1. Clustering Trajectories of Moving Objects in an Uncertain World
Nikos Pelekis1, Ioannis Kopanakis2, Evangelos E. Kotsifakos1,
Elias Frentzos1, Yannis Theodoridis1
1 Dept. of Informatics,
Univ. of Piraeus, Greece
2 Tech. Educational Institute of Crete, Greece
IEEE International Conference on Data Mining (ICDM 2009), Miami, FL, USA, 6-9 December, 2009

2. Outline Related work Motivation Our contribution Experimental study
From Trajectories to Intuitionistic Fuzzy Sets
A similarity metric for Uncertain Trajectories (Un-Tra)
Cen-Tra: The Centroid Trajectory of a bunch of trajectories
TR-I-FCM: A novel clustering algorithm for Un-Tra
Experimental study
Conclusions & future work
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

3. Related Work on Mobility Data Mining
Trajectory clustering

4. Trajectory Clustering
Questions:
Which distance between trajectories?
Which kind of clustering?
What is a cluster ‘mean’ or ‘centroid’?
A representative trajectory?
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

5. Which distance? ò Average Euclidean distance
“Synchronized” behaviour distance
Similar objects = almost always in the same place at the same time
Computed on the whole trajectory
Computational aspects:
Cost = O( |1| + |2| ) (|| = number of points in )
It is a metric => efficient indexing methods allowed, e.g. [Frentzos et al. 2007]
Timeseries-based approaches: LCSS, DTW, ERP, EDR
Trajectory-oriented approach:
(time-relaxed) route similarity vs. (time-aware) trajectory similarity and variations (speed-pattern based similarity; directional similarity; …) [Pelekis et al. 2007] | )) ( ), ) , 2 1 T dt t d D ò =
distance between moving objects 1 and 2 at time t
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World" 5

6. Which kind of clustering?
K-means
HAC-average
Reachability plot
(= objects reordering for distance distribution)
T-OPTICS [Nanni & Pedreschi,
2006]
 threshold
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World" 6

7. TRACLUS: A Partition-and-Group Framework
[Lee et al. 2007]
Discovers similar portions of trajectories (sub-trajectories)
Two phases: partitioning and grouping
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

8. What about usage of Mobility Patterns?
Visual analytics for mobility data

9. Visual analytics for mobility data
[Andrienko et al. 2007]
What is an appropriate way to visualize groups of trajectories?
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

10. Summarizing a bunch of trajectories
1) Trajectories  sequences of “moves” between “places”
2) For each pair of “places”, compute the number of “moves”
3) Represent “moves” by arrows (with proportional widths)
Many small moves
Major flow
Minor variations
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

11. A word on uncertainty

12. Handling Uncertainty Handling uncertainty is a relatively new topic!
A lot of research effort has been assigned
Developing models for representing uncertainty in trajectories. The most popular one [Trajcevski et al. 2004]:
a trajectory of an object is modeled as a 3D cylindrical volume around the tracked trajectory (polyline)
Various degrees of uncertainty
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

13. Coming back to our approach

14. Motivation
Challenge 1: Introduce trajectory fuzziness in spatial clustering techniques
The application of spatial clustering algorithms (k-means, BIRCH, DBSCAN, STING, …) to Trajectory Databases (TD) is not straightforward
Fuzzy clustering algorithms (Fuzzy C-Means and its variants) quantify the degree of membership of each data vector to a cluster
The inherent uncertainty in TD should taken into account.
Challenge 2: study the nature of the centroid / mean / representative trajectory in a cluster of trajectories.
Is it a ‘trajectory’ itself?
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

15. Our contribution
I-Un-Tra: An intuitionistic fuzzy vector representation of trajectories
enables clustering of trajectories by existing (fuzzy or not) clustering algorithms
DUnTra: A distance metric of uncertain trajectories
Cen-Tra: The centroid of a bunch of trajectories
using density and local similarity properties
TR-I-FCM: A novel modification of FCM algorithm for clustering complex trajectory datasets
exploiting on DUnTra and Cen-Tra.
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

