August 30, 2004STDBM 2004 at Toronto Extracting Mobility Statistics from Indexed Spatio-Temporal Datasets Yoshiharu Ishikawa Yuichi Tsukamoto Hiroyuki Kitagawa University of Tsukuba

Outline Background and objectives Markov transition probability Indexing method for moving trajectories Proposed methods naïve algorithm CSP-based algorithm Experimental results Conclusions

Background Moving object databases stores and manages information on a huge number of moving objects supports queries on moving trajectories and/or moving status Research issues spatio-temporal indexes extraction of statistics (e.g., selectivities) Statics in spatio-temporal databases used for query optimization also useful in mobility analysis

Objective: extracting mobility statistics from spatio- temporal databases Target: trajectory data indexed using R-trees Statistics to be extracted ： Markov transition probability target space is decomposed in cells estimating transition probabilities between cells using the indexed trajectory data Features search problem is formalized as constraint satisfaction problem (CSP) efficient processing using R-trees Our Approach

Outline Background and objectives Markov transition probability Indexing method for moving trajectories Proposed methods naïve algorithm CSP-based algorithm Experimental results Conclusions

Markov Transition Probability (1) Assumption: target space is decomposed in cells Example 1: What is the estimated probability that an object currently in cell c 0 moves in cell c 1 in a unit time later? First-order Markov transition probability Pr(c 1 |c 0 ) t =τ A t =τ+1 A c1c1 c0c0

Markov Transition Probability (2) Example 2: What is the probability that an object which moves from c 0 to cell c 1 in a unit time moves to cell c 2 in the next unit time? Second-order transition probability Pr(c 2 |c 0, c 1 ) Extension to order-n Markov transition probability Pr(c n |c 0, …, c n-1 ) is easy t =τ A t =τ+1 A t =τ+2 A c1c1 c0c0 c2c2

Markov Transition Probability Conventional technique in traffic data analysis Upton & Fingleton, 1989 [13] Special kind of association rules probability corresponds to the confidence factor difference: existence of order Usage trajectory estimation estimates where a moving object moves to in the next period simulation of movement status given status of moving objects at t = , we can estimate the change of the status at t =  + 1,  + 2, …

Assumptions Movement patterns obeys stationary process movement tendency does not change as time passes Cell decomposition each cell is a rectangle cell size is arbitrary: non-uniform decomposition is allowed cell decomposition can be specified dynamically Unit time length unit time can be specified as arbitrary length (e.g., one minuite, 10 minuites, …) but a unit time length should be a multiple of sampling time length

Formalization of Probability (1) Target data: trajectory data from t = 0 to t = T Definition of first-order Markov transition probability objs(c i, t) : set of objects which were in cell c i at t denominator: no. of objects which were in cell c 0 at arbitrary t (0 ≤ t ≤ T  1) numerator: no. of objects each of which contained in denominator and moved cell c 1 at t + 1

Formalization of Probability (2) Definition of order-n Markov Transition Probability denominator: no. of objects each of which was in cell c 0 at t (0 ≤ t ≤ T  1), in cell c 1 at t + 1, …, and in cell c n  1 at t + n  1 numerator: no. of objects each of which is contained in Dominator and moved cell c n at t + n

Generalized Transition Probability Estimation Problem (1) Given n + 1 cell sets for each of arbitrary cell combinations output Pr(c n |c 0,…,c n-1 ) Derives transition probability according to the specified cell sets at once

Generalized Transition Probability Estimation Problem (2) Example: Given C 0 = {c 0, c 1 }, C 1 = {c 1, c 2 }, C 2 = {c 1, c 2, c 3 }, estimate second-order probabilities Algorithm outputs 12 probabilities Pr(c 1 |c 0, c 1 ), Pr(c 2 |c 0, c 1 ), …, Pr(c 3 |c 1, c 2 ) c0c0 c1c1 c2c2 c3c3

Outline Background and objectives Markov transition probability Indexing method for moving trajectories Proposed methods naïve algorithm CSP-based algorithm Experimental results Conclusions

Indexing Methods for Trajectories R-tree-based approach is assumed Point-based representation: trajectories is represented as a set of points ( d+1 )-dimension R-tree is used (e.g., 3D R-tree) incorporating temporal dimension

０ (=T) x (d +1)-D R-tree-based Representation Sampling-based representation A B root abc ０ (=T) x １ ２ ４ ５ ３ ６ a b c root

Outline Background and objectives Markov transition probability Indexing method for moving trajectory data Proposed methods naïve algorithm CSP-based algorithm Experimental results Conclusions

Naïve Algorithm (1) Based on the definition of the Markov transition probability Example: Estimating Pr(c 2 |c 0, c 1 ) Determine objs(c 0,  ) and objs(c 1,  + 1) using the R-tree objs(c i, t) : the set of objects which were in cell c i at time t Take intersection of two sets; the cardinality of the intersection is added to Scount If the intersection is not empty objs(c 2,  + 2) is determined using the R-tree Take intersection of objs(c 0,  ), objs(c 1,  + 1), objs(c 2,  + 2) ; the cardinality of the result is added to Qcount This process is repeated for each  (0 ≤  ≤ T – n) Calculate Pr(c 2 |c 0, c 1 ) based on Scount, Qcount No. of search on R-tree is proportional to T

