Travel Time Estimation of a Path using
Sparse Trajectories
Dr. Yu Zheng
[email protected]
Lead Researcher, Microsoft Research
Chair Professor at Shanghai Jiao Tong University
r1
r5
r2
r3
r6
r7
r4
Tr1
Tr2
Tr3
Tr4
Goal
• Estimate the travel time of any given path
– on road network instantly
– using historical and current trajectories generated by a
sample of vehicles
D
S
Challenges
• Data sparsity
• Trajectory concatenation
– Multiple ways to combine sub-trajectories
– Length of a sub-trajectory and its support
• Scalability and efficiency
– A citywide estimation
– E.g. Beijing has over 100,000 road segments
S
r1
r5
𝑟1 → 𝑟2 → 𝑟3 → 𝑟4
r2
r3
r6
r7
r4
D
Tr1
Tr2
Tr3
Tr4
𝑟1 + 𝑟2 + 𝑟3
(𝑟1 → 𝑟2 ) + 𝑟3
𝑟1 + (𝑟2 → 𝑟3 )
Methodology
• Framework
– Context-Aware Tensor Decomposition (CATD)
– Optimal Concatenation (OC)
MapMatching
Trajectories
Tensor
Decomposition
Tensor
Construction
Ar
Road
Networks
Trajectory
Database
Frequent Trajectory
Pattern Mining
Context Feature
Extraction
Features
Arec
Patterns
Optimal
Concatenation
Path
cost
Methodology
• Supplementing missing values
g1
g2
g3
g4
g5
g6
g7
g8
g9
R10
g11
g12
MG=
g13
g14
g15
g16
g1 g2 g3
tj
tk
t'j
t'k
g16
p2 0
21 0 0 1 0
6 17 9 0
14
tj
0
t1
t2
22 0
16 8
0
0
0
27
ti
tj
0 0 11 0
0 42 15 0
0 0 0 31
0
0
0
tn
0
r1 0
tj
0
0
35
0
p3
r4
0
0 42
g1 g2 g3
Ar
g1 g2
g16
p
t
M G= t i
i+1
g16
7
r2
Tr1
v1 v2
v3
r1 v4
v4
v6 r2
v5
r4
r3
0
r5
r4
Tr3
trk1: (v1, v2, v4) r2: (v4, v5, v6t')k r3: (v5, v7)
t'j
Y
r4: (v2, v3)
uM
r1
r2
u2
u1
rN
r1 r2
rN
r2
r3
Xh
Xr
r3
r1
d1
v5
v7
Tr2
r6
r1 r2
rN
f1 f2
fr
fq
fp
Methodology
• Supplementing missing values
Ah
X
tj
tk
t'j
t'k
g1 g2
g16
Xh
