The presentation deals with the clustering of trajectories of moving objects. A k-means-like algorithm based on a Euclidean distance between piece-wise linear curves is used. The main novelty of the paper is the opportunity of considering in the clustering procedure a step that automatically weights the importance of sub-trajectories of the original ones. The algorithm uses an adaptive distances approach and a cluster-wise weighting. The proposed algorithm is tested against some workbench trajectory datasets. Presented at SIS 2019, Milan.

1. Trajectory clustering using adaptive Euclidean
distances
Antonio Irpino, PhD
19/6/2019
Sis 2019
Dept. of Mathematics and Physics
Caserta, Italy
19/6/2019 Sis 2019 1 / 47

2. Introduction

3. Outline
• We aim at clustering trajectories of
moving objects.
• A k-means-like algorithm based on a
Euclidean distance between piece-wise
linear curves is used. Each trajectory is
decomposed into sub-trajectories.
• The importance of each sub-trajectory is
automatically computed in the clustering
algorithm using an adaptive distances
approach.
• The proposed algorithm is tested against
some workbench trajectory datasets
Some trajectories
Examples of clustered
trajectories
19/6/2019 Sis 2019 2 / 47

4. Trajectory
A trajectory 𝑃𝑖 is a collection of ordered pairs of data (s𝑖
𝑗, 𝑡𝑖
𝑗),
𝑗 = 1, … , 𝑇, sampled in 𝑇 time-points where s𝑖
𝑗 is a spatial location
(namely. a 2D or a 3D vector of spatial coordinates) and 𝑡𝑖
𝑗 is a
time-stamp. A trajectory can be enriched with other data recorded at
each time-point, but we don’t consider this case. Considering the order
provided by the time-stamps, a trajectory 𝑃 is described as a curve in a
2D (or 3D space).
Trajectories are everywhere
Trajectories of
• pedestrians
• animals
• vehicles
• hurricanes
• …
Sensed by:
• GPS systems
• GSM
networks
• RFID and
WiFi
• …
Clustering and classification are
useful applications in
• Transportation
• Urban planning
• Business
• …
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5. Clustering trajectories
Clustering aims at grouping objects such that
• similar objects are grouped together
• different objects belongs to different groups
Trajectories clustering looks for groups of trajectories, or of
sub-trajectories, such that they represent a movement pattern in the data.
When are two trajectories similar?
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6. Different approaches to trajectory clustering
In the literature, the following two approaches represent the state of the
art of trajectory clustering:
• Lee et al. [6] propose a distance between sub-trajectories, and an
algorithm implements an extension of a density based clustering for
grouping set of sub-trajectories.
• Ferreira et al. [4] estimate 𝑘 vector fields associated with 𝑘 groups of
trajectories observed in a 2D space. This application, is inspired by
the problem of monitoring and predicting storm or hurricane paths.
• Another approach is provided by functional data analysis where a
trajectory is considered as a curve in a 2D or 3D space. Sangalli et
al. [8] proposed a k-means type algorithm using an alignment step.1
1We do not consider alignment in this paper
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7. What if trajectories have different time-length? Two choices.
Consider sub-trajectories:
sub-trajectories of equal lengths can be
selected and then compared
• Pro: time length is preserved
• Cons: computational cost can be high
Normalize lengths
time lengths are set equal to 1
• Pros: trajectories are considered as a
single objects, distances are more
interpretable, computational cost is
acceptable. Averaging trajectories is
possible.
• Cons: if trajectories have very different
time-lengths some biases arise
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8. K-means of trajectories with
adaptive distances

9. Distances between trajectories
Some assumptions
• We consider normalized trajectories.
• We consider trajectories as piece-wise linear curves.
• For each piece, we assume a constant relative speed.
Under these assumptions, we can consider the Euclidean distance between
two 2D trajectories2
having the same 𝑘 time-stamps normalized in [0, 1].
Given two normalized trajectories
𝑃1 = {{(𝑥1
0, 𝑦1
0), 0}, … , {(𝑥1
𝑗 , 𝑦1
𝑗 ), 𝜏1
𝑗 }, … , {(𝑥1
𝑇1
, 𝑦1
𝑇1
), 1}} and
𝑃2 = {{(𝑥2
0, 𝑦2
0), 0}, … , {(𝑥2
𝑗 , 𝑦2
𝑗 ), 𝜏2
𝑗 }, … , {(𝑥2
𝑇2
, 𝑦2
𝑇2
), 1}},
where 𝜏 𝑖
𝑗 =
𝑡 𝑖
𝑗−𝑡 𝑖
0
𝑡 𝑖
𝑇 𝑖
−𝑡 𝑖
0
2The trajectory is on a plane, but the extension to 3D spaces is straightforward.
