Predictive Nearest Neighbor Queries Over Uncertain Spatial-Temporal Data

Transcription

1 Predictive Nearest Neighbor Queries Over Uncertain Spatial-Temporal Data Jinghua Zhu, Xue Wang, and Yingshu Li Department of Computer Science, Georgia State University, Atlanta GA, USA, Abstract. Predictive nearest neighbor queries over spatial-temporal data have received significant attention in many location-based services including intelligent transportation, ride sharing and advertising. Due to physical and resource limitations of data collection devices like R- FID, sensors and GPS, data is collected only at discrete time instants. In-between these discrete time instants, the positions of the monitored moving objects are uncertain. In this paper, we exploit the filtering and refining framework to solve the predictive nearest neighbor queries over uncertain spatial-temporal data. Specifically, in the filter phase, our approach employs a semi-markov process model that describes object mobility between space grids and prunes those objects that have zero probability to encounter the queried object. In the refining phase, we use a Markov chain model to describe the mobility of moving objects between space points and compute the nearest neighbor probability for each candidate object. We experimentally show that our approach can filter out most of the impossible objects and has a good predication performance. Keywords: spatial-temporal data; uncertain data; predictive query;markov chain;semi-markov model 1 Introduction With the emerging and proliferation of GPS enabled mobile devices and wireless communications, processing and managing spatial-temporal data becomes vital for many location-based services including intelligent transportation, ride sharing and advertising [1]. Predictive queries over spatial-temporal data are proved to be vital in these applications such as nearest neighbor queries, e.g., find the nearest companions of a moving tourist 1 hour later, range queries, e.g., find customers who are predicted to be within ten miles of my moving store in the next 30 minutes and aggregate queries, e.g., find areas with predictive traffic jam before it happens [2][3][4][5]. However, due to physical and resource limitations of data collection devices like RFID, sensors and GPS, data is collected only at discrete time instants. Inbetween these discrete time instants, the positions of the tracked moving objects are uncertain. As shown in Fig.1, the trajectory in Fig.1(a) has no uncertainty,

2 Fig. 1. Trajectory in Space and Time and the trajectory in Fig.1(b) has uncertainty. Unlike most of the existing works regarding predictive queries that assume the deterministic behavior of object movement, few research works assume uncertainty about these trajectories. Basically, uncertainty deals with stochastic process with respect to object locations and velocities at different time instants. In this paper, we assume that the uncertain movement of an object between consecutive observations can be described by a Markov Chain Model, which captures the time dependencies between consecutive locations. The work in [6] shows that the possible worlds (trajectories between consecutive observations) can be efficiently analyzed and probability query evaluation can be facilitated by integrating pruning mechanisms into Markov Chain matrics. All these are sufficient for the case where there are few queried objects following similar movements; however, if there is a large number of objects in the database, with different movements, especially the query objects are moving as other objects, evaluating a predictive probability spatial-temporal query directly against each object individually would be very expensive. The work in [7] proposes hierarchical index UST tree to bound the uncertain movement of objects. It allows for efficient and effective filtering during query evaluation. However, the storage cost and time cost for UST construction and reconstruction will increase with the number of possible states. In this work, we exploit the filtering and refining framework to evaluate predictive nearest neighbor queries over uncertain spatial-temporal data. In the filtering phase, a semi-markov model that describes objects mobility between space grids and prunes those objects that have zero probability to encounter the queried object. In the refinement phase, we use a Markov chain model to describe the mobility of moving objects between space points and compute the nearest neighbor probability for the small amount of candidate objects. The main contributions of our work are as follows: We introduce the predictive probability nearest neighbor query (PPNN) over uncertain spatial-temporal data; both the query objects and the common objects are moving randomly in the space. We exploit the filtering and refining framework to process the proposed PPN- N queries. We use a semi-markov model to describe the object movement between space grids and effectively prune the impossible objects. We use a Markov Chain model to describe the object movement between space points and compute the PPNN probability for each candidate object.