Naïve Algorithm (2) ０ (=T) x cell c 1 Example: estimation of Qcount += 1 No. of search on R-tree is proportional to T Output = Qcount Scount Scount += 1 cell c 0 cell c 2

Outline Background and objectives Markov transition probability Indexing method for moving trajectories Proposed methods naïve algorithm CSP-based algorithm Experimental results Conclusions

Basic Idea (1) Estimation of Pr(c n |c 0, …, c n-1 ) based on three steps: 1. Count the no. of objects which were in c 0, …, c n-1 at each unit time using an R-tree 2. Count the no. of objects which were in c 0, …, c n at each unit time using an R-tree 3. Compute Pr(c n |c 0, …, c n-1 ) by [result of step 2] / [result of step 1] Benefits step 1 & 2 can be processed using the same algorithm algorithm for step 1 is given by setting n → n – 1 requires only two searches on R-tree

Basic Idea (2) ０ (= T ) x cell c 2 Example: estimation of Pr(c 2 |c 0, c 1 ) cell c 1 cell c 0 Step 1: count objects which moved from c 0 to c 1 within a unit time Scount = 2 Step 2: count objects that moved as c 0, c 1, c 2 at each unit time Qcount = 1 Pr(c 2 |c 0, c 1 ) = ― ―――― Step 3: compute probability

Counting Using R-tree (1) How can we compute no. of objects which were in c 0, …, c n at each unit time? Idea: the problem is formalized as a constraint satisfaction problem (CSP) An object satisfying the constraint fulfills the following constraints for some  it was in cell c 0 at t =  it was in cell c 1 at t =  + 1 … it was in cell c n at t =  + n Search objects that satisfy all n + 1 constraints

Counting Using R-tree (2) Effective use of R-tree is necessary We extend the CSP solution search method using R-trees (Papadias et al, VLDB’98) [7] considers spatial constraints Example: find all spatial objects x, y, z that satisfy overlap(x, y) and north(y, z) search CSP solutions from the root to leaves Use of pruning and backtracks Reduce search space using constraints enumerates all solutions with one R-tree access

Example of Counting (1) ０ (=T) x １ ２ ４ ５ ３ ６ a b c root c１c１ c２c２ For C 0 = {c 1 }, C 1 = {c 1, c 2 }, C 2 ={c 2 }, derive probabilities for (C 0, C 1, C 2 ) Derive two probabilities at once Pr(c 2 |c 1, c 1 ) : the probability that an object which have moved as c 1  c 1 next moves to c 2 Pr(c 2 |c 1, c 2 )

Example of Counting (2) root a bc R-tree ０ (=T) x １ ２ ４ ５ ３ ６ a b c root c1c1 c2c2

Pruning Method (1) Pruning condition 1: Movement between two R-tree nodes which do not temporary consecutive is impossible Candidates can be deleted ０ (=T) x a c b Example: - movement such as a  b and b  c are allowed - movement a  c is impossible

Pruning Method (2) ０ (=T) x cell c 1 Pruning condition 2: Trajectory is not contained in the target cell Example: When we are counting for c 1  c 1, we should consider only nodes that overlaps with c 1

Pruning Method (3) ０ (=T) x 2 1 distance between MBRs Pruning condition 3: If [max distance an object can move] < [distance between MBRs] then an object cannot move from a node to next node

Query Processing Example cell c 1 cell c 2 cell c 1 cell c 2 tree level = 2 cell c 1 cell c 2 x t root pruning a b c 1 2 tree level = 1 pruning tree level =0 backtrack An object that moved as c 1  c 1  c 2 is found and counted There is no objects that moved as c 1  c 1  c 2 c 1  c 2  c 2 Targets: c 1  c 1  c 2 c 1  c 2  c 2

Outline Background and objectives Markov transition probability Indexing method for moving trajectory data Proposed methods Naïve algorithm CSP-based algorithm Experimental results Conclusions

Dataset (1) Generated using the moving object simulator made by Brinkoff [1] Simulates car movement situation on actual city road network Oldenburg city, Germany (about 2.5km x 2.8km) no. of initial moving objects: 5 5 objects are created in a minute on average 100 objects are moving in the map at a time data is generated for T = 1000 minutes 120K points are stored in 3-D R-tree

Dataset (2) ｃ０ ｃ３ ｃ６ ｃ１ ｃ４ ｃ７ ｃ２ ｃ５ ｃ８ Example for estimating using 3 x 3 cells

Experimental Result (1) Map is decomposed into 30 x 30 cells First-order Markov transition probabilities Randomly 3 x 3 cells are selected

Experimental Result (2) Estimation of second-order transition probabilities Other parameters are same to the former case

Experimental Result (3) Estimation of third-order transition probabilities Other parameters are similar to the former case

Experimental Result (4) The case when CSP-based approach is not effective Target space is decomposed into 20 x 20 cells Estimation of second-order transition probabilities Since cell decomposition is coarse, the pruning cannot reduce candidates

Conclusions and Future Work Conclusions mobility statistics based on Markov transition probability proposals of two algorithms naïve approach CSP-based approach CSP-based approach effectively utilizes R-tree structure Future Work adaptive cell decompositions extension to non-stationary Markov transitions