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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.

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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

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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

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