3 We conduct thorough experimental evaluation of the proposed method to valid its performance. The remainder of this paper is organized as follows: in Section 2, we discuss the related works about uncertain spatial-temporal queries. Section 3 gives the predictive probability query definition and assumptions. Section 4 and Section 5 introduce the Semi-Markov and Markov model which are used to describe the uncertain movement in two space levels. We describe the PPNN algorithm in Section 6 and evaluate its query performance using real and synthetic data in Section 7. Finally, Section 8 concludes this paper. 2 Related Works The management of spatial-temporal data has gained increasing interests in many location based applications. A typical aim of these applications is to answer users queries such as range queries, nearest neighbor queries and aggregate queries [2][3][4][5]. Another important set of location based services focuses on predicative spatial-temporal queries[3][9]. A mobility model[4]is used to predict the coming path of each of the underlying objects and employ the prediction results to evaluate predictive range queries. RangeSearch and KNNSearchBF [9]are introduced to traverse a spatial-temporal index tree to find the nodes that intersect with the query circular region for Range and KNN queries, respectively. A comprehensive technique [10] that employs adaptive multi-dimensional histogram, historical synopsis, and a stochastic method is used to provide an approximate answer for aggregating spatio-temporal queries for the future. Unlike most of the existing works in predictive queries that assume the deterministic behavior of object movements, few research works assume uncertainty about these movements. To index uncertain motions of a set of moving objects, the Bx-tree[11] is enhanced and two movement inference techniques are introduced to obtain anticipated object locations in a non-deterministic manner. A Spatio-Temporal Prediction tree, STP-tree[12], is introduced to index uncertain movement patterns and to answer predictive queries. Recent approaches model the uncertain movement of objects based on s- tochastic processes. In particular, in [13][14][15] trajectories are modeled based on Markov chains. This approach permits correct consideration of possible world semantics in the trajectory domain. 3 Problem Definition 3.1 Assumptions In this paper, we assume that the given space is divided into L virtual grids G = {g 1, g 2,..., g L }, where L is a system parameter representing the length of the single grid. At some certain time, one moving object can only be in one grid and there may be multiple moving objects in the same grid. Furthermore,

4 one moving object could stay in a specific grid for a while. We also assume a discrete state space of possible locations S = {s 1, s 2,..., s S } and a discrete time domain T = {0, 1,..., n}. Thus, a spatial-temporal database D stores quadruples (o i, t, l 1, l 2 ) where o i is a unique object identifier, t T is a timestamp and l 1 G and l 2 S are the grid location and space location of o i respectively. Each such quadruple corresponds to an observation that o i has been seen at some location at time t. Let D = {o 1, o 2,..., o D } be a trajectory database containing uncertain D moving objects. For each object o D we store a set of observations Θ o = {< t 1, l o 1(t 1 ), l o 2(t 1 ) >, < t 2, l o 1(t 2 ), l o 2(t 2 ) >,..., < t Θo, l o 1(t Θo ), l o 2(t Θo ) >}. We assume that at any timestamp, the grid location of an object is certain, while the space location of an object between two observations is uncertain. 3.2 Predictive Probability Nearest Neighbor Queries In this paper, we focus on the predictive nearest neighbor query over uncertain moving objects. Both the queried object and the common objects are randomly moving in the space. DEFINITION 1 (PPNN QUERY)A predictive probability query retrieves all objects o D which have a sufficient high probability to be the nearest neighbor of query q at the future timet T,formally, P P NNQ(D, q, t, τ) = {o i D\q, P P NN(D, o i, q, t) τ} where P P NN(D, o i, q, t) = P { o j D\o i : dist(l i 2(t), l q 2 (t)) dist(lj 2 (t), lq 2 (t)} and l i 2(t) is the space location of object o i at time t,dist(x,y) is a distance function defined on spatial points, typically the Euclidean distance. 4 Semi-Markov Model 4.1 Model We model the mobility of object o between the virtual grids with a time homogenous semi-markov (G o n, Tn) o with discrete time. The states are represented by the grids G = {g 1, g 2,..., g L }. An object that moves between two grids transits in the Markov process between the corresponding states. We assume the transition probabilities between states have the Markov memory-less property, meaning that the probability of object o transiting from state G o i to G i + 1 o is independent of state G i 1 o. Thus, process (G o n) is a standard discrete time Markov Chain. The random variable Tn o represents the time instance of the transition G o n G o n+1. The random variable Tn+1 o Tn o represents the sojourn time of object o in grid n.the associate Semi-Markov model is defined as follows: Q o ij (t) = P (Go n+1 = j, T o n+1 T o n t G o 0, G o 1,..., G o n; T o 0, T o 1,..., T o n) = P (G o n+1 = j, T o n+1 T o n t G o n = i) (1) t Qo ij Let p o ij be transition probability that object o moves from grid i to grid j,and p o ij = lim (t), i, j G(2). In this paper, we assume that the transition probability p ij are given which can be attained by the history information or given