Ar
t'k
t'j
Y
tk
tj
uM
Xr
r1
r2
𝜆3
2
2
+ 𝑅
fp
2
+ 𝑈
A = Ar || Ah
rN
1
𝒜 − 𝑆 ×𝑅 𝑅 ×𝑈 𝑈 × 𝑇 𝑇
2
𝑆
fr
fq
rN
u2
u1
r1 r2
ℒ 𝑆, 𝑅, 𝑈, 𝑇, 𝐹, 𝐺 =
f1 f2
2
+ 𝑇
2
+
2
𝜆1
𝑋 − 𝑇𝐺
2
+ 𝐹
2
+ 𝐺
2
+
2
𝜆2
𝑌 − 𝑅𝐹
2
2
+
Methodology
• Framework
MapMatching
Trajectories
Tensor
Decomposition
Tensor
Construction
Ar
Road
Networks
Trajectory
Database
Frequent Trajectory
Pattern Mining
Context Feature
Extraction
Features
Arec
Patterns
Optimal
Concatenation
Path
cost
Methodology
• Optimal Concatenation
Suppose 𝑷 is decomposed as 𝑃1 ||𝑃2 || ⋯ ||𝑃𝑘
true travel time: 𝜇𝑷 . estimated travel time: 𝑡𝑃1 + 𝑡𝑃2 + ⋯ + 𝑡𝑃𝑘
𝐿𝑆𝐸𝑷,𝑃1,𝑃2,⋯,𝑃𝑘 ≜ 𝐸 𝜇𝑷 − 𝑡𝑃1 − 𝑡𝑃2 − ⋯ − 𝑡𝑃𝑘
argmin𝑃1,𝑃2,⋯,𝑃𝑘 𝐿𝑆𝐸𝑷,𝑃1,𝑃2,⋯,𝑃𝑘 ,
subject to 𝑃1 ||𝑃2 || ⋯ ||𝑃𝑘 = 𝑷
S
𝑡𝑃1
𝑡𝑃2
𝑡𝑃𝑘 D
𝑃1
𝑃2
𝑃𝑘
𝜇𝑷
2
Methodology
• Optimal Concatenation
𝐿𝑆𝐸𝑷,𝑃1 ,𝑃2 ,⋯,𝑃𝑘 = 𝐸 𝜇𝑷 − 𝑡𝑃1 − 𝑡𝑃2 − ⋯ − 𝑡𝑃𝑘
= 𝐸 𝜇𝑃1 + 𝜇𝑃2 + ⋯ 𝜇𝑃𝑘 − 𝑡𝑃1 − 𝑡𝑃2 − ⋯ − 𝑡𝑃𝑘
𝑘
𝑖=1
=𝐸
𝜇𝑃𝑖 − 𝑡𝑃𝑖
2
𝑘
𝑖=1
+
𝑘
𝑘
𝑗=1(𝜇𝑃𝑖
𝑘
𝑖=1
2
− 𝑡𝑃𝑖 )(𝜇𝑃𝑗 − 𝑡𝑃𝑗 )
𝑘
𝐸(𝜇𝑃𝑖 − 𝑡𝑃𝑖 )2 +
=
2
𝐸 (𝜇𝑃𝑖 − 𝑡𝑃𝑖 )(𝜇𝑃𝑗 − 𝑡𝑃𝑗 )
𝑖=1 𝑗=1
assuming 𝑡𝑃𝑖 and 𝑡𝑃𝑗 are independent
𝐸 (𝜇𝑃𝑖 − 𝑡𝑃𝑖 )(𝜇𝑃𝑗 − 𝑡𝑃𝑗 ) = 𝐸 𝜇𝑃𝑖 − 𝑡𝑃𝑖 𝐸(𝜇𝑃𝑗 − 𝑡𝑃𝑗 )=0
𝐿𝑆𝐸𝑷,𝑃1,𝑃2,⋯,𝑃𝑘 =
𝐸(𝜇𝑃𝑖 − 𝑡𝑃𝑖 )2 = 𝐸(𝜇𝑃𝑖 −
=
1
2
𝑛𝑃
𝑛 𝑃𝑖
𝑖
𝑗=1
𝐸(𝜇𝑃𝑖 −
1
𝑛𝑃𝑖
𝑘
𝑖=1 𝐸(𝜇𝑃𝑖
− 𝑡𝑃𝑖 )2
𝑛 𝑃𝑖
𝑗=1
𝟏
𝑡𝑃𝑖 ,𝑗 )2 =
𝒏 𝑷𝒊
𝑡𝑃𝑖 ,𝑗 )2 =
1
𝐸
𝑛𝑃2 𝑖
𝑽𝒂𝒓(𝒕𝑷𝒊 ,𝒋 )
𝑛 𝑃𝑖
(𝜇𝑃𝑖 − 𝑡𝑃𝑖 ,𝑗 )2
𝑗=1
Methodology
• Support 𝑛𝑃𝑖 vs. Variance 𝑉𝑎𝑟 𝑡𝑃𝑖 ,𝑗
– The bigger the support is, the smaller the error is
– The bigger the variance the bigger the error
argmin𝑃1,𝑃2,⋯,𝑃𝑘
1
𝑘
𝑖=1 𝑛
𝑃𝑖
𝑉𝑎𝑟(𝑡𝑃𝑖,𝑗 )
subject to 𝑃1 ||𝑃2 || ⋯ ||𝑃𝑘 = 𝑷
• Solved by dynamic programming
1
– Denote 𝑔 𝑃𝑖 = 𝑛
𝑃𝑖
𝑉𝑎𝑟(𝑡𝑃𝑖 ,𝑗 )
argmin𝑃1,𝑃2,⋯,𝑃𝑙
𝑙
𝑗=1 𝑔
𝑃𝑗
subject to 𝑃1 ||𝑃2 || ⋯ ||𝑃𝑙 = 𝑃′.