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10. Euclidean distance between two trajectories i
It is possible to express the two trajectories with a common set of 𝜏’s by
a linear interpolation. Once the two trajectories are registered such that
they have the same normalized 𝐿 ∈ [𝑚𝑖𝑛(𝑇1, 𝑇2), (𝑇1 + 𝑇2)] time-stamps
we compute the squared Euclidean distance between 𝑃1 and 𝑃2 as
follows:
𝑑2
𝐸 (𝑃1, 𝑃2) =
1
∫
0
[(𝑥1(𝜏) − 𝑥2(𝜏))
2
+ (𝑦1(𝜏) − 𝑦2(𝜏))
2
]𝑑𝜏 =
=
𝐿
∑
ℓ=1
(𝜏ℓ − 𝜏ℓ−1) {
| ̄𝑥1(ℓ) − ̄𝑥2(ℓ)|
2
+ | ̄𝑦1(ℓ) − ̄𝑦2(ℓ)|
2
+
+1
3 [| ̇𝑥1(ℓ) − ̇𝑥2(ℓ)|
2
+ | ̇𝑦1(ℓ) − ̇𝑦2(ℓ)|
2
]
}
(1)
where:
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11. Euclidean distance between two trajectories ii
• ̄𝑥1(ℓ) = 𝑥1(𝜏ℓ)+𝑥1(𝜏ℓ−1)
2 , ̄𝑥2(ℓ) = 𝑥2(𝜏ℓ)+𝑥2(𝜏ℓ−1)
2 ,
̄𝑦1(ℓ) = 𝑦1(𝜏ℓ)+𝑦1(𝜏ℓ−1)
2 , and ̄𝑦2(ℓ) = 𝑦2(𝜏ℓ)+𝑦2(𝜏ℓ−1)
2 . The points
( ̄𝑥1(ℓ), ̄𝑦1(ℓ)) and ( ̄𝑥2(ℓ), ̄𝑦2(ℓ)) are, respectively, the centers of the
segment that starts from (𝑥1(𝜏ℓ−1), 𝑦1(𝜏ℓ−1)) and arrives at
(𝑥1(𝜏ℓ) , 𝑦1(𝜏ℓ)), respectively, the centers of the segment that starts
from (𝑥2(𝜏ℓ−1) , 𝑦2(𝜏ℓ−1)) and arrives at (𝑥2(𝜏ℓ), 𝑦2(𝜏ℓ));
• ̇𝑥1(ℓ) = 𝑥1(𝜏ℓ)−𝑥1(𝜏ℓ−1)
2 , ̇𝑥2(ℓ) = 𝑥2(𝜏ℓ)−𝑥2(𝜏ℓ−1)
2 ,
̇𝑦1(ℓ) = 𝑦1(𝜏ℓ)−𝑦1(𝜏ℓ−1)
2 , and ̇𝑦2(ℓ) = 𝑦2(𝜏ℓ)−𝑦2(𝜏ℓ−1)
2 . The value
( ̇𝑥1(ℓ), ̇𝑦1(ℓ)) and ( ̇𝑥2(ℓ) , ̇𝑦2(ℓ)) are, respectively, the pairs of the
signed component-wise half widths of the segment that starts from
(𝑥1(𝜏ℓ−1) , 𝑦1(𝜏ℓ−1)) and arrives at (𝑥1(𝜏ℓ) , 𝑦1(𝜏ℓ)), respectively, of
the segment that starts from (𝑥2(𝜏ℓ−1) , 𝑦2(𝜏ℓ−1)) and arrives at
(𝑥2(𝜏ℓ) , 𝑦2(𝜏ℓ)).
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12. A k-means algorithm
The (squared) Euclidean distance allows the Fréchet means of a set of
trajectories, thus a k-means algorithm can be applied.
Can we compare different sub-trajectories?
• In several scenarios, if could be useful to consider how mobile-objects
enter in place, how they exit and their intermediate paths.
• We propose to extend the k-means algorithm by considering a
trajectory as a sequence of 𝑀 common (w.r.t. the normalized time)
sub-trajectories which is a partition of the original one.