5 by the experts. Assume Si o (t) is the probability that object o will leave grid i on or before time t. It also indicates the distribution of the dwell time of object o in grid i. S o i (t) = P (T o n+1 T o n t G o n = i) = L S ij (t) = P (t ij < t) = t 1 n=0 j=1 Q o ij (t) (3) P (t ij = n) (4) We assume that the mobility history provides a representative sample from which the sojourn time distribution can be drawn. Then, using the assumption that the dwell time random variables are independent from the embedded state transition process (G ij ), we derive Q ij (t) = P (G n+1 = j, T n+1 T n t G n = i) = P (G n+1 = j G n = i) P (T n+1 T n t G n = i, G n+1 = j) = p ij S ij (t) (5) 4.2 Meeting Probability If we know that an object o is currently in grid i, after t time units, it will be in grid j with probability φ o (t). This provides the prediction of object grid ij location. To determine φ o (t),we start with a special case when an object is in ij grid i and there is no movement between time 0 and t. P (G t = i, G 0 = i, T 1 t) = P (T 1 T 0 t G 0 = i) = 1 S i (t) If the object transits at least once between time 0 and t, we consider on time k of the first transition from state i and state r to which the object moves immediately after state i. The process can be formulated as follows: (6) P (G t = i, G 0 = i, atleastonetransition) = l t 1 (Q ir (k) Q ir (k 1))φ rj (t k)) (7) r=1 k=1 Putting them together, we obtain: φ ij (t) = (1 S i (t))φ ij (0) + l t 1 r=1 k=1 (Q ir (k) Q ir (k 1))φ rj (t k)) (8) We note that φ can be computed iteratively. Next, we show how to compute the meeting probability of two moving objects. Assume that the trajectories of moving objects are independent of each other. The probability that objects a and b will meet in grid k at future time t is m k ab (t) = ϕa sa,k (t t a) ϕ b sb,k (t t b).here, sa/sb is the most recent grid location of object a/b at time ta/tb. Then the probability that a and b will meet at any grid at time t is:

6 m ab (t) = L k=1 m k ab (t) (9) The PPNN algorithm in our paper will use this meeting probability to filter those impossible result objects. We will introduce the filtering process in Section 6 in details. 5 Markov Model 5.1 Uncertain Trajectory Model The semi-markov model in Section 4 describes the movement of objects between space grids. We can filter the impossible result objects by computing their meeting probability. However, as for the remaining candidate objects, we have to compute their nearest neighbor probability. We can describe the location of an uncertain moving object at time t as a random variable o(t).given a time interval [0,t], the locations of moving objects can be represented by a family of correlated random variables, i.e. a stochastic process. In this paper, we explore the Markov Model as a specific instance of a stochastic process. As introduced in Section 3.1, the state space of the Markov Model is the spatial domain S, and state transitions are defined over time domain T. The Markov chain model is based on the assumption that the position o(t + 1) of an uncertain object o only depends on the location o(t) at time t. The condition probability Mij o (t) = P (o(t + 1) = s j o(t) = s i ) (10) is the transition probability of uncertain object o from state i to state j at a given time t. Transition probabilities are stored in a matrix M o (t),called transition matrix of object o at time t.let s(t) = (s 1, s 2,..., s S ) T (11) be the distribution vector of object o at time t, where s(t) = P{o(t) = s i }. Without any further knowledge, the distribution s(t + 1) can be obtained from s(t) by the following formula s(t + 1) = M o (t) T s(t) (12). 5.2 Nearest Neighbor Probability We start by computing the probability that a candidate object o is the NN of query q at time t. Let J (t) be the joint probability matrix of o and o.j ijk (t) = P (o(t) = s i o (t) = s j q(t) = s k ) (13) denotes the probability that at time t object o is in state s i,object o is in state s j and query q is in state s k.we assume that the movement of uncertain objects are dependent of each other, therefore, matrix J (t) can be computed by J(t) = s o (t) s o (t) T s q (t) T (14).From this joint probability matrix, we can compute the probability that o is closer to q than o at time t as follows.we need to define an indicator matrix I : I ijk (t) = { 1, if dist(si, s k ) dist(s j, s k ) 0, otherwise (15)

7 The indicator matrix I (t) can describe for each state pair, which state is closer to q. Therefore, we can determine the aggregated probability of o being closer than o by evaluating H(t) = J(t) I(t). H (t) represents all the possible worlds still satisfying the query predicated at time t, and for each entry H ijk (t), we have to aggregate over all states object o, o and q can come from the previous time step t 1, as the following formula shows: H ijk (t) = l m n Therefore, (16) can be rewritten as (H lmn (t 1) Mli o o (t 1) Mmj (t 1) M q nk (t 1)) I ijk(t) (16) H(t) = (H(t 1) M o (t 1) M o (t 1) M q (t 1)) I(t) (17) Then the following holds: P P NN(o, q, {o }, t) = i (H ijk (t)) (18) Due to the stochastic independence of moving objects, the NN probability of o in the candidate sets CS can be computed as following: P P NN(o, q, CS, t) = P P NN(o, q, {o }, t) (19) 6 PPNN Algorithm o CS In this section, we present the detailed processing algorithm for a PPNN query. In the tracking system, we first maintain for each moving object two models, the Semi-Markov and Markov models as introduced in Section 4 and Section 5. We can construct these models based on the history trajectories. Then we can process a predictive query by employing the PPNN algorithm. Algorithm 1 and algorithm 2 are the pseudocodes summarizing the above process for processing a PPNN query. PPNN-Filter, by employing the Semi-Markov model and computing the contact probabilities for objects o and q, we can prune a great number of impossible results and get an candidate set CS. As shown in line 3 in Algorthm1, the contact probability is computed according to Formula (9) in Section 4. PPNN-Refine, by employing the Markov model and computing the NN probability for each object in the candidate set, we can obtain the final objects satisfying the query predicates. As shown in Algorithm 2, given the trajectory database D, the candidate set CS, the future time te and the query object q, Algorithm 2 will output the predictive possible nearest neighbor of q. Line 1 and line 2 first generate the indicator matrix for each discrete time stamp from now to the future time te. Lines 3 through 9 use a double loop to compute the PPNN probability for each candidate object when there is only one object o in database D. Then in line 10, due to the independence of each object trajectory, we can compute the final NN probability of object o according to Formula (19) in Section 5. j k

