DOI 10.1007/s10619-012-7115-1
QS-STT: QuadSection clustering and spatial-temporal
trajectory model for location prediction
Po-Ruey Lei· Shou-Chung Li · Wen-Chih Peng
Published online: 20 October 2012
© Springer Science+Business Media New York 2012
Abstract Location prediction is a crucial need for location-aware services and
ap-plications. Given an object’s recent movement and a future time, the goal of location prediction is to predict the location of the object at the future time specified. Differ-ent from traditional location prediction using motion function, some research works have elaborated on mining movement behavior from historical trajectories for loca-tion predicloca-tion. Without loss of generality, given a set of trajectories of an object, prior works on mining movement behaviors will first extract regions of popularity, in which the object frequently appears, and then discover the sequential relationships among regions. However, the quality of the frequent regions extracted affects the ac-curacy of the location prediction. Furthermore, trajectory data has both spatial and temporal information. To further enhance the accuracy of location prediction, one could utilize not only spatial information but also temporal information to predict the locations of objects. In this paper, we propose a framework QS-STT (standing for QuadSection clustering and Spatial-Temporal Trajectory model) to capture the movement behaviors of objects for location prediction. Specifically, we have devel-oped QuadSection clustering to extract a reasonable and near-optimal set of frequent regions. Then, based on the set of frequent regions, we propose a spatial-temporal trajectory model to explore the object’s movement behavior as a probabilistic suf-fix tree with both spatial and temporal information of movements. Note that STT is
Communicated by Mohamed Mokbel. P.-R. Lei
Department of Electrical Engineering, Chinese Naval Academy, Kaohsiung, Taiwan e-mail:[email protected]
S.-C. Li· W.-C. Peng (
)National Chiao Tung University, Hsinchu, Taiwan e-mail:[email protected]
S.-C. Li
not only able to discover sequential relationships among regions but also derives the corresponding probabilities of time, indicating when the object appears in each re-gion. Based on STT, we further propose an algorithm to traverse STT for location prediction. By enhancing the quality of the frequent region extracted and exploring both the spatial and temporal information of STT, the accuracy of location predic-tion in QS-STT is improved. QS-STT is designed for individual locapredic-tion predicpredic-tion. For verifying the effectiveness of QS-STT for location prediction under the differ-ent spatial density, we have conducted experimdiffer-ents on four types of real trajectory datasets with different speed. The experimental results show that our proposed QS-STT is able to capture both spatial and temporal patterns of movement behaviors and by exploring QS-STT, our proposed prediction algorithm outperforms existing works.
Keywords Trajectory pattern· Movement behavior mining · Location prediction ·
Frequent region· Spatial-temporal data
1 Introduction
With the growth of location aware technologies such as mobile devices and GPS applications, it has become possible to track and collect an increasing amount of tra-jectory data from moving objects. A tratra-jectory is generally considered as the path left behind by a moving object in space and time, i.e, a sequence of location points where each point is a spatial-temporal data corresponding a position in space at a certain timestamp. As a huge amount of trajectories accumulated, there may be some trajec-tories frequently reappear in the data. Such trajectrajec-tories are called trajectory patterns. Due to the trajectory data records an object’s real movements, the trajectory patterns are supposed to reflect the object’s movement behaviors. The trajectory patterns play a fundamental role in helping to analyze the popularity of regions and traveling routes [3,13,16,17,19], the movement flows of users and location-based social networks [2,6,18,23,29–31]. In this paper, we target on utilizing trajectory patterns to predict a moving object’s location. The problem of location prediction can be generally for-mulated as: given an object’s recent movements and one query time in the future, the location of this object at the future time can be estimated. The location prediction is useful in many applications, such as prefetching for the location-based services, in-ferring the crowd of a region for tourism recommendations, and estimating the traffic status for transportation management.
The traditional method for location prediction is based on motion function. Given an object’s recent movements, one could predict future location by recent movement speed and direction. However, the motion function may provide assistance to pre-dict location in near future time and give an prepre-dicted location with large error in far future time. For example, if we know a user was at classroom at 10:00 and he is passing a post office at 10:10 currently, motion function could provide a reason-able predicted location at 10:20. For the location at query time 12:00, the motion function may give an incorrect location. To enhance the prediction accuracy, several works proposed that utilizing trajectory patterns to predict a moving object’s future
Fig. 1 An example of trajectory patterns
location [10,18,20,31]. An object’s movements are supposed to follow some trajec-tory patterns and then those patterns can be utilized as a movement model to predict the future location and improve the prediction accuracy. For example, users usually have their habitual routes from their home to workplace and animals have the annual migration. Without loss of generality, prior works on mining movement behaviors will first extract regions of popularity, in which the object frequently appears, and then discover the movement relationships among regions. It is shown in [10] that by exploring trajectory patterns in form of association rules, the accuracy of loca-tion predicloca-tion with a far future time is improved compared to the tradiloca-tional method (i.e., motion function approaches). However, most of the existing works only focus on trajectory pattern discovery in spatial domain. Prediction with spatial based trajec-tory pattern may result in biased prediction. Consider an example in Fig.1(a), where a user has two trajectory patterns (i.e., T1 and T2 without considering temporal in-formation). In T1, the user usually has the routine path from his home (i.e., region A) to his study place (i.e., region C) along the main street (i.e., region B) in week-days. In T2, the user often goes to shopping center (i,e., region E) along the main street from his home in weekends. The sequential patterns compute their support to indicate the number of times they appear in a trajectory dataset. In this example, T1 (i.e., A→ B → C) has higher support than T2. Assume that the user’s recent move-ments are regions A and B, and this user’s current location is region B around 10:00 am. By these two sequential patterns, the next location of this user is always predicted as region C since T1 has higher support in spatial domain. However, if the trajectory patterns are discovered by considering both spatial and temporal information. As an example shown in Fig.1(b), one could imply that if a user moves to C, the time will be 11:00 (i.e., the sum of the current time and the time interval between B and C). The appearing probability at region C at 11:00 via the routing path A→ B is zero. On the other hand, one could infer that this user may appear in region E at 12:00 with the probability larger than zero. we can predict the user has the higher oppor-tunity of moving to region E. Therefore, to further enhance the accuracy of location prediction, we address that one could utilize not only spatial information but also temporal information to estimate the object’s future location. Based on above
obser-vations, there are two major issues for our pattern-based location prediction problem as follows:
(1) We suppose that an object’s movement behavior follows some patterns and these patterns can be used as predictor to estimate the future location. The represen-tative and effectiveness of the trajectory patterns are critical for prediction accu-racy. For accuracy enhancement, the trajectory patterns should be designed and explored by considered in both spatial and temporal domain.