16. From Fuzzy sets to Intuitionistic fuzzy sets
Definition 1 (Zadeh, 1965). Let a set E be fixed. A fuzzy set on E is an object of the form
Definition 2 (Atanassov, 1986; Atanassov, 1994). An intuitionistic fuzzy set (IFS) A is an object of the form
where
where
and
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

17. Hesitancy For every element
The hesitancy of the element x to the set A is
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

18. Vector representation of trajectories
Assume a regular grid G(m  n) consisting of cells ck,l , a trajectory
and a target dimension p << ni,
The “approximate trajectory”
consists of p regions (i.e. sets of cells) crossed by Ti during period pj
The “Uncertain Trajectory” is the ε-buffer of
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

19. Intuitionistic Uncertain Trajectories
membership = inside cell with 100% probability (i.e. thick portions)
non-membership = outside cell with 100% probability (i.e. dotted portions)
hesitancy = ignorance whether inside or outside the cell (i.e. solid thin portions)
A cell ck.l
ck.l ε
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

20. Intuitionistic Uncertain Trajectories
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

21. Proposed similarity metric (1/2)
The distance between two I-UnTra A and B is:
where
and
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

22. Proposed similarity metric (2/2)
Assuming two intuitionistic fuzzy sets on it, A = (MA, ΓA, ΠA) and B = (MB, ΓB, ΠΒ), with the same cardinality n, the similarity measure Z between A and B is given by the following equation:
where z(A’,B’) for fuzzy sets A' and B' (e.g. for MA, MB) is defined as:
and similarly for ΓA, ΓB and ΠA, ΠB.
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

23. C is more similar to A than B
An example
A={x, 0.4, 0.2}, B={x, 0.5, 0.3}, C={x, 0.5, 0.2}
C is more similar to A than B
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

24. The Centroid Trajectory
The idea (similarity-density-based approach):
adopt some local similarity function to identify common sub-trajectories (concurrent existence in space-time),
follow a region growing approach according to density
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

25. Algorithm CenTra: An example
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

26. The Centroid Trajectory
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

27. Fuzzy C-Means algorithm
The FCM objective function:
Given that to be minimized requires:
and
The FCM algorithm minimizes intra-cluster variance, but shares the same problems as k-means.
It does not ensure that it converges to an optimal solution, while the identified minimum is local and the results depend on the initial choice of centroids.
FCM tries to partition the dataset by just looking at the data vectors and as such it ignores the fact that these vectors may be accompanied by qualitative information which may be given per feature (i.e. dimension)!
1. Determine c (1 < c < N), and initialize V(0), j=1,
2. Calculate the membership matrix U(j),
3. Update the centroids’ matrix V(j),
4. If |U(j+1)-U(j)|>ε then j=j+1 and go to Step 2.
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

28. Ignore update centroid step
CenTR-I-FCM algorithm
The FCM objective function:
Given that to be minimized requires :
and
Ignore update centroid step
and instead use CenTra
1. V(0) = c random I-UnTra; j=1;
2. repeat
3. Calculate membership matrix U(j)
4. Update the centroids’ matrix V(j) using CenTra;
5. Compute membership and non-membership degrees of V(j)
6. Until ||Uj+1-Uj||F≤ε; j=j+1;
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

29. Experiments (1/2) Dataset: ’Athens trucks’ MOD (www.rtreeportal.org)
50 trucks, 1100 trajectories, position records
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

30. “Round trips” clusters
Experiments (2/2)
Use CommonGIS [Andrienko et al., 2007] to identify real clusters
“Round trips” clusters
“Linear” clusters
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

31. Results (Clustering accuracy scaling cell size, ε )
Fix density threshold to δ=2% of the total number of trajectories
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

32. Results (Clustering accuracy scaling density threshold, δ)
Fix uncertainty to ε=1
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