• While k-means of 𝑁 trajectories is like to do a k-means on one
(functional) variable, k-means of 𝑁 trajectories on 𝑀
sub-trajectories is like to do a k-means on 𝑀 (functional) variables.
– If the 𝑀 sub-trajectories have the same time-length, k-means of
trajectories and on sub-trajectories return the same results! So,
whats’ new!
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13. Using adaptive distances
The hypothesis is that each sub-trajectory could have a different
importance in the clustering process.
Adaptive distances (or weighted distances)[3]
• We can consider a system of weights for each sub-trajectory
reflecting the importance in the clustering process.
• Using adaptive distances in a k-means algorithm as suggested in [3],
we extended the k-means algorithm for trajectories such that a
system of weights is the solution of the minimization of the criterion
function (or the cost function) of the k-means algorithm.
• We propose a global weighting system (a weight for each
sub-trajectory) and a cluster-wise (a weight for each cluster) one.
Note that, weights are computed by the algorithm and not provided
by the user.
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14. The optimized criteria
Method Criterion
K-means 𝑊 𝐾𝑚 =
𝐾
∑
𝑘=1
∑
𝑖∈𝑘
𝑑2
𝐸(𝑃𝑖, ̄𝑃 𝑘)
SK-means 𝑊 𝑆𝐾𝑚 =
𝐾
∑
𝑘=1
𝑀
∑
𝑚=1
∑
𝑖∈𝑘
𝑑2
𝐸(𝑃𝑖,𝑚, ̄𝑃 𝑘,𝑚)
SKADAG (Glob.
W.)
𝑊 𝐴𝐺 =
𝐾
∑
𝑘=1
𝑀
∑
𝑚=1
∑
𝑖∈𝑘
𝜆 𝑚 ⋅ 𝑑2
𝐸(𝑃𝑖,𝑚, ̄𝑃 𝑘,𝑚) s. a
𝑀
∏
𝑚=1
𝜆 𝑚 = 1
SKADAL (Cl.-wise
W.)
𝑊 𝐴𝐿
𝐾
∑
𝑘=1
𝑀
∑
𝑚=1
∑
𝑖∈𝑘
𝜆 𝑘,𝑚 ⋅ 𝑑2
𝐸(𝑃𝑖,𝑚, ̄𝑃 𝑘,𝑚) s. a
𝑀
∏
𝑚=1
𝜆 𝑘,𝑚 = 1 ∀𝑘 ∈ 1, … , 𝐾
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15. The algorithm: Initialization
0. Input: A dataset 𝑃 of normalized and registered trajectories cut at some
predefined 𝑀 normalized time-stamps. A predefined 𝐾 number of
clusters.
1. Initialization Set 𝑡 = 0
1.1 Centers selection Select randomly 𝐾 trajectories and store
them in 𝐺(0)
1.2 Fix initial weights Fix Λ(0) = 1
1.3 Assign Assign data to clusters according to a minimum
distance criterion, and generate the initial partition
of trajectories 𝒫(0)
1.4 Compute initial criterion Compute 𝑊
(0)
𝐴𝐺 (SKADAG) or 𝑊
(0)
𝐴𝐿
(SKADAL)
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16. The algorithm: Iterative optimization
2. Repeat Set 𝑡 = 𝑡 + 1
2.1 Centers selection Fixed 𝒫(𝑡−1) and Λ(𝑡−1), compute the average
trajectories for each cluster and store them in 𝐺(𝑡)
2.2 Compute weights Fixed 𝒫(𝑡−1) and 𝐺(𝑡), compute Λ(𝑡)
according to the constrained minimization of 𝑊
(𝑡)
𝐴𝐺
(SKADAG) or 𝑊
(𝑡)
𝐴𝐿 (SKADAL), using the
Lagrange multiplier method.
2.3 Assign Fixed 𝐺(𝑡) and Λ(𝑡), assign trajectories to clusters
according a minimum distance criterion w.r.t. the
average trajectories, and store the partition of
trajectories in 𝒫(𝑡).
2.4 Compute the new criterion Compute 𝑊
(𝑡)
𝐴𝐺 (SKADAG) or
𝑊
(𝑡)
𝐴𝐿 (SKADAL) .