(2) Given the object’s recent movements and one query time in the future, how to utilize these discovered spatial-temporal trajectory patterns to predict the user’s location at specified future time is a considerable issue.
In this paper, we propose a framework QS-STT (QuadSection clustering and Spatial-Temporal Trajectory model) to explore the movement behavior in form of spatial and temporal model for location prediction. Prior to STT model exploring, the frequent regions where the user often passed by are first discovered by the QuadSec-tion clustering. Then, the algorithm of spatial-temporal trajectory model construcQuadSec-tion is developed, in which both sequential relationships among regions and the corre-sponding appearance time information of objects are captured. In light of QS cluster-ing based STT, we further propose an STT prediction algorithm to traverse STT for location prediction.
In frequent region discovery, grid-based clustering is a conventional approach. A cell is identified as a frequent region if the cell contains a sufficient number of trajectories which passed through it. However, determining a proper cell size is the challenging issue of grid-based clustering. The cell size affects the number of patterns and the accuracy of the location prediction [14]. In order to discover the possible number of movement patterns, as many frequent regions as possible are expected to be extracted. Given a minimum number of trajectories, clustering by setting a bigger cell size may easily detect the frequent region but lose the granularity of movement behavior modeling. On the contrary, a smaller cell size may earn granularity but be difficult in frequent region detection and result in fewer regions being discovered. To approach the reasonable and near-optimal set of frequent regions, we propose QuadSection clustering (abbreviated as QS clustering). QS clustering is based on the concept of divisive clustering [25] to detect the maximum number of regions within the optimal cell size in order to improve the completeness of the movement behavior modeling. Based on the frequent regions discovered by QS clustering, The spatial-temporal trajectory model (abbreviated as STT) is designed to discover the sequential relationships among regions in the spatial domain and derives the corresponding time probabilities of the objects which appear in the temporal domain.
The spatial-temporal trajectory model is represented as a variant of the probabilis-tic suffix tree with both spatial and temporal information of movements. Furthermore, the nature of the probabilistic suffix tree can convert a large number of sequential pat-terns into a compact model and generate a probability of next movement occurring in the recent sequences. Consequently, STT can reflect the moving behavior of an object in both spatial and temporal features, and its structure associated probability can be a predictor for the future location estimation. To evaluate the performance of the proposed QS-STT and location prediction algorithm, we conduct comprehensive
experiments on real data. Specifically, for verifying the effectiveness of QS-STT for location prediction under the different spatial density, we use four types of real tra-jectory datasets with different speed, including Walk, Run, Bike, and Car. The exper-imental results show that the proposed QS-STT is able to reflect an object’s moving behavior and can predict future movements at a specified time slot efficiently and accurately. The main contributions of this paper are summarized as follows:
1. We propose a location prediction framework QS-STT, which extracts the frequent region using QuadSection clustering and explores the movement behavior in the form of Spatial-Temporal Trajectory model.
2. For frequent region discovery, we propose QuadSection clustering to approximate the reasonable and optimal solution for cell size determining in grid-based cluster-ing and to earn the maximum number of discovered frequent regions to improve the movement behavior modeling.
3. We propose Spatial-Temporal Trajectory model (STT) to capture movement be-havior in both spatial and temporal domains for location prediction.
4. The STT is an efficient compression scheme that converts a large trajectory data into a compact but representative model which reduces the storage size of the trajectory patterns compared to the association rules proposed.
5. We propose an STT Prediction Algorithm that provides accurate predictions. 6. The effectiveness and accuracy of our proposed framework QS-STT for location
prediction has been demonstrated via extensive experiments on real trajectory datasets.
The rest of the paper is organized as follows. Section2discusses related works. An overview of the proposed framework QS-STT is given in Sect.3. Section4describes the STT construction and the algorithm of location prediction using STT model is shown in Sect.5. Section6 discusses comprehensive experimental evaluations and Sect.7concludes the paper.
2 Related works
Traditional location prediction that uses motion functions to predict next locations of users only has good accuracy if the future time specified in a predictive query is close to the current time [10]. To achieve a better accuracy of location prediction for query time far away from the current time, prior works in [10] have elaborated on min-ing trajectory pattern for location prediction. The experimental results in [10] show that by exploring trajectory patterns, the accuracy of location prediction for both near time or distant future time is significantly improved. Many researches have put much effort on movement behavior analysis and proposed some algorithms to mine move-ment patterns. Movemove-ment behaviors are represented as different kinds of trajectory patterns in prior works. For example, movement behaviors are defined as sequential patterns [5,7,12,17,18] and association rules [10,20]. Without loss of generality, given a set of trajectories, algorithms of mining trajectory pattern will first extract some regions with a certain degree of popularity, which are referred to as frequent re-gions. Then, original trajectories are transformed into sequences of frequent rere-gions.
With given sequences of frequent regions, movement behaviors are thus defined as trajectory patterns that frequently appear among sequences of frequent regions. Those trajectory patterns can imply that objects usually follow similar movement behaviors. Specifically, the trajectory pattern discovery is a key role of the pattern-based location prediction.