33. Results (scaling the number of clusters)
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

34. Results (scaling the dataset cardinality)
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

35. Results (Quality of CenTra)
Representative Trajectories vs. Centroid Trajectories
cell size=1.3%, ε=0, δ=0.09
cell size=1.3%, ε=0, δ=0.09, cell size=2.8%, ε=0, δ=0.02
Figure 6(a) illustrates the outcome of TRACLUS. Evidently, the cluster representative (red line) does not fit the real movement, mainly due to its averaging technique. Recall at this point that TRACLUS clusters segments rather than whole trajectories (even considering this, the algorithm does not compass the turn occurring at the bottom of the figure). On the other hand, Figure 6(b) and Figure 6(c) illustrate CenTra, produced with variable cell size, ε and density δ. It turns out that CenTra not only resides on the data traces, but also vanishes the non-interesting movement details (the ‘noisy’ infrequent parts are not part of the centroid), it catches turns, and it becomes thicker in portions where something interesting (i.e. dense-similar subtrajectories) happens.
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

36. Conclusions
We proposed a three-step approach for clustering trajectories of moving objects, motivated by the observation that clustering and representation issues in TD are inherently subject to uncertainty.
1st step: an intuitionistic fuzzy vector representation of trajectories plus a distance metric consisting of
a metric for sequences of regions and
a metric for intuitionistic fuzzy sets
2nd step: Algorithm CenTra, a novel technique for discovering the centroid of a bundle of trajectories
3rd step: Algorithm CenTR-I-FCM, for clustering trajectories under uncertainty
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

37. Future Work
Devise a clever sampling technique for multi-dimensional data so as to diminish the effect of initialization in the algorithm;
Exploit the metric properties of the proposed distance function by using an distance-based index structure (for efficiency purposes);
Perform extensive experimental evaluation using large trajectory datasets
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

38. Acknowledgements
Research partially supported by the FP7 ICT/FET Project MODAP (Mobility, Data Mining, and Privacy) funded by the European Union. URL:
a continuation of the FP IST/FET Project GeoPKDD (Geographic Privacy-aware Knowledge Discovery and Delivery) funded by the European Union. URL:
Some slides are from:
Fosca Giannotti, Dino Pedreschi, and Yannis Theodoridis, “Geographic Privacy-aware Knowledge Discovery and Delivery”, EDBT Tutorial, 2009.
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

39. Back up slides

40. Examples of mobility patterns exploitation
Trajectory Density-based queries
Find hot-spots (popular places) [Giannotti et al. 2007]
Find T-Patterns [Giannotti et al. 2007]
Find hot motion paths [Sacharidis et al. 2008]
Find typical trajectories [Lee et al. 2007]
Identify flocks & leaders [Benkert et al. 2008] δt ε X Y T
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

41. Which kind of clustering?
General requirements:
Non-spherical clusters should be allowed
E.g.: A traffic jam along a road = “snake-shaped” cluster
Tolerance to noise
Low computational cost
Applicability to complex, possibly non-vectorial data
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World" 41

42. Temporal focusing
Different time intervals can show different behaviours
E.g.: objects that are close to each other within a time interval can be much distant in other periods of time
The time interval becomes a parameter
E.g.: rush hours vs. low traffic times
Already supported by the distance measure
Just compute D(1 , 2) |T on a time interval T’  T
Problem: significant T’ are not always known a priori
An automated mechanism is needed to find them
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World" 42

43. TRACLUS – representative trajectory
The representative trajectory of the cluster:
Compute the average direction vector and rotate the axes temporarily .
Sort the starting and ending points by the coordinate of the rotated axis.
While scanning the starting and ending points in the sorted order, count the number of line segments and compute the average coordinate of those line segments.
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"

44. Trajectory Uncertainty vs. Anonymization
Never Walk Alone [Bonchi et al. 2008]
Trade uncertainty for anonymity: trajectories that are close up the uncertainty threshold are indistinguishable
Combine k-anonymity and perturbation
Two steps:
Cluster trajectories into groups of k similar ones (removing outliers)
Perturb trajectories in a cluster so that each one is close to each other up to the uncertainty threshold
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World" 44 44

45. Qualitative evaluation of Z
Pelekis et al. "Clustering Trajectories of Moving Objects in an Uncertain World"