1. Robust Similarity Measures for Mobile Object Trajectories
Michalis Vlachos (UCR), Dimitrios Gunopulos (UCR), George Kollios (BU)
MDDS ‘02

2. Introduction Problem: Discover similar trajectories of moving objects
Examples:
Features extracted from video-clips
Animal Mobility Experiments (GPS data)
Sign Language Recognition, etc.

3. Applications & Requirements
Clustering
Classification
What do we need?
Similarity Measure (robust to noise)
Indexing Scheme
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4. Outline Related Work (Euclidean Distance, Time Warping)
Extension of LCSS model to 2d trajectories
Algorithms for Computing the new similarity model
Flexible Sigmoidal Matching
Comparison with Lp-Norms and DTW distance
Conclusions, Future Work
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5. Related Work – Euclidean Distance
Lp–Norm: LP=(Σ(xi-yi)p)1/p
L2: Euclidean Distance
L1: Manhattan Distance
Disadvantages
Small Robustness to outliers
Sensitive to time axis displacement
Does not support variable lengths
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6. Related Work – DTW Time Warping Disadvantages
Allows stretching in time axis
Difficult Indexing
Disadvantages
Computationally intensive, O(n*m)
Has to match ALL elements
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7. Requirements for new Similarity Model (1)
We need to address the following issues:
Different Sampling Rates or Different Speeds
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8. Requirements for new Similarity Model (2)
We need to address the following issues:
Similar Motions in different space Regions
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9. Requirements for new Similarity Model (3)
We need to address the following issues:
Outliers
Random Peaks
Noise Everywhere
Non Recoverable Part
Different Lengths
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10. Longest Common Subsequence (LCSS)
Arithmetic Example:
t1=[0, 4, 6, 8, 7, 4, 6, 5, 6, 4, 6]
t2=[0, 3, 4, 6, 7, 6, 3, 6, 4, 6 ]
Dynamic Programming Solution
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11. Extending LCSS (1)
We extend the LCSS to 2-dimensions and add more flexibility:
Similarity of 2 seq/s with length n & m:
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12. Extending LCSS – Example2ε
Rigid matching
Points marginally outside matching region are ignored
Set parameter epsilon
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13. Extending LCSS – Flexible Matching
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14. Sigmoidal Matching
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15. Computation Algorithms for new models (S1)
Computing Similarity S1
Lemma 1: Given two trajectories A and B, with |A|=n and |B|=m, we can find the SigmoidSimδ(Α,Β) in O(δ(n+m)) time
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16. Extending LCSS (2) S1 cannot detect parallel movements,
f(B)
Time B Y X
So, we define S2:
S2 can detect parallel movements
Better accuracy than simple normalization
Distance D1= 1-S1 & distance D2 = 1-S2
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17. Exact Algorithm for similarity function S2
For trajectories A, B with length n we want to find:
translation fc,d that maximizes SigmoidSim between A and fc,d (B)
Not infinite translations.
Each dimension separately
A translation in 1D: fc(bi) = bi + c (line with slope 1)
fc(bi) will allow bi to be matched to all aj: |i-j|<δ & ai-ε ≤ fc(bi) ≤ (bi, aj+ε)
Transform into a stabbing problem
Translations : O(δ2n2)
LCSS : O(δn)
Total : O(δ3n3)
y=x+2
y=x 1 2 3 4 5 6 1 2 3 4 5 6
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18. Approximate Algorithm for similarity function S2
A translation corresponds to a line fc(x) = x+c.
Sort translations by c  THEY DIFFER IN HOW MANY SEGMENTS?
If we can afford to be within β of max(Sim)  we can afford to lose βn elements
Don’t take all translations  we can examine every βn translations each time
So, if we examine every βn, we lose at most βn elements (1D)
So, for 2D, we can skip every βn/2 translations
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19. Approximate Algorithm for similarity function S2
Theorem: Given two trajectories A and B, with |A| = n and |B|=n, and a constant 0<β<1, we can find an approximation AS2δ,β(A,B) of the similarity S2(δ,ε,A,B) such that S2(δ,ε,A,B) - AS2δ,β(A,B) < β in O(nδ3/ β2) time.
Example:
|A| = |B| = 1000, δ=2, β=0.04=>b=0.04*1000/2=20
total # translations: 2δn = 4000, {-100, -98, -95,…, -30, -10, 0,…, 0.1, 2, 3.3, ..}
# translations we consider: 2δn/b = 200; in 2d 400 times less translations
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20. Approximate Algorithm for similarity function S2 (cont/d)
Theorem: Given two trajectories A and B, with |A| = n and |B|=n, and a constant 0<β<1, we can find an approximation AS2δ,β(A,B) of the similarity S2(ε,A,B) such that S2(δ,ε,A,B) - AS2δ,β(A,B) < aβ in O(nδ3/ β2) time, for a constant a.
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21. Clustering Accuracy C1 C2 C3 C4 C5 Datasets: MobileLong MobileShort
MobileShort + Noise
Test clustering accuracy using Hierarchical Clustering C1 C2 C3 C4 C5
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22. DTW SIGMOIDSIM Clustering Accuracy Lp–Norm: LP=(Σ(xi-yi)p)1/p
DTW = Lp + min((Head(A), B), (A,Head(B)), (Head(A), Head(B)))
SigmoidSim without translation
DTW
SIGMOIDSIM
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23. Clustering Accuracy (MobileLong)
Number of Correct Clusterings out of 10
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24. Clustering Accuracy (MobileShort)
Number of Correct Clusterings out of 21
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25. Clustering Accuracy (MobileShort + Noise)
Number of Correct Clusterings out of 21
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26. Conclusions, Future Work
Sigmoid Similarity provides best results under noise
Optimal translation can be found
Approximate solutions with provable performance bounds
FUTURE WORK
Improve LCSS performance
Trajectory Segmentation
Add Scaling & Rotation
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27. MDDS ‘02