Most research works for trajectory pattern mining use sequential pattern mining techniques [5,7,17]. From a set of trajectories, a set of frequent regions are extracted. Then, based on frequent regions, raw trajectories are represented as sequences of fre-quent regions. Thus, sefre-quential pattern mining techniques are able to discover se-quential relationships among frequent regions. Different from most of the existing literature in sequential pattern mining that focuses on the dependent order of fre-quent regions in the sequence without temporal information, the authors in [4,5] proposed Temporally-Annotated Sequences (abbreviated as TAS) as trajectory pat-terns. As such, the results of sequential patterns contain a transition time between consecutive frequent regions along with TAS. Based on the above concept, the au-thors in [18] utilize TAS for location prediction. The input of location prediction in [18] is the set of recent movements without any query time, which is different from our prediction query. As such, the proposed method in [18] cannot deal with our prediction query.
In [10], the authors proposed a hybrid prediction model. The association rules among frequent regions are explored to represent trajectory patterns. Then, a hybrid prediction model is developed that combines trajectory patterns with motion function to forecast future location of a moving object. If the query time is close to the current time, one could predict the next location by the recent movement speed and direction, i.e, motion function. On the other hand, when the future time is far away from the current time, the authors discovered trajectory patterns in the form of association rules to predict the locations of objects. Hence, mining movement behaviors could provide a powerful way to enhance the accuracy of location predictions. However, the number of ruled patterns are significantly increased with a increasing number of trajectory data.
In the previous works on location prediction, the Markov chain models also have been used to predict the next movement of moving objects [9,11,26,27]. The whole space is divided into cells and the Markov transition probability among cells is de-rived from a set of trajectories. Given a current location within a certain cell, the next cell is predicted by the transition probability of the Markov chain model that cap-tures movement behaviors. Furthermore, the authors in [11] address the granularity problem of the space partitioning scheme. In particular, the authors utilize density-based clustering techniques to derive frequent regions. Then, the movements could be referred from cells and frequent regions. Thus, a prediction model is based on the hidden Markov process among the frequent regions and cells. However, the above works do not deal with predictive queries in which both the spatial and the temporal predicates are given.
To sum up the related works, there are some deficiencies. First, trajectory pattern is a key role in pattern-based location prediction. Most of the existing works only focus on trajectory pattern discovery in spatial domain. Prediction with spatial based trajectory pattern may result in biased prediction. In order to enhance the prediction
accuracy, both spatial and temporal information should be take into consideration on trajectory pattern discovery. Second, with a huge number of trajectory data, the number of trajectory patterns are significantly increased. To develop a representative compact model is a considerable issue on pattern-based location prediction.
3 The framework of QS-STT for location prediction
In this section, the overview of our proposed framework QS-STT for location-prediction is presented. Given an object’s trajectories, our objective is to derive a spatial-temporal trajectory model to capture the movement behaviors of the object. In light of our trajectory model, given recent movements and a future time query, we intend to predict its locations at the specified future time. Thus, our proposed framework is shown in Fig.2. As can be seen in Fig.2, the proposed framework con-sists two modules: STT model construction and STT-location prediction. In the STT model construction module, given a set of raw trajectories, there are three steps as follows:
Step 1. Frequent Region Discovery: In this step, we extract frequent regions from a set of trajectories. Since locations of raw trajectories are GPS data points with uncertain properties in terms of both spatial and time domains, to capture movement behaviors, we need to extract frequent regions. A frequent region contains a sufficient number of trajectories whose data points are within the corresponding region. QuadSection Clustering is proposed to extract good quality frequent regions for movement behav-ior modeling.
Fig. 2 The framework QS-STT
Step 2. Trajectory Transformation: According to the set of frequent regions deter-mined by Step 1, each raw trajectory is transformed into a region-based moving se-quence. Location points that are not in frequent regions will be regarded as noise. As such, our STT could capture movement behaviors among frequent regions.
Step 3. STT Model Construction: In this step, we adopt a probabilistic suffix tree with spatial and temporal information to discover movement behaviors. Our proposed STT model contains not only transition probabilities among frequent regions (referred to as the spacial feature of the STT model) but also appearance probabilities within frequent regions (referred to as the temporal feature of the STT model).
In the STT-location prediction module, given the current movements of an object and a future time query, we use the STT model as a location predictor and propose an STT-location prediction algorithm to traverse the model and estimate the future location at the query time.
4 Spatial-temporal trajectory model construction
In this section, we detail the procedure for the discovery of frequent regions and the transformation of the raw trajectories to region-based moving sequences. Finally, we describe the process that how to construct an STT model to capture the moving behavior.
4.1 Frequent region discovery and trajectory transformation
An object’s trajectory is generally represented as a sequence of spatial-temporal points sampled by a positioning sensor or device. Let a trajectory of a moving object be represented as T = {p1, p2, . . . , pn}, where n is the total number of points and a
point pi= (li, ti)is denoted as a location li = (xi, yi)at a timestamp ti (0≤ i ≤ 1).
We cannot directly use such trajectory data for spatial-temporal trajectory model con-struction. The reason is that the object will not repeat exactly the same location in ev-ery timestamp of each trajectory even if such a set of trajectories has a similar spatial route. Thus, we consider a trajectory to be a sequence of frequent regions. A frequent region is a region which an object often visits. For the discovery of frequent regions from an object’s trajectories, density based clustering is adopted by many previous works [5,10,11,17]. In the density based approach, a cluster is viewed as a frequent region if the number of trajectories visited is larger than a predefined threshold. For example, we can apply DBSCAN [8] to discover the clusters, i.e. frequent regions. For each point of a cluster, the neighborhood of a given radius (i.e., Eps) has to con-tain at least a minimum number of points (i.e., MinPts), that is, the density in the neighborhood has to exceed a predefined threshold.