𝑜𝑝𝑡𝑛 = min (𝑜𝑝𝑡𝑖 + 𝑔(𝑃𝑟𝑖+1||𝑟𝑖+2⋯||𝑟𝑛 )
1≤𝑖<𝑛
Methodology
• Make optimal concatenation more efficient
• Frequent trajectory pattern mining
– Not necessary to check every combination
– Using suffix-tree-based method
P: r1 r2
Root
2
r1
r5
r2
r3
r6
r7
r4
Tr1
Tr2
Tr3
Tr4
r1
r2
2
2 1
r2
1
r3
3
r3
1
r6
3 1
(1
1
r3
r5
1
1
r7
r2
1
r6
r7
Pattern
r
1
r6
r3
A) An example of suffix-tree
B) Filling
Methodology
• Combining the suffix tree with tensors
– Searching for frequent trajectory pattern from the suffix tree
– Find the travel time of a particular user from the tensor
P: r1
Root
3
3 1
r3
r4 tr3
(1)
1
r3
r5
1
1
r7
r2
t r4,u2,k , t r4,u3,k , t r4,u4,k
1
r6
r4,u2,k = tr3,u2,k+t r4,u2,k
tk
r7
(3)
tj
uM
Patterns: (1)
r2 r3, r3
r2
1
r3r
r1
r2
5
mple of suffix-tree
r3
r6
r7
r4
Tr1
Tr2
Tr3
Tr4
r4
Tr2,Tr3,Tr4
(u2, u3, u4)
u2
u1
r1 r2
(3)
rN
Arec
B) Filling in the missing time for a pattern
N
Methodology
• Deal with efficiency and scalability
–
–
–
–
data-driven space partition
an element-wise optimization algorithm
Use trajectory patterns as concatenation candidates
Indexing recent trajectories for a fast online retrieval
Root
g1
g2
g1
tr
tr
1
g2
1
tr
tr
g4
g3
g4
1
r2
1
r2
Tr1
Tr2
r1
Tr1
Tr2
r2
tr
1
r2
r3
Tr2
r6
Tr1
r2
Tr2
Tr2
2
2
r3
g3
tr
tr
tr
2
r3
tr
1
r3
r6
r3
r6 t r Tr 1 Tr 1 tr
Tr2
Tr 1 tr
r2
3
r6
6
2
r6
Experiments
• Datasets
– Taxi Trajectories:
•
•
•
•
•
Generated by 32,670 taxicabs in Beijing
From Sept. 1 to Oct. 31, 2013.
673+ million GPS points
Total length: over 26 million km.
Sampling rate: 96 seconds per point.
– Road networks:
• 148,110 nodes and 196,307 edges.
• Covers a 40×50km spatial range
• Total length of road segments: 21,985km.
– POIs:
• Th273,165 POIs of Beijing
• 195 tier two categories.