8 Algorithm 1: PPNN Filter Input: (1) Uncertain trajectory database D, (2) future query time t, and (3) query object q Output: Candidate result object set CS 1: CS := ; 2: for each node o in D q do 3: Compute m oq (t) = L m k oq(t); 4: if m oq(t) > 0 then 5: Insert o into CS; 6: Return CS; k=1 Algorithm 2: PPNN Refine Input: (1) Uncertain trajectory database D, (2) future query time te, (3) query object q,(4)candidate query result set CS Output: query result object set RS 1: RS := ; 2: for t=0 to te do 3: Generate I (t); 4: for each o in CS do 5: for each o in CS-o do 6: J(0) = s o (0) s o (0) T s q (0) T 7: H(0) = (M o (0) M o (0) M q (0)) I(0); 8: for t=1 to te do 9: H(t) = (H(t 1) M o (t 1) M o (t 1) M q (t 1)) I(t); 10: P P NN(o, q, {o }, t) = (H ijk (t)); i j k 11: p = P P NN(o, q, CS, t) = P P NN(o, q, {o }, t); 12: if p > 0 then 13: Insert o into RS; 14: Return RS; o CS 7 Experiment Evaluation In this section we report on the experimental results on different datasets. We describe the relevant settings in Section 7.1 and present the experimental results in Section Experimental Settings We conduct a set of experiments to verify both the effectiveness and efficiency of the proposed solutions, using a desktop computer having an Intel i CPU at 2.40GHz and 4GB of RAM. We use the data generator to construct a two dimensional Euclidean state space, consisting of n states. Each of these states

9 is drawn uniformly from the [0, 1] 2 square. The space is further divided into L virtual grids. We assume that the mobility history provides a representative sample from which the sojourn time distribution can be drawn. In order to construct a transition matrix, we derive a graph by introducing edges between any point and its neighbors. We then set the transition probability of this entry indirectly proportional to the distance between the two connected vertices. 7.2 Experimental Results Varying the number of grids We set the number of moving objects to be in the first experiment and evaluate the filtering capability when varying the number of grids from 100 to As shown in Fig.2(a), the more the virtual grids, the fewer the candidates left. It means that the filtering capability increases with the number of grids. In fact, the filtering capability will be more obvious when the number of grids is large. However, the computation cost will increase with the number of grids as shown in Fig.2(b). Therefore, there is a tradeoff between the number of grids and the computation cost. Fig. 2. (a)filtering capability VS #grids (b)cpu time VS #grids Varying the number of space states The CPU time increases with the number of states in the refinement phase. However, after the filtering phase, there are only a small number of candidate objects left. Hence, the CPU time will not increase too much with the number of the space states. As shown in Fig.3(a), the CPU time only increases a little when increasing the number of states. As shown in Fig.3(b), the size of the candidate set is relative stable when varying the number of states. It is related to the size of grids. Varying the number of objects As shown in Fig.4 and Fig.5, the number of objects leads to a decreasing performance as well. The more objects stored in a database with the same underlying motion model, the more candidates objects are found during the filtering step. This leads to an increasing number of probability calculations during refinement, and hence a higher query cost.

10 Fig. 3. (a)filtering capability VS #states (b) CPU time VS #states number of candidates Cpu time(sec) number of objects number of objects Fig. 4. Filtering capability VS #objects Fig. 5. CPU time VS #objects 8 Conclusions In this paper, we address the problem of processing predictive NN queries over uncertain spatial temporal data. We use the Markov chain to model the uncertain movement of objects based on stochastic processes. To cope with the large trajectory database, we exploit the filtering and refinement framework to speed up processing a predictive NN query. In the filtering phase, we divide the space into L virtual grids and model the movement of objects as a Semi-Markov chain. By computing the meeting probability of each object to the queried object, we can prune a large number of non-result objects and obtain a small number of candidate result objects. Then in the refinement phase, we model the movement of objects as a Markov chain and compute the exact NN probability of the candidate objects. The experimental evaluation shows that our two-level Markov model can effectively describe the movement of objects and can efficiently process a predictive NN query over uncertain spatial-temporal data. Acknowledgment This work was supported in part by the National Science Foundation for Young Scholars of China( , ), and the Research Foundation of Harbin for Youth Innovative Talents(2011RFQXG028, 2012RFQXG096).

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