However, the clusters that are detected by DBSCAN cannot be applicable to our STT because the clustering parameters are universal. In reality, the density of clusters extracted from trajectories varies widely because the speed of a moving object is not always constant. For example, the location point density of a trajectory for highway driving is lower than that for local city driving. The parameter setting is the con-straint of DBSCAN for various density clustering. As an example in Fig.3(a), if the
Fig. 3 An example of DBSCAN results with various Eps
Eps value is set to a smaller value, some points are classified as “noise” and some information would be missed. Using a larger value for Eps causes the granularity problem to represent the trajectory, which is shown in Fig.3(b).
The key concept of this research is to establish an STT model to capture the mov-ing behavior by describmov-ing the relations between frequent regions. The granularity of clusters can affect the explanation of STT considerably. Therefore, we use grid based clustering to discover the frequent regions. The trajectory moving space C is parti-tioned into k grid cells of the same size. Thus, each region can be represented as a cell
ci with a universal size. The density of a cell is computed by taking the number of
tra-jectory segments which pass through the cell. A tratra-jectory segment Ts= {pi, pi+1},
where piis the ith location and 1≤ i < i +1 ≤ n, is a contiguous sub-trajectory of T .
We formally define the frequent region as follows:
Definition 1 A Frequent Region is a grid cell that contains at least MinTs number of
trajectory segments passing by the grid cell.
An example of frequent region discovery is provided in Fig.4(a), where Trajec-tory1, Trajectory2 and Trajectory3 are trajectories of a moving object passing through a certain space. Given MinTs= 2, the frequent regions are regions with gray color. Note that we use the number of segments passing through the cell instead of the num-ber of points enclosed in a cell to decide if the cell is a frequent region. The reason is that the density of points varies with an object’s moving speed meaning that it may miss some trajectory information. For example, the frequent regions c43, c41, c30 are
not detected if the minimum point is set to 2 in a cell as shown in Fig.4(b).
Furthermore, in order to construct a good STT model to capture the movement behavior precisely, as many frequent regions as possible have to be explored. Intu-itively, a proper grid cell size can generate many frequent regions and potentially achieve good prediction precision. However, a large number of patterns may improve the prediction precision but increase the storage requirements. Therefore, it is neces-sary to find a good balance between the accuracy of the prediction and the storage requirements.
Fig. 4 An example of frequent region detection by grid-based clustering
Fig. 5 The experimental result
of frequent region discovery by greedy testing under various MinTs
4.2 QuadSection clustering
One main challenging issue of grid file management for frequent region discovery is to determine the cell size. The cell size affects the number of patterns and prediction accuracy. Obviously, a large number of explored movement patterns may capture the movement behavior precisely and improve the prediction precision. In order to discover the possible number of movement patterns, as many frequent regions as possible are expected to be extracted. The naive approach is the greedy testing of the grid file with different cell sizes and selecting the one that results in the maximum number of frequent regions. Figure5shows the experimental results of greedy testing for frequent region discovery with various cell sizes. Given a MinTs, the cell size with the maximum number of discovered frequent regions is selected to be the optimal cell size.
However, greedy testing is an inefficient method to find the optimal cell size. Given a user-defined MinTs, the optimal cell size is defined as the cell size can result in max-imal number of frequent regions. The greedy method for frequent region discovery
detects the number of frequent regions under the various cell size by greedy test-ing and bottom-up clustertest-ing. For each cell size, the greedy method will compute all the data point and then extract the number of frequent regions. In this paper, we de-velop a simplified method, called QuadSection clustering, to approach the optimum solution of determining the cell size and detecting the frequent region under a given MinTs. QuadSection clustering (abbreviated as QS clustering) is based on divisive hierarchical clustering [25]. A top-down cluster hierarchy is generated using the di-visive clustering method. For frequent region detection, STT with QS clustering has advantages over STT with greedy-based grid clustering, including: the hierarchical approach can reduce and speed up the computation; the top-down clustering is more efficient than bottom-up clustering because we may not have to generate a complete hierarchy all the way down to individual data points.
Given a trajectory dataset, QS Clustering starts at the top rectangular space with all data points as one cluster. The cluster is then split by subdividing it into four cells. A cell is identified as a frequent region and becomes a leaf node if the cell has a maximum capacity, i.e., equal or greater than a user defined MinTs. This procedure is applied recursively to leaf nodes of each level until the number of discovered frequent region reaches the maximum value. The level with the maximum number of nodes is determined as the optimal grid file and the nodes of selected level are retrieved as the frequent regions. The cut-cost is proposed to measure the stopping criterion for divi-sive clustering if the cut-cost value is equal to or greater than zero. For QuadSection clustering, the cut-cost for level Li in a divisive hierarchy Hd is defined as follows:
Cut-cost(Li)= log Ni Ni+1 (1) In (1), the Ni is the total number of cells that become frequent regions for level Li.
The QuadSection Clustering algorithm is shown in Algorithm1. Given a set of his-torical trajectory data D and user defined MinTs, we first bound the data space in rectangular space as the root cluster C0. C0is initialized as level L0. The algorithm
then executes the splitting operation to divide the cluster into four equal rectangular sub-cells. A sub-cell is identified as a frequent region if the number of trajectories contained in the region is equal or greater than MinTs and becomes a leaf node of the next level in the hierarchy Hd. The function cut-cost(Li) computes the stopping
criterion for the splitting operation in each iteration. If the cut-cost(Li) is equal to or
greater than zero, the nodes belonging to Li are retrieved and outputted as frequent
regions.
