robust similarity measures for mobile object trajectories
DESCRIPTION
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 PresentationTRANSCRIPT
![Page 1: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/1.jpg)
MDDS ‘02 1
Robust Similarity Measures for Mobile Object Trajectories
Michalis Vlachos (UCR), Dimitrios Gunopulos (UCR), George
Kollios (BU)
![Page 2: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/2.jpg)
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.
![Page 3: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/3.jpg)
MDDS ‘02 3
Applications & Requirements
What do we need?What do we need?• Similarity Measure (robust to
noise)• Indexing Scheme
ClassificationClustering
![Page 4: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/4.jpg)
MDDS ‘02 4
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
![Page 5: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/5.jpg)
MDDS ‘02 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
![Page 6: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/6.jpg)
MDDS ‘02 6
Related Work – DTW
• Time Warping
– Allows stretching in time axis
– Difficult Indexing
• Disadvantages
– Computationally intensive, O(n*m)
– Has to match ALL elements
![Page 7: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/7.jpg)
MDDS ‘02 7
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
![Page 8: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/8.jpg)
MDDS ‘02 8
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
![Page 9: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/9.jpg)
MDDS ‘02 9
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
![Page 10: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/10.jpg)
MDDS ‘02 10
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 ]
![Page 11: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/11.jpg)
MDDS ‘02 11
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:
![Page 12: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/12.jpg)
MDDS ‘02 12
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
![Page 13: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/13.jpg)
MDDS ‘02 13
Extending LCSS – Flexible Matching
![Page 14: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/14.jpg)
MDDS ‘02 14
Sigmoidal Matching
![Page 15: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/15.jpg)
MDDS ‘02 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
![Page 16: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/16.jpg)
MDDS ‘02 16
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:
![Page 17: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/17.jpg)
MDDS ‘02 17
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)
![Page 18: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/18.jpg)
MDDS ‘02 18
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?
![Page 19: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/19.jpg)
MDDS ‘02 19
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.
![Page 20: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/20.jpg)
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.
![Page 21: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/21.jpg)
MDDS ‘02 21
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
![Page 22: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/22.jpg)
MDDS ‘02 22
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
![Page 23: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/23.jpg)
MDDS ‘02 23
Clustering Accuracy (MobileLong)
• Number of Correct Clusterings out of 10• Number of Correct Clusterings out of 10
![Page 24: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/24.jpg)
MDDS ‘02 24
Clustering Accuracy (MobileShort)
• Number of Correct Clusterings out of 21• Number of Correct Clusterings out of 21
![Page 25: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/25.jpg)
MDDS ‘02 25
Clustering Accuracy (MobileShort + Noise)
• Number of Correct Clusterings out of 21• Number of Correct Clusterings out of 21
![Page 26: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/26.jpg)
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
![Page 27: Robust Similarity Measures for Mobile Object Trajectories](https://reader036.vdocument.in/reader036/viewer/2022062309/56814388550346895db004eb/html5/thumbnails/27.jpg)
MDDS ‘02 27