1. ICDE, San Jose, CA, 2002 Discovering Similar Multidimensional Trajectories Michail VlachosGeorge KolliosDimitrios Gunopulos UC RiversideBoston UniversityUC Riverside

2. Outline Introduction Similarity Measures Compute the Similarity Indexing Trajectories Experimental Evaluation Related Work Conclusion

3. Introduction The trajectory of a moving object is typically modeled as a sequence of consecutive locations in a multidimensional Euclidean space An appropriate and efficient model for defining the similarity for trajectory data will be very important for the quality of the data analysis tasks.

4. Examples of 2D trajectories

5. Hierarchical clustering of 2D series (displayed as 1D for clariry)

6. Similarity Measures Let A and B be two trajectories of moving objects with size n and m respectively, where A = ((a x,1, a y,1 ),…,(a x,n, a y,n )) and B = ((b x,1, b y,1 ),…,(b x,m, b y,m )). For a trajectory A, let Head(A) be the sequence Head(A) = ((a x,1, a y,1 ),…,(a x,n-1, a y,n-1 ))

7. Definition 1 Given an integer δ and a real number 0 < ε <1, we de fi ne the LCSS δ,ε (A; B) as follows:

8. Definition 2 We de fi ne the similarity function S1 between two trajectories A and B, given δ and ε, as follows:

9. A region of δ & ε for a trajectory

10. Definition 3 Given δ, ε and the family F of translations, we de fi ne the similarity function S2 between two trajectories A and B, as follows:

11. Translation of trajectory B

12. Definition 4 Given δ, ε and two trajectories A and B we de fi ne the following distance functions:

13. Compute the Similarity Similarity function S1 Given two trajectories A and B, with |A| = n and |B| = m, we can find the LCSS δ,ε (A, B) in O( δ (n + m)) time. Similarity function S2 Given two trajectories A and B, with |A| = n and |B| = m, we can compute the S2( δ,ε, A, B) in O((n+m) 3 δ 3 ) time.

14. Approximate Algorithm

15. Indexing Structure For every node C of the tree we store the medoid (M C ) of each cluster. The medoid is the trajectory that has the minimum distance (or maximum LCSS) from every other trajectory in the cluster:

16. Time and Accuracy Experiments Similarity values and running times from SEALS dataset

17. Experiment 1 - Video tracking data

18. Experiment 2 & 3 - Australian Sign Language Dataset (ASL)

19. Evaluating the indexing technique

20. Related Work Use a p-norm distance to de fi ne the similarity measure. [2, 37, 18, 14, 10, 32, 10, 20, 24, 23] Based on the time warping technique.[5, 25, 28, 33] Find the longest common subsequence (LCSS) of two sequences.[3, 7, 11] Define time series similarity are based on extracting certain features.[13, 17, 29, 31]

21. Conclusion Efficient techniques to accurately compute the similarity between trajectories Approximate algorithms with provable performance bounds ef fi cient index structure

22. Comments Good approach for similarity queries Use real GPS trajectory data? …

23. Dynamic Time Warping Sequences are similar but accelerate differently along the time axis Enforcing a temporal constraint δ on the warping window size improves computation efficiency and accuracy Application : Speech recognition ( Berndt and Clifford, 1996)

24. 1 Longest Common Subsequence Similarity Dissimilarity: Tolerance: 2 5 4 5 3 1 8 12345171234517 0 0 0001 1 1 11 1 11 1 1 11 1 22 2 1 1 22 2 22 2 2 3 3 33 1 1 2 2 3 3 44 2 2 3 3 44 2 4 5 1 Match 2 sequences by allowing some elements to be unmatched C = {1,2,3,4,5,1,7} and Q = {2,5,4,5,3,1,8} Longest is {2,4,5,1} Application : Bioinformatics Vlachos et al., 2002

25. 1 Longest Common Subsequence Similarity for i := 1..m for j := 1..n if C[i] = Q[j] L[i,j] := L[i-1,j-1] + 1 else: L[i,j] := max(L[i,j-1], L[i-1,j]) return L[m,n] Input sequences C[1..m] and Q[1..n] Compute LCS btwn C[1..i] and Q[1..j] for all 1 ≤ i ≤ m and 1 ≤ j ≤ n Stores it in L[i,j] L[m,n] = length of the LCS 2 5 4 5 3 1 8 12345171234517 0 0 0001 1 1 11 1 11 1 1 11 1 22 2 1 1 22 2 22 2 2 3 3 33 1 1 2 2 3 3 44 2 2 3 3 44 2 4 5 1 Vlachos et al., 2002