Figure6is an example of QuadSection clustering for MinTs= 2. The data space is first bounded as a rectangular region as root cluster C0. The algorithm then splits
C0 into four equal sub-cells. C2, C3 and C4 are identified as frequent regions and
become leaf nodes of L1. Cut-cost(L0)= log(13)= −0.477 and repeats the
split-ting operation for L1. L2only has two frequent regions C42and C13and meets the
stopping criterion due to cut-cost(L1)= log(32)= 0.176. The nodes (C2, C3and C4)
belonging to Level 1 are outputted as frequent regions under the optimal cell size. An Effectiveness comparison of QuadSection clustering and the naive approach (grid clustering by greedy approach)is conducted as shown in Table1. The effectiveness
Algorithm 1: QuadSection Clustering Input : a set of trajectory data D, MinTs Output: a set of frequent regions Rf
C0← {C|C contains points ∈ D and count(D) ≥ MinTs};
1
Let a Hd has a single root C0and C0∈ L0;
2
foreach node n∈ Li do
3
Sc← Split(n);
4
foreach sub cell s in Scdo
5
if count(s)≥ MinTs then
6 Add node s to Li+1; 7 end 8 end 9 end 10 if cost-cut(Li) < 0 then 11 Repeat line 3; 12 else 13 Rf ← {n|n ∈ Li}; 14 end 15
Fig. 6 QuadSection clustering example for MinTs= 2
Table 1 Comparison of
QuadSection clustering and grid clustering by greedy approach
QuadSection Greedy approach
MinTs= 2 3240 3299
MinTs= 3 1707 1833
MinTs= 4 1215 1240
Fig. 7 An example of frequent region detection and trajectory transformation
is measured by the average approaching rate. The average approaching rate is de-fined as the average ratio between the number of frequent regions discovered by QS clustering and naive approach. The average approaching rate of Table1is 96.8 %. The experimental result shows that QS clustering is able to effectively discover the frequent regions with the approaching optimal cell size.
After extracting the frequent regions, each raw trajectory is transformed into a sequence of frequent regions. Such a sequence is called a region-based moving se-quence and is defined as follows:
Definition 2 A Region-based Moving Sequence (moving sequence) is a sequence of
frequent regions Sr = {(r1, t1), (r2, t2), . . . , (rn, tm)} with time constraint t1< t2<
· · · < tm, where (ri, tj)indicates that object visits the frequent region ri at
times-tamp tj.
As can be seen in Fig.4(a), three trajectories are transformed into region-based moving sequences {(c16, t1), (c26, t2), (c25, t3), (c35, t4), (c45, t5), (c44, t6), (c43, t7),
(c42, t8), (c41, t9), (c31, t10), (c30, t11)}.
Figure7 shows a running example of frequent region discovery and trajectory transformation. There are six trajectories of a data set. The whole space is partitioned into cells. Given a MinTs, a frequent region is extracted if a cell is passed by MinTs
Fig. 8 An example of the STT model
trajectories. Finally, the raw trajectories are transformed into region-based moving sequences.
4.3 Spatial-temporal trajectory model construction
After trajectories are transformed into region-based moving sequences, we construct a spatial-temporal trajectory model to capture the moving behavior of an object. STT model is a variant of the probabilistic suffix tree. PST is a compact representation of a variable-order MarKov chain which draws the probability distribution for discrete events occurring in sequences. PST can be used to predict the next event by given the preceding sub-sequence [1,22,24,28]. The original PST only considers the order of the sequence in the spatial domain but disregards the temporal information. For dealing with our location prediction at a specific future time, we propose the spatial-temporal trajectory model, which is an extension of PST.
Figure8shows an example of the STT model, which builds an STT over a symbol set = {r1, r2, . . . , rn}, where symbol ri is represented as a frequent region and n is
the total number of frequent regions. Each edge in the tree is labeled by a frequent region that indicates one movement from one frequent region to the other. Each tree node is labeled by a sequence which represents a path from the node to the root. For example, a tree node which is labeled as rk· · · r2r1can be located by traversing from
the root along the path root→ r1→ r2→ · · · → rk. A predictive table is associated
with each node to maintain both spatial and temporal correlation between the region and the next movement: the spatial probability Prospatialand the temporal probability
Protemporal. Prospatial is denoted as the conditional probability P (rk+1| r1r2· · · rk)
of the next frequent region rk+1that follows the label of the tree node r1r2· · · rk.
tk+1. The transition interval is defined as vector ik+1= (mean, sd), where mean is
the average transition interval from rk to rk+1 and sd is the standard deviation of
the transition interval. Representative timestamp tk+1is donated as the representative
time when the moving object often passes by rk+1. For example, the predictive table
of node ab in Fig.8shows that the next movement region is region c, P (c| ab) = 1, ic= (60, 5.8) and tc= 9:30, 18:00, i.e., the conditional probability in the spatial
domain that from ab moving to c is 0.6, the object often spends 60 minutes from ab moving to c and the standard deviation is 5.8 minutes, and the time to pass by c is usually 9:30 and 18:00.
Now, we will present how to construct the STT model. STT construction includes two steps: (1) for spatial domain, construct a PST to discover the frequent sequential patterns over frequent regions. (2) for the temporal domain, extract the transition in-terval vector and representative time of the next movement region for each tree node. Before the construction of STT, minimal support, denoted as MinSup, is specified to decide whether a frequent region rk should generate a tree node as a child of a
parent node or not. At the beginning of the first step, we hold an STT consisting of a single root node with the counts of each frequent region appearing in the trajec-tories. If the count of frequent region rk is larger than the predefined MinSup, one
tree node labeled as rk will be created as a child node of the root. Then, tree node
rkwill maintain the conditional probability of the frequent region rk+1with the
pre-fix segment of node rk in the predictive table. For each sequence of frequent regions
r1r2· · · rk, if a frequent region rk+1appears behind it, the statistical information of
all nodes labeled with the suffix of r1r2· · · rk should be updated accordingly.
