1. Yoshiharu Ishikawa (Nagoya University) Yoji Machida (University of Tsukuba) Hiroyuki Kitagawa (University of Tsukuba) A Dynamic Mobility Histogram Construction Method Based on Markov Chains

2. 1 Outline Background and Objectives Modeling Movement Patterns Mobility Histogram: Logical Structure Mobility Histogram: Physical Structure Experimental Results Conclusions

3. 2 Background Advance of GPS and communication technology enabled tracking of moving objects –Example: A taxi company in Tokyo monitor >200 taxi cabs continually Movement data is delivered as a data stream Data Stream Movement Data Moving Object Database Moving Objects

4. 3 Objectives Construction and maintenance of a mobility histogram –Compact summary of movement data for a specific time period –Used for mobility analysis and estimation Problems –Concrete definition of a mobility histogram How to model movement patterns –Compact representation Tradeoff with accuracy –Efficient construction and maintenance Incremental processing for streamed data

5. 4 Movement Data (as a Data Stream) Mobility histogram Histogram Maintenance Module Incremental updates Mobility Analysis / estimation Module Query for estimation … Request for analysis / estimation Results Basic Idea

6. 5 Outline Background and Objectives Modeling Movement Patterns Mobility Histogram: Logical Structure Mobility Histogram: Physical Structure Experimental Results Conclusions

7. 6 Approach 2-D movement area Uniform cell decompositions –But allow multiple spatial granularities (e.g., 4 x 4, 16 x 16) Movement pattern is represented as a sequence of cell numbers Based on the Markov chain model –Treats a movement pattern as a Markov chain sequence –Well-known model in traffic modeling

8. 7 Movement Patterns: Example (1) Movement pattern of A Movement pattern of B Movement pattern of C 2  2  0  0 3  3  1  1 0  2  2  3 0 1 23 A B C

9. 8 Movement Patterns: Example (2) Cell partitioning with different granularities Movement pattern of A 11  9  3  1 A 0 2 8 10 1 3 9 11 4 6 12 14 5 7 13 15

10. 9 Cell Numbering Scheme (1) Based on Z-ordering method –Simple encoding method –Assign similar values to neighboring cells –Translation to different granularities is easy 0 2 8 10 1 3 9 11 4 6 12 14 5 7 13 15

11. 10 Cell Numbering Scheme (2) 0 (2) 0000 1 (2) 0001 2 (2) 0010 3 (2) 0011 Level-1 (2 1 x2 1 ) decompositionLevel-2 (2 2 x2 2 ) decomposition

12. 11 Markov Chain Model (example: order = 2) Step 0Step 1Step 2 2 (1)  3 (1)  1 (1) 9 (2)  12 (2)  6 (2)

13. 12 Outline Background and Objectives Modeling Movement Patterns Mobility Histogram: Logical Structure Mobility Histogram: Physical Structure Experimental Results Conclusions

14. 13 Mobility Histogram as a Data Cube Representing order-n Markov chain statistics as a (n +1)-d data cube Example: 1 (1)  1 (1)  0 (1)

15. 14 Movement Data Mobility histogram Histogram Maintenance Module Incremental updates Mobility Analysis / Estimation Module Query for analysis … Histogram Maintenance … Periodical reconstruction –To cope with non-stationary movement patterns –Ease of maintenance –Old histograms are written to disk

16. 15 Outline Background and Objectives Modeling Movement Patterns Mobility Histogram: Logical Structure Mobility Histogram: Physical Structure Experimental Results Conclusions

17. 16 Mobility Histogram: Physical Structure Problems in logical structure: huge space –2GB (!) for a typical parameter setting –Needs multiple cubes for multiple spatial granularities –Data cubes are sparse: most of mobility patterns are hard to occur Solution: tree-based representation –Unification of quad-tree, k-d tree, and trie –Integration of cubes in multiple granularities –Selective allocation of nodes Saves memory space

18. 10 root x : counter 00 01 11 01 11 10 01 level 1 level 2 Binary representation Step 0: Step 1: Step 2: 0011 0110 1100 (=3) (=6) (=12) : visited edge : non-visited edge 00 10 11 00 01 11 00 0110 01 00 11 00 11 +1 step 0  step 1  step 2 10 Insertion of 3 (2)  6 (2)  12 (2) : BASE method

19. 18 Approximated Histogram (APR) Problem of the BASE method –Memory size requirement is still high Approximated method (APR) –Compact histogram construction by adaptive tree expansion Allocate a buffer for each leaf node If skew is observed, the leaf node is expanded  2 statistics is used to check the non-uniformity –Inherited the idea from decision tree construction from streamed data (e.g., VFDT)

