts Institute of Technology (MIT) is used [17]. This dataset contains 40,453 trajectories of moving objects in a parking lot scene that are captured for five days. As the trajectory dataset is huge and requires a significantly high computation time, 100 randomly selected trajectories are used. Fig. 3 shows the plot of collected trajectories from two perspectives. Each of the plotted trajectories in blue consists of an array of tracked points. From the Fig. 3, the clustering of the trajectories is necessitated to provide spatial definition by finding trajectories that are identical and grouped together as a cluster.

Results of FCM clustering on the trajectories with different number of clusters are shown in Fig. 4. It can be seen that trajectories are successfully grouped into the clusters with the parameter of ɛ = 10 and δ = 50. Each of the clusters provides specific shape. Trajectories in cluster 1 are share the same route and located at the bottom area of the parking lot. As for the trajectories in cluster 2, they are in the centre part of the parking lot and most of the trajectories share the u-shape route. Trajectories that does not share the same route shapes of trajectories in cluster 1 and cluster 2 are therefore grouped in cluster 3. As the number of clusters for FCM clustering is a required input, it may have ambiguity for the number of clusters to be determined depending how much details are to be acquired in the surveillance system. In other words, the number of route types can be related closely to number of clusters to be obtained.

The parameter setting of ɛ and δ give influence to the performance of similarity function which can impact the FCM clustering. The ɛ is a threshold value depicts the

maximum Euclidean distance between paired trajectories to be allowed. Hence, trajectories that are distributed in a way that they are far apart each other would require bigger value of ɛ. However, trajectories that are not constraint by geometrical properties such that only travel at certain lanes on road, are likely not useful. This is due to there can be no distinctive gap (example such as road divider) to distinguish

Figure 3. Plot of 100 randomly selected trajectories.

(a) Image of parking lot scene

(b) x-y plot of trajectories x

y

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100

150

200

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300

350

(c) Trajectories with background image x

y

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Trajectory

First Tracked Point

Last Tracked Point



Figure 4. Trajectory clustering result using FCM with ɛ = 10 and δ = 50.

the trajectories travelling with different route. The setting of δ relies on the sampling rate and length of a trajectory. In this case of parking lot scene, the trajectories are captured in high frequency rate such that the location of a vehicle is almost tracked in every one pixel it moves. This requires a high δ for appropriate trajectory similarity computation.

Clustering based on spatiality alone does not reflect the dynamic changes of the moving objects [18]. To reflect on the dynamic changes, clustering with the time domain should be considered when executing the clustering algorithm. Fig. 5 shows the trajectories plotted in 3 dimensions. Trajectories

that share similar route are seemingly topping up as the time goes.

Clustering results in five different time frames with the same parameters used in as in Fig 4 are shown in Fig. 6. Comparing the results in Fig. 4 and Fig. 6, the clusters vary greatly in the context of the shape and spatiality. As in Fig. 4, the trajectories in cluster 1 occupied at the bottom part of the parking lot. However, in b part of Fig. 6 (second time frame), 3 clusters are formed due to the trajectories are far apart each other. This clearly concludes that the two approaches trajectory clustering: spatial and spatial-temporal, requires different parameter setting and number of clusters to be generated.

VI. CONCLUSION

Trajectory clustering is an important task before performing pattern learning and prediction for surveillance purpose. It is useful in application with the purpose of finding semantic region for route and manoeuvre specification of the moving vehicles in a monitored scene.

The clustered trajectories can potentially provide visualisation system of traffic flow. This helps to assist operators in transportation surveillance to reduce manual work in providing traffic parameters for monitoring specific activities such as traffic conflicts that can cause collision.

Nevertheless, it is vital to understand that the spatial and spatial-temporal clustering approaches lead to different application-specific goals. Broadly speaking, determining the clustering type by considering whether the application is intended to find similar trajectories in a set of time frames or to find the sets of time frames with similar online trend is crucial. In a simplification of this, the spatial clustering is suitable for shape-based offline clustering. Whereas, the spatial-temporal clustering suits for online clustering based on correlation.

Figure 5. Plot of the 100 trajectories in 3 dimensions

(a) Cluster 1 x

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(b) Cluster 2 x

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(c) Cluster 3



ACKNOWLEDGMENT

The authors would like to acknowledge the Ministry of Higher Education Malaysia (KPT) for supporting this research under Exploratory Research Grant Scheme (ERGS), grant no. ERG0047-ICT-1/2013, Fundamental Research Grant Scheme (FRGS), grant no. FRG0405-TK-2/2014 and MyPhD scholarship under MyBrain15 program.

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Figure 6. Result of trajectory clustering in 5 different time frames.



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Trajectory Pattern Mining via Clustering based on Similarity ...traffic data, that is vehicle trajectory, is selected as reliable data for discovering the underlying pattern via trajectory