Af-ter constructing the relationship between the frequent regions in the spatial feature from the historical trajectory dataset, the STT construction enters the second step to retrieve the temporal information of moving behavior. Each predictive table of tree nodes of STT is extended by aggregating the relevant temporal information. In this paper, the time distribution of a region for a moving object frequently visited is ap-proximately either a single Gaussian distribution or mixture Gaussian distribution. Therefore, we assume that the time distribution of a frequent region can be approx-imated as a Gaussian mixture model (GMM), where each component of the GMM represents a temporal feature of the frequent cluster. The determination of the transi-tion interval and representative timestamp of each frequent region can be approached by the GMM parameter estimation. In general, the parameters of the GMM model can be extracted by an Expectation-Maximization (EM) algorithm. However, using the EM algorithm for analysis of the GMM model may result in some drawbacks: it does not guarantee completeness due to unfortunate initialization; some smaller dense regions may be absorbed by larger and denser regions if they are too close to them. In order to extract the temporal feature, we build a histogram of the frequency distri-bution to approach the time distridistri-bution of each frequent region. Finally, we estimate the parameters of each component as a transition interval vector and representative timestamp. The whole process of STT construction is described in Algorithm2. The algorithm of STT construction extracts significant sequential patterns and prunes in-frequent patterns during tree construction, and then generates a STT. The input of the algorithm includes the STT parameter MinSup and a set of region-based sequence tp.
Algorithm 2: STT Construction
Input : a set of region-based sequence tp, MinSup Output: a STT model ¯T
Initialization: Let a ¯T has a single root and k= 1; 1
Sk← {σ |σ ∈ tpand count(σ )≥ MinSup};
2
while Skis not empty do 3
foreach element s in Skdo
4
Add node s to ¯T; 5
// Build the predictive table of node s
SpatialPrability (σ|s); 6 TransitionInterval(σ , s); 7 RepretativeTimes(σ , s); 8
if there exists a σ∈ tpsuch that count(σs)≥ MinSup then
9 Add σsto Sk+1; 10 end 11 end 12
node of a subsequence s. The algorithm starts by initializing ¯T and then extracts the set Sk of candidate patterns with length 1 (Lines 1–2). As shown in Lines 3–12,
the algorithm checks whether each candidate is qualified to be a node in the tree. If a candidate s is qualified, the functions SpatialPrability, TransitionInterval, and RepretativeTimes (Lines 6–8) return the spatial-temporal information and update the corresponding data in predictive table. Next, the algorithm extends the candidates, as shown in Lines 9–10, and iterates the procedure until the candidate set is empty.
Although the execution efficiency for QS-STT is dependent on the size of trajec-tory data, QS-STT has some advantages to improve the execution efficiency. In QS clustering, the top-down hierarchical clustering approach can reduce and speed up the computation. Furthermore, STT model is a variant of PST, which is a successful and efficient model to capture the significant sequential patterns and organize those patterns into a tree structure.
5 Location prediction using STT model
In this section, we present how to predict future location using the trajectory mov-ing profile for a given movmov-ing object’s recent movements and query time. The STT represents a keyword dictionary of frequent trajectory patterns which are associating with conditional probability entries for each possible next movement. The concept of our prediction is to find the best next movement literally. We first encode the recent movements into a query sequence. Specifically, the query sequence sqis a sequence
of frequent regions which the object has visited. The tree node of STT will be lo-cated by the best movement similarity of its labeled pattern and sq. The predictive
table associated with the node will decide moving potential of each candidate for next movement in the table. The candidate with the highest moving potential will be
the next movement region and added into sq. Such prediction procedure will be
re-peated literally until the query time is reached. The last movement is selected as the possible location at query time.
5.1 Movement similarity
The movement similarity is used to measure the similarity of a labeled sequence of a tree node nkof STT and the moving sequence sq. The objective is to search a node of
STT model whose labeled sequence is the most similar to the moving sequence. In-tuitively, the more recent movements have greater effect on future movements. Thus, we assign more weight to the frequent region where it is closer to the query time. The movement similarity of a tree node labeled nk and the query sequence sq is defined
follows: MS(nk, sq)= Size(nk&sq) i=1 i2 Size(sq) j=1 j2 (2) where 0≤ MS ≤ 1 and (nk&sq)is the longest common suffix of nk and sq.
For example, assume sq= abc and there are the patterns a, b, c, bc and ab in the
tree. Considering the similarity between abc and those nodes, the similarity values are 0.07, 0.28, 0.64, 0.93, and 0.36 respectively. The node labeled bc has the best movement similarity with the query sequence abc.
5.2 Moving potential
After the best similar node is located, the moving potential of each next movement candidate is calculated to decide the next movement. The region with larger spatial probability in the predictive table has a greater chance of an object moving to it. How-ever, such prediction would not be an effective prediction for a moving object because it only considers the sequential relationship in the spatial domain and ignores the ef-fect of the temporal domain. For example, the probability that a user starting from home to school around 7:00 am is 0.7 and the probability of going from home to the park around 4:00 pm is 0.3. If the user’s recent movement is home at 3:50 pm, the answer of location prediction is always school when we consider the spatial domain only. However, the correct location prediction should be the park. Thus, by consider-ing both spatial and temporal information, the accuracy of location prediction could be enhanced. To reflect this idea, we propose a moving potential that takes both spa-tial and temporal information into consideration for location prediction.
For a next movement rk+1of node nk, the moving potential ProST, is measured as
follows:
ProST= Prospatial× Protemporal (3)
Prospatialis computed according to the conditional probabilities of seeing the
sym-bol rk+1right after the node nklabeled string r1r2· · · rkin the dataset.
To take the time feature into account for location prediction, we have to mea-sure Protemporal. We use Chebyshev’s inequality [21] to compute the upper bound
of probability of time error as a temporal similarity measurement. For a candidate of next movement rk+1of node nk, we estimate the arrival time of next movement
te which is the sum of the current time tc and the average transition interval mean.