20. 19 Node Expansion 00 01 10 11 10 11 trans_seq[0] trans_seq[1] … buffer 00 01 10 11 10 11 00 01 10 11 buffer expansion skew is detected root internal node leaf node internal or leaf node 00 01 10 11 Quit expansion when no. of nodes has reached a given constant

21. 20 Example: 100 sequences in the buffer Non-uniformity Check Use of  2 test for goodness of fit Null hypothesis: distribution is uniform If  2 value > 7.815, the distribution is non-uniform at the significance level 5% Buffer … 5 (2)  12 (2)  9 (2) 7 (2)  13 (2)  15 (2) 4 (2)  12 (2)  6 (2 ） 2223 2728 1020 5020 UniformNon-uniform x 00 x 01 x 10 x 11 Distribution of next steps

22. 21 Problems in Statistical Test Problems:  2 value is not reliable –when the total number is small –when some value(s) is close to 0 Solution: use non-parametric statistics while  2 value is not reliable –Detail is shown in the paper 12 14 Total number = 1 + 2 + 1 + 4 = 8 010 2025 These situations are common in our case

23. 22 Minor improvement to the APR method –Use a small bitmap cube in addition to a tree- structured histogram –Represent “correct” summary in some coarse level –Improvement of precision Use of Bitmap Cube (APR-BM) level = 1 level = 2 10 00 01 10 11 10 01 00 10 11 00 01 11 00 0110 01 00 11 00 11 10 11 25336 13821 4351 1293 538 299 53 38 Tree-based histogram (APR method) + Small bitmap cube in a coarse level Example: When partition level = 3, Markov order = 2, bitmap size = 32KB Accurate estimation for some queries

24. 23 Outline Background and Objectives Modeling Movement Patterns Mobility Histogram: Logical Structure Mobility Histogram: Physical Structure Experimental Results Conclusions

25. 24 Dataset and Environments Experimental data –Used moving objects simulator by Brinkoff –1024×1024 in finest granularities –1,000 moving objects are on the map at every time instance Environments –CPU ： Pentium4 3.2GHz –Memory ： 1GB RAM –OS ： Cygwin

26. 25 Histogram Size Settings –Data Size: 1K, 10K, 50K –Order-2 Markov transition Results –BASE method requires huge storage BASEAPRAPR-BM 1K0.350.010.04 10K2.70.100.13 50K9.40.520.55 Data Size Histogram Size (MB)

27. 26 Construction Time Comparison of BASE and APR –M: maximal partitioning level (granularity of input sequences) Results –BASE has small construction cost –APR has nearly O(n 2 ) cost due to non-uniformity check, but still has small processing cost (less than 0.15 ms per input sequence) M = 5, BASE M = 5, APR M = 10, BASE M = 10, APR M = 5, BASE M = 5, APR M = 10, BASE M = 10, APR Construction Time Construction Time per Sequence

28. 27 Query Processing Time Two types of queries –Fine level: Issue queries on the most fine partitioning level (M = 10) –Mixed-level: Issue queries on randomly mixed partitioning levels Results –Comparison of BASE and APR –No difference –Quite fast BASE APR fine-level query mixed-level query

29. 28 Accuracy : Histogram Plot (1) Order-1 Markov chain histograms Partition level = 2 BASE ( “ true ” count) APR

30. 29 Accuracy: Histogram Plot (2) Diff Count = |Base count – APR count| Histogram Difference

31. 30 Precision: Evaluation Measures Distance Relative Error ACT i : Actual cell value (BASE method) EST i : Estimated cell value (APR and APR- BM methods)

32. 31 Evaluation of Precision Comparison of APR and APR-BM –Using “Distance” and “Relative Error” Results –Similar results for Distance –APR-BM is better in terms of Relative Error APR-BM can estimate small cell values accurately Distance Relative Error

33. 32 Outline Background and Objectives Modeling Movement Patterns Mobility Histogram: Logical Structure Mobility Histogram: Physical Structure Experimental Results Conclusions

34. 33 Conclusions Mobility histogram construction method –Based on Markov chain model –Handling streamed trajectory sequences –Logical histogram: data cube –Physical histogram: tree structure (quad tree + k-d tree) Adaptive tree growth Approximated representation method Use of nonparametric statistics for exceptional cases Use of a bitmap cube to enhance precision