2.5 Verify the stopping rule If 𝑊
(𝑡)
𝐴𝐺 < 𝑊
(𝑡−1)
𝐴𝐺 (SKADAG) or
𝑊
(𝑡)
𝐴𝐿 < 𝑊
(𝑡)
𝐴𝐿 (SKADAL) then go to 2. else go to
3..
3. Return solution Return 𝒫(𝑡), 𝐺(𝑡), Λ(𝑡).
19/6/2019 Sis 2019 14 / 47

17. Experiments and results

18. Two datasets
We apply the algorithm to two datasets3
CROSS (road) dataset
CROSS is a dataset of 1, 900 trajectories of
vehicles approaching to a crossroad. The
trajectories are labeled into 19 different
types.
LABOMNI
It is a dataset describing 15 (𝐾) sets of
trajectories of 209 people in a laboratory.
3available from http://cvrr.ucsd.edu/LISA/Datasets/TrajectoryClustering/CVRR_d
ataset_trajectory_clustering.zip
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19. Main results: external validity indices
Table 2: CROSS and LABOMNI datasets: ARI= Adjusted Rand Index,
PUR=Purity, NMI=Normalized Mutual Information
CROSS LABOMNI
N=1,900 k=19 N=209 K=15
Methods ARI PUR NMI ARI PUR NMI
K-means 0.8163 0.8389 0.9405 0.6715 0.8373 0.8248
cuts (0.15,0.85) cuts (0.005, 0.15,0.85,0.995)
K-means pieces 0.8210 0.8411 0.9443 0.8772 0.9234 0.9118
SKADAG 0.8192 0.8400 0.9426 0.8930 0.9330 0.9230
SKADAL 0.8200 0.8405 0.9433 0.8273 0.9139 0.8998
Note: algorithms based on sub-trajectories perform slightly better for CROSS and
significantly better for the LABOMNI. According to [7], we note that CROSS has less
complex trajectories than LABOMNI. Indeed, CROSS contains more regular
trajectories, since they show cars behaviors at a crossroad, while LABOMNI consists in
trajectories of people that walk almost freely in a laboratory.
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20. LABOMNI clustering results i
In the following slides, we focus on LABOMNI dataset.
• For each ground-truth class (top-left), we show the closest cluster
(bottom-left).
• On the right, it is reported the respective Variance function, namely
𝑉 𝑎𝑟 𝑘(𝑝) =
1
𝑁 𝑘
∑
𝑖∈𝐶 𝑘
[𝑃𝑖(𝑝) − ̄𝑃𝑖(𝑝)]
2
For comparing results, we plot the square root of the function.
• We report the log of relevance weights (that sum to zero because of
the log transformation)
19/6/2019 Sis 2019 17 / 47

21. SKADAG Traj 1 –> Old cla 1 (8) Newcla 13 (9)
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Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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22. SKADAG Traj 2 –> Old cla 2 (25) Newcla 15 (20)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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23. SKADAG Traj 3 –> Old cla 3 (8) Newcla 4 (11)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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24. SKADAG Traj 5 –> Old cla 5 (30) Newcla 9 (29)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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25. SKADAG Traj 6 –> Old cla 6 (36) Newcla 2 (34)
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Vark(p)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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26. SKADAG Traj 7 –> Old cla 7 (28) Newcla 5 (27)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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27. SKADAG Traj 9 –> Old cla 9 (11) Newcla 7 (12)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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28. SKADAG Traj 10 –> Old cla 10 (4) Newcla 10 (8)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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29. SKADAG Traj 11 –> Old cla 11 (20) Newcla 6 (19)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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30. SKADAG Traj 13 –> Old cla 13 (4) Newcla 10 (8)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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31. SKADAG Traj 14 –> Old cla 14 (22) Newcla 3 (22)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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32. SKADAG Traj 15 –> Old cla 15 (4) Newcla 12 (4)
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log(𝑤) -0.18 -0.63 -1.42 1.11 1.12
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33. SKADAL Algorithm
Let’s see the SKADAL (cluster-wise adaptive distances)
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34. SKADAL Traj 1 –> Old cla 1 (8) Newcla 1 (11)
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0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) 2.39 2.04 -1.84 -1.12 -1.47
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35. SKADAL Traj 2 –> Old cla 2 (25) Newcla 9 (20)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) -0.01 -2.23 -0.51 1.45 1.30