The minimum difference of teand the representative time tk+1of the next movement
candidates is defined as temporal error. Protemporal=
sd2
| min{te− tk+1}|2
(4) For example, assume that temporal features of the next movement with node nk
are ik+1= (5, 2) and tk+1= {12:00, 15:00, 17:00}. If the current time is 11:52, we
evaluate that arrival time of region rk+1of the moving object is 11:57 because the
mean of the transition interval is 5. Minimum error among tk+1is|11:57−12:00| = 3
minutes because 12:00 is the closest time to the arrival time. Therefore, Protemporal
is bound by 0.44. To ensure the sum of probability is 1, we normalize the estimation value to reassign the adjusted value as its temporal probability after estimating the upper bound of probability of time.
5.3 Location prediction
Given a recent movement and query time, the algorithm initializes the query sequence by encoding the recent movements into a sequence of frequent regions. The tree node of STT with the best movement similarity of its labeled pattern and sq will be
lo-cated. The predictive table associated with the node will decide the moving potential of each candidate for next movement in the table. The candidate with the highest moving potential will be the next movement region and added into sq. Such
predic-tion procedure will be repeated literally until the query time is reached. Therefore, we design a prediction algorithm which is a dynamic programming with time complexity of O(nd) where n is the number of tree nodes and d is the number of time intervals. Let F (c, t) be the highest probability to reach the next region among the candidates of next node. The recursive solution and initial condition are defined as follows:
F (c, t )=
1, for t≤ 0 (5)
maxc(F (cc, t− mc)× Proc), for t > 0, (6)
where c is denoted as a query sequence, t is traveling time, cis one candidate of the next nodes of c, and mc is the mean of the transition time between them.
The sub-problem, F (cc, t− mc), is calculated recursively. The initial condition
is set F (c, t) to 1 for t is too short to go continuously. An example is shown in Fig.9. If we want to find the answer of F (c1,10), we have to calculate F (c2,5) and
F (c3,5) first. The value of F (c1,10) will be set to the maximum probability between
F (c2,5) and F (c3,5), the value of F (c3,5) is decided by F (c7,0), and so on. As a
result, F (c1,10) is 0.4.
The STT prediction processing algorithm follows a general backward recursion procedure of a tabular method and is shown in Algorithm3. In the beginning, we initialize the query sequence with recent movements and locate the node c with best movement similarity with sq. Then, we run the procedure STT-prediction(c, tq) to
lo-cate the predicted node c. For each problem STT-prediction(c, time) with time larger than 0, we first check whether it has computed or not. If it has not been computed yet, we scan all of the candidate next node cof c to find the answer to the problem and return the computed result.
Fig. 9 An example of recursive solution for STT prediction
Algorithm 3: STT Location Prediction Input : query pattern sq, query time tq Output: predicted location
Set the best match node c by searching STT corresponding to sq;
1
Get predicted node cby calculating STT-Prediction(c, tq);
2
Return Location(c); 3
// A backward recursion function
function STT-Prediction(c, time)
4 if time≤ 0 then 5 Return pattern c; 6 else 7
if STT-Prediction(c, time) hasn’t compute then
8
Return Max(STT-Prediction(cc, time-meanc), Proc’), where
9
c∈ nextc; else
10
Return the computed result; 11 end 12 end 13 end function 14 6 Experiments
In this section, extensive experiments are performed to evaluate the effectiveness and efficiency of our proposed location prediction method.
6.1 Experimental setting
In our experiments, we use real trajectory datasets from CarWeb [15], which is a traffic data collection platform, in which users record their own location every five seconds and upload their trajectories to the CarWeb server. We extract the data of
Fig. 10 Number of discovered patterns comparison by different clustering methods
one car of which the moving behavior has more than one similar path data as our major dataset for the experiments. The movements of a vehicle were obtained by a GPS equipped car while it followed the traffic network in Hsinchu city, Taiwan, over a period of two months. Since CarWeb is our own platform, the ground truth is easy to verify. Moreover, in order to evaluate the effectiveness and efficiency of our pro-posed method under various conditions, we use three trajectory data sets in different kinds of moving behaviors from RunSaturdayhttp://www.runsaturday.com, which is a website server to collect training paths of sports hobbyists and the uploading rate is around tens of seconds to several minutes. The three data sets are summarized as follows:
– Walk: A walker’s movements were collected over a period of seven months and 24 trajectories were extracted. The man has two types of walking behavior: walk-ing in the street near his home and gowalk-ing hikwalk-ing in Pohjois-Savo, Ita-Suomi, Fin-land.
– Run: A runner’s movements were collected over a period of two months and 35 trajectories were extracted. The man has three types of running behavior: running in the playground near his home, running along an urban outer road, and running along the coastline.
– Bike: A bike’s movements were collected over a period of two months and 16 trajectories were extracted. The data was recorded while the bike went
Fig. 11 Number of discovered regions comparison by different clustering methods
from a start position, cruising, and back to the start position. The moving be-haviors on weekdays are different from on weekends such that the time pe-riods on weekends were longer than on weekdays, so the trajectories around a larger circle. Those movements were recorded over a period of almost two hour.
The larger datasets are generated by perturbing the original datasets: every trajec-tory is duplicated with a small (≤ ±5 %) random value added to the original value. The number of trajectories (points) is 100 (20000). The test dataset was randomly selected from 20 % of the original trajectories.
