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Robust Similarity Measures for Mobile Object Trajectories. Michalis Vlachos (UCR), Dimitrios Gunopulos (UCR), George Kollios (BU). Introduction. Problem: Discover similar trajectories of moving objects Examples: Features extracted from video-clips Animal Mobility Experiments (GPS data) - PowerPoint PPT Presentation
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MDDS ‘02 1
Robust Similarity Measures for Mobile Object Trajectories
Michalis Vlachos (UCR), Dimitrios Gunopulos (UCR), George
Kollios (BU)
2
Introduction
Problem:Problem: Discover similar trajectories of moving objectsExamples:Examples:• Features extracted from video-clips• Animal Mobility Experiments (GPS data)• Sign Language Recognition, etc.
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Applications & Requirements
What do we need?What do we need?• Similarity Measure (robust to
noise)• Indexing Scheme
ClassificationClustering
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Outline
• Related Work (Euclidean Distance, Time Warping)Related Work (Euclidean Distance, Time Warping)
• Extension of LCSS model to 2d trajectoriesExtension 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 WorkConclusions, Future Work• Conclusions, Future WorkConclusions, Future Work
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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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Related Work – DTW
• Time Warping
– Allows stretching in time axis
– Difficult Indexing
• Disadvantages
– Computationally intensive, O(n*m)
– Has to match ALL elements
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Requirements for new Similarity Model (1)
We need to address the following issues:We need to address the following issues:
• Different Sampling Rates or Different Speeds
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Requirements for new Similarity Model (2)
We need to address the following issues:We need to address the following issues:
• Similar Motions in different space Regions
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Requirements for new Similarity Model (3)
We need to address the following issues:We need to address the following issues:
• Different Lengths
• Outliers
Random PeaksRandom Peaks Noise EverywhereNoise Everywhere Non Recoverable PartNon Recoverable Part
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Longest Common Subsequence (LCSS)
• Dynamic Programming Solution
• Dynamic Programming Solution
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 ]
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 ]
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Extending LCSS (1)
We extend the LCSS to 2-dimensions and add more flexibility:We extend the LCSS to 2-dimensions and add more flexibility:
Similarity of 2 seq/s with length n & m:
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Extending LCSS – Example
• Rigid matching• Points marginally outside matching region are ignored• Set parameter epsilon
• Rigid matching• Points marginally outside matching region are ignored• Set parameter epsilon
2ε
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Extending LCSS – Flexible Matching
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Sigmoidal Matching
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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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Extending LCSS (2)
• S1 cannot detect parallel movements,• S1 cannot detect parallel movements,
BTime
X
Y
c d
f(B)
• S2 can detect parallel movements • Better accuracy than simple normalization• Distance D1= 1-S1 & distance D2 = 1-S2
• S2 can detect parallel movements • Better accuracy than simple normalization• Distance D1= 1-S1 & distance D2 = 1-S2
• So, we define S2:• So, we define S2:
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Exact Algorithm for similarity function S2
• Transform into a stabbing problem• Transform into a stabbing problem
y=x+2
y=x
112233445566
112233445566
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)
Translations : O(δ2n2)LCSS : O(δn) Total : O(δ3n3)
Translations : O(δ2n2)LCSS : O(δn) Total : O(δ3n3)
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Approximate Algorithm for similarity function S2
• If we can afford to be within β of max(Sim) we can afford to lose βn elements• 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
A translation corresponds to a line fc(x) = x+c.
• Sort translations by c THEY DIFFER IN HOW MANY SEGMENTS?
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Approximate Algorithm for similarity function S2
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
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.
MDDS ‘02 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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Clustering Accuracy
Datasets:
• MobileLong• MobileShort• MobileShort + Noise
Datasets:
• MobileLong• MobileShort• MobileShort + Noise
Test clustering accuracy using Hierarchical ClusteringTest clustering accuracy using Hierarchical Clustering
C1
C2
C3
C4
C5
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Clustering Accuracy
DTWSIGMOIDSIM
• 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
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Clustering Accuracy (MobileLong)
• Number of Correct Clusterings out of 10• Number of Correct Clusterings out of 10
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Clustering Accuracy (MobileShort)
• Number of Correct Clusterings out of 21• Number of Correct Clusterings out of 21
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Clustering Accuracy (MobileShort + Noise)
• Number of Correct Clusterings out of 21• Number of Correct Clusterings out of 21
MDDS ‘02 26
Conclusions, Future Work
FUTURE WORK
• Improve LCSS performance– Trajectory Segmentation
• Add Scaling & Rotation
FUTURE WORK
• Improve LCSS performance– Trajectory Segmentation
• Add Scaling & Rotation
• Sigmoid Similarity provides best results under noise• Optimal translation can be found• Approximate solutions with provable performance
bounds
• Sigmoid Similarity provides best results under noise• Optimal translation can be found• Approximate solutions with provable performance
bounds
MDDS ‘02 27