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## 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) MDDS ‘02**Introduction**Problem:Discover similar trajectories of moving objects Examples: • Features extracted from video-clips • Animal Mobility Experiments (GPS data) • Sign Language Recognition, etc.**Applications & Requirements**Clustering Classification What do we need? • Similarity Measure (robust to noise) • Indexing Scheme MDDS ‘02**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 MDDS ‘02**Related Work – Euclidean Distance**• Disadvantages • Small Robustness to outliers • Sensitive to time axis displacement • Does not support variable lengths • Lp–Norm: LP=(Σ(xi-yi)p)1/p • L2: Euclidean Distance • L1: Manhattan Distance MDDS ‘02**Related Work – DTW**• Time Warping • Allows stretching in time axis • Difficult Indexing • Disadvantages • Computationally intensive, O(n*m) • Has to match ALL elements MDDS ‘02**Requirements for new Similarity Model (1)**We need to address the following issues: • Different Sampling Rates or Different Speeds MDDS ‘02**Requirements for new Similarity Model (2)**We need to address the following issues: • Similar Motions in different space Regions MDDS ‘02**Requirements for new Similarity Model (3)**We need to address the following issues: • Outliers Non Recoverable Part Noise Everywhere Random Peaks • Different Lengths MDDS ‘02**Longest Common Subsequence (LCSS)**• 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 ] MDDS ‘02**Extending LCSS (1)**We extend the LCSS to 2-dimensions and add more flexibility: Similarity of 2 seq/s with length n & m: MDDS ‘02**Extending LCSS – Example**2ε • Rigid matching • Points marginally outside matching region are ignored • Set parameter epsilon MDDS ‘02**Extending LCSS – Flexible Matching**MDDS ‘02**Sigmoidal Matching**MDDS ‘02**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 MDDS ‘02**f(B)**c d 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 MDDS ‘02**6**6 5 5 4 4 3 3 2 2 1 1 Exact Algorithm for similarity function S2 For trajectories A, B with length n we want to find: • translation fc,dthat 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 MDDS ‘02**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 βof max(Sim) we can afford to lose βn elements • Don’t take all translations we can examine every βntranslations each time • So, if we examine every βn, we lose at mostβnelements (1D) • So, for 2D, we can skip every βn/2 translations MDDS ‘02**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 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. MDDS ‘02**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. MDDS ‘02**Clustering Accuracy**Datasets: • MobileLong • MobileShort • MobileShort + Noise Test clustering accuracy using Hierarchical Clustering C1 C2 C3 C4 C5 MDDS ‘02**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 MDDS ‘02**Clustering Accuracy (MobileLong)**• Number of Correct Clusterings out of 10 MDDS ‘02**Clustering Accuracy (MobileShort)**• Number of Correct Clusterings out of 21 MDDS ‘02**Clustering Accuracy (MobileShort + Noise)**• Number of Correct Clusterings out of 21 MDDS ‘02**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 MDDS ‘02