6.2 Performance comparison of clustering algorithm
In this section, in order to verify that the prediction accuracy of STT can be im-proved by using QuadSection clustering (QS-STT), the performance of the proposed QS-STT, STT prediction model with density-based clustering (DBSCAN-STT), and grid-based clustering (Grid-STT) [14] is compared. Under various MinpTs, the best experimental Eps is set for DBSCAN-STT and the greedy testing is conducted to find the optimal cell size with maximum discovered regions for Grid-STT while the near-optimal cell size of QS-STT is able to be auto-decided. Given MinSup= 2 for
Fig. 12 Prediction error with different clustering methods
STT construction, we compare the number of regions and patterns discovered and the prediction error of different clustering methods. The number of regions discovered by the different clustering methods is compared in Fig.10. The number of discovered regions decreases as the value of MinpTs grows. The more frequent regions that are extracted, the more trajectory patterns may be generated. As shown in Fig.11, the ex-perimental result indicates that the QuadSection clustering is able to detect the more patterns than STT-DBSCAN and Grid-STT.
Prediction accuracy is measured by the prediction errors, which is defined as the distance between a prediction location and its true location at a given query time. We tested 150 queries by giving different query times and averaging their errors. The pre-diction accuracy comparison with varied MinTs is shown in Fig.12. For each MinTs, the prediction errors under different query time are averaged. The experimental result shows that QS-STT is more precise than other methods. QS-STT has very low errors regardless of prediction length while the errors of DBSCAN-STT rise significantly as the prediction length changes and this is because that region (cluster) discovered by DBSCAN-STT results in various sizes, which may cause granularity problems and lose accuracy when the predictive location falls in a big region. QS-STT is able to detect the frequent region with reasonable and near-optimal cell size compared with Grid-STT. This implies that QS-STT has more precise and complete information than DBSCAN-STT does in capturing moving behavior. Thus, the QuadSection clustering
Fig. 13 Prediction model comparison
Fig. 14 Effect of MinTs on number of patterns
method preserves the moving behaviors well for the STT location prediction model. Note that the average errors of bike is higher than those of car is because the bike has smaller number of patterns. The smaller number of patterns will reduce the prediction accuracy.
Fig. 15 Effect of MinTs on prediction error
6.3 Performance comparison of location prediction
We compare the location prediction accuracy and storage requirements of our method (QS-STT) with that of the hybrid prediction model (HPM) [10], which uses the as-sociation rule-based pattern prediction approach. The parameters are set for best per-formance in terms of accuracy based on the experimental results. The QS-STT model is constructed by the near-optimal cell size under MinTs= 13. The HPM for our dataset: the frequent regions are decided by DBSCAN Eps= 0.0055, MinPts = 4, and minimum confidence= 0.3 for association rule discovery. The prediction com-parison is under the various prediction temporal lengths. As expected, Fig.13(a) indi-cates that QS-STT has lower errors than HPM in location prediction. HPM has higher errors caused by the fact that the frequent regions discovered by DBSCAN cluster-ing may result in clusters of arbitrary shapes and sizes while the error of QS-STT is restricted by the fixed cell size.
We next study the storage requirements comparison under various MinSup. As ex-pected, Fig.13(b) demonstrates that our method has smaller storage size than HPM. While the storage size of HPM dramatically grows as the number of frequent re-gions increases, our method QS-STT still maintains a small storage size with tiny changes. The reason is because HPM using association rule based patterns generates an exponential number of rules as the number of frequent regions increases. On the
Fig. 16 Effect of MinSup on number of patterns
other hand, QS-STT using uses the suffix tree data structure which can compress the number of sequential patterns into a compact model.
6.4 Sensitivity analysis of parameters
In this section, we examine the effect of MinTs and MinSup to our model and predic-tion. MinTs is the parameter of QuadSection clustering and MinSup is the parameter of STT construction. We first study the effect of MinTs on frequent region discovery and the accuracy of the location prediction. In our definition, a frequent region is de-cided by MinTs number of trajectories which pass the region in a cell size. Therefore, a high value of MinTs may cause a small number of frequent regions and trajectory patterns. Prediction based on trajectory patterns could be affected by MinTs. Accord-ing to the experimental results shown in Fig.14, the number of trajectory patterns is reduced as the number of MinTs increases. The prediction error increases signif-icantly due to the small number of trajectory patterns as shown in Fig.15. We also investigate the effect of MinSup. For STT construction, a movement sequence is de-fined as a trajectory pattern if the sequence has MinSup number of same movement sequences appearing in a trajectory data set. Figure16presents the experimental re-sults with MinSup varied. The number of trajectory patterns decreases dramatically as the value of MinSup grows. Furthermore, the prediction error affected by MinSup
Fig. 17 Effect of MinSup on prediction error
is provided in Fig.17. The prediction error will potentially rise as the value of MinTs grows.
In general, MinTs and MinSup are two parameters which mainly determine the number of frequent regions and trajectory patterns which can be discovered. The more frequent regions that are extracted, the more trajectory patterns may be gen-erated. From the experimental results above we can observe that the model can po-tentially achieve good prediction precision as many frequent regions and trajectory patterns which are discovered.
7 Conclusion
In this paper, we presented a pattern-based approach to predicting an object’s future locations. We not only focus on how to discover frequent movement patterns and manage these patterns to answer predictive queries but also aim to propose a model that can reduce the pattern storage size. To achieve this goal, we proposed Quad-Section clustering to extract the maximum number of frequent regions with the rea-sonable and near-optimal cell size, and develop a spatial-temporal trajectory model to capture the movement behaviors of objects. QS-STT model could be a predictor for location prediction and enhance the prediction accuracy. The experimental results show that the QS-STT model is able to reflect an object’s moving behavior with a
smaller storage size compared to existing patten-based approaches, while still guar-anteeing the accuracy of location prediction.
Acknowledgements Wen-Chih Peng was supported in part by the National Science Council, Project No. 100-2218-E-009-016-MY3 and 100-2218-E-009-013-MY3, by Taiwan MoE ATU Program, by ITRI JRC, Project No. B301EA3300, by D-Link and by Microsoft.
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