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36. SKADAL Traj 3 –> Old cla 3 (8) Newcla 5 (11)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) -0.26 -1.11 -2.29 2.05 1.60
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37. SKADAL Traj 5 –> Old cla 5 (30) Newcla 3 (25)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) 0.76 0.21 -1.93 0.58 0.38
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38. SKADAL Traj 6 –> Old cla 6 (36) Newcla 10 (12)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) -3.30 -3.15 -1.60 3.70 4.36
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39. SKADAL Traj 6 –> Old cla 6 (36) Newcla 15 (31)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) 1.59 1.54 -1.88 -0.94 -0.32
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40. SKADAL Traj 7 –> Old cla 7 (28) Newcla 2 (22)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) -1.57 -1.47 -1.41 2.20 2.25
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41. SKADAL Traj 7 –> Old cla 7 (28) Newcla 13 (12)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) 0.52 -1.20 -1.84 1.40 1.12
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42. SKADAL Traj 9 –> Old cla 9 (11) Newcla 2 (22)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) -1.57 -1.47 -1.41 2.20 2.25
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43. SKADAL Traj 11 –> Old cla 11 (20) Newcla 6 (23)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) 2.99 3.47 -1.66 -2.37 -2.43
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44. SKADAL Traj 14 –> Old cla 14 (22) Newcla 7 (22)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) -0.65 -0.64 -1.54 1.92 0.91
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45. SKADAL Traj 15 –> Old cla 15 (4) Newcla 8 (7)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
50
100
150
200
100 150 200 250
x
y
0
20
40
60
0.00 0.25 0.50 0.75 1.00
p
Vark(p)
0-0.005 0.005-0.15 0.15-0.85 0.85-0.995 0.995-1
log(𝑤) -2.11 -1.78 -0.30 2.22 1.96
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46. Comments on LABOMNI
• We see that some clusters are strange! This is due to misaligned
trajectories.
• While alignment is solvable in a hierarchical clustering approach (for
example, distances matrices can be computed using warping), in a
k-mean approach is less feasible.
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47. Conclusions
• In this work
– We presented a new k-means clustering algorithm for trajectories
– We observed that using sub-trajectories improves clustering results
• In perspective
– How many cuts and where to cut?
– A possible solution is to cut everywhere! I.e., we are developing and
testing a new algorithm where the 𝜆 are continuous.
– Alignment of trajectories in a k-means framework. Recently, some
proposal have been introduced for time-series.
– Relaxing hypothesis without losing interpretability and performances.
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48. References i
[1] Demšar, U., Buchin, K., Cagnacci, F., Safi, K., Speckmann, B.,
Weghe, N.V., Weiskopf, D., Weibel, R.: Analysis and visualisation of
movement: an interdisciplinary review. Movement ecology. 3:5,
(2015)
[2] Diday, E.: The dynamic clusters method in nonhierarchical
clustering. International Journal of Computer and Information
Sciences 2: 61 (1973) doi: 10.1007/BF00987153
[3] Diday, E. and Govaert, G.: Classification Automatique avec
Distances Adaptatives. R.A.I.R.O. Informatique Computer Science,
11 (4), 329-349 (1977)
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49. References ii
[4] Ferreira, N. , Klosowski, J. T., Scheidegger, C. E. and Silva, C. T.:
Vector Field k‐Means: Clustering Trajectories by Fitting Multiple
Vector Fields. Computer Graphics Forum, 32: 201-210. (2013) doi:
10.1111/cgf.12107
[5] Jiang Bian, Dayong Tian, Yuanyan Tang, Dacheng Tao. A review of
moving object trajectory clustering algorithms. Artif Intell Rev
(2016) doi: 10.1007/s10462-016-9477-7
[6] Lee, J., Han, J., Whang, K.: Trajectory clustering: a
partition-and-group framework. Proceedings of the 2007 ACM
SIGMOD international conference on Management of data, pp.
593-604 (2007)
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50. References iii
[7] Morris, B. T., Trivedi, M. M.:Learning Trajectory Patterns by
Clustering: Experimental Studies and Comparative Evaluation, in
Proc. IEEE Inter. Conf. on Computer Vision and Pattern Recog.,
Maimi, Florida, (2009)
[8] Sangalli, L.M., Secchi, P., Vantini, S., Vitelli, V.: K-mean alignment
for curve clustering, Computational Statistics & Data Analysis, 54,5,
1219–1233 (2010)
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51. Thank you for your attention.
Any questions?