Robust Similarity Measures for Mobile Object Trajectories

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Robust Similarity Measures for Mobile Object
Trajectories

• Michalis Vlachos (UCR), Dimitrios Gunopulos
  (UCR), George Kollios (BU)

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Introduction

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

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Applications Requirements
Classification
Clustering

• What do we need?
• Similarity Measure (robust to noise)
• Indexing Scheme

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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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Related Work Euclidean Distance

• LpNorm LP(S(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(nm)
• Has to match ALL elements

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Requirements for new Similarity Model (1)
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

• Similar Motions in different space Regions

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Requirements for new Similarity Model (3)
We need to address the following issues

• Outliers

Non Recoverable Part
Noise Everywhere
Random Peaks

• Different Lengths

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Longest Common Subsequence (LCSS)

• Dynamic Programming Solution

• Arithmetic Example
• t10, 4, 6, 8, 7, 4, 6, 5, 6, 4, 6
• t20, 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
Similarity of 2 seq/s with length n m
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Extending LCSS Example
2e

• Rigid matching
• Points marginally outside matching region are
  ignored
• Set parameter epsilon

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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
  An and Bm, we can find the SigmoidSimd(?,?)
  in O(d(nm)) time

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Extending LCSS (2)

• S1 cannot detect parallel movements,

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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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-jltd ai-e fc(bi) (bi, aje)

• Transform into a stabbing problem

Translations O(d2n2) LCSS O(dn) Total
O(d3n3)
yx2
yx
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Approximate Algorithm for similarity function S2

• A translation corresponds to a line fc(x) xc.
• Sort translations by c ? THEY DIFFER IN HOW MANY
  SEGMENTS?

• If we can afford to be within ß of max(Sim) ? we
  can afford to lose ßn elements

• Dont take all translations ? we can examine
  every ßn translations each time
• So, if we examine every ßn, we lose at most ßn
  elements (1D)
• So, for 2D, we can skip every ßn/2 translations

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Approximate Algorithm for similarity function S2
Theorem Given two trajectories A and B, with A
n and Bn, and a constant 0ltßlt1, we can find
an approximation AS2d,ß(A,B) of the similarity
S2(d,e,A,B) such that S2(d,e,A,B) - AS2d,ß(A,B) lt
ß in O(nd3/ ß2) time.
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Approximate Algorithm for similarity function S2
(cont/d)
Theorem Given two trajectories A and B, with A
n and Bn, and a constant 0ltßlt1, we can find
an approximation AS2d,ß(A,B) of the similarity
S2(e,A,B) such that S2(d,e,A,B) - AS2d,ß(A,B) lt
aß in O(nd3/ ß2) time, for a constant a.
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Clustering Accuracy

• Datasets
• MobileLong
• MobileShort
• MobileShort Noise

Test clustering accuracy using Hierarchical
Clustering
C1
C2
C3
C4
C5
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Clustering Accuracy

• LpNorm LP(S(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

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Clustering Accuracy (MobileShort)

• Number of Correct Clusterings out of 21

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Clustering Accuracy (MobileShort Noise)

• Number of Correct Clusterings out of 21

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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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