• Chose the top 10 categories that occur around
road segments
Download data here
Experiments
0.74
– 𝒜r /(5×5): 4,736×12,674×4
– 0.09%
300
MAE
Time Cost
MAE (min)
0.73
MAE (min)
RMSE
250
200
0.72
150
100
0.71
Time Cost (minute)
• Performance of CATD
50
𝑇𝐷
0.747
1.646
0.70
𝑇𝐷 + 𝐻
𝐶𝐴𝑇𝐷 (𝑇𝐷 + 𝐻 + 𝐶)
0.732
1.629
0.714
1.613
0
2
3
4
5
Number of time slices L
1.0
180
160
𝑖 |𝑦𝑖 −𝑦𝑖 |
𝑀𝐴𝐸 =
tj
g2
g1
g2
𝑖 |𝑦𝑖 −𝑦𝑖 |
𝑀𝑅𝐸 =
𝑖 𝑦𝑖
u2
•
g1
𝑛
u
𝑅𝑀𝑆𝐸1 =
r r
1
2
MAE (min)
120
tk
uM
•
140
0.8
100
80
0.7
60
40
0.6
2
𝑖(𝑦𝑖 −𝑦𝑖 )
𝑛rN
g4
g3
g4
20
g3
0
0.5
1
2
3
4
5
The number of partitions (hxh)
Time cost (min)
Metrics
•
MAE (min)
Time cost (min)
0.9
Experiments
• Query paths
– From taxi trajectories
•
•
•
•
12,384 queries, 50 paths per hour/day
Traveled by at least two drivers
Total length 76,412.6km
Effective time span: 2,734 hours
– In the field study
•
•
•
•
114 queries
from Sept. 1 to Oct. 30, 2013
Total length 999km
Effective time span: 62 hours
B) Distribution of the length
A) Geographical distribution
C) Distribution of time length
Experiments
• Baselines
–
–
–
–
Speed-Constraint-based (SC) method
Trajectory-based Simple Concatenation (TSC) method.
Optimal Concatenation with Historical Travel Time (OC+H) method.
Optimal Concatenation with Nonnegative Matrix Factorization (OC+MF)
Query paths from taxi data
𝑆𝐶
𝑇𝑆𝐶
𝑂𝐶 + 𝐻
𝑂𝐶 + 𝑀𝐹
𝑷𝑻𝑻𝑬
MAE (min)
8.808
5.244
3.245
3.061
2.545
MRE
0.665
0.396
0.245
0.231
0.192
MAE/L (min/km)
1.428
0.850
0.526
0.496
0.412
In-the-field study
𝑆𝐶
𝑇𝑆𝐶
𝑂𝐶 + 𝐻
𝑂𝐶 + 𝑀𝐹
𝑷𝑻𝑻𝑬
MAE (min)
18.193
11.300
4.990
4.052
3.771
MRE
0.561
0.349
0.154
0.125
0.116
MAE/L (min/km)
2.075
1.289
0.569
0.462
0.430
Experiments
• Efficiency
Components
Time
Memory (MB)
Building matrix 𝑋, 𝑌
34ms
9
𝒜𝑟
44ms
4.4
𝒜ℎ
233ms
14.6
5×5
6.31min
6.4min
2.3ms
116
144
995
w/o trajectory patterns
8.6ms
877
w/o index
12.2s
714
Decomposition
Total
Best OC
800
700
2
600
Size of Index (MB)
P: r1 r2 r3 r4 tr3
Root
r1
r2
2
500
400
r2
300
r3
1
200
3
2 1
r3
r6
3 1
1
r3
r5
1
1
r7
8
1
r6
Patterns:
6
r2 r3, r3 r4
1
r6
7
(3)
tj
Time/Query (ms)
MAE
2.8
uM
(1)
r2
1
t r4,u2,k , t r4,u3,k , t r4,u4,k
tk
r7
3.0
r4,u2,k = tr3,u2,k+t r4,u2,k
9
(1)
Time/Query (ms)
Optimal
Concatenation (OC)
Tr2,Tr5
3,Tr4
(u2, u3, u4)
r3
A) An example of suffix-tree
2.6
u2
u1
r1 r2
(3)
rN
Arec
2.4
4
B) Filling in the missing time for a pattern
3
100
2.2
2
0
0
200
400
600
Support
800
1000
1
2.0
0
200
400
600
Support
800
1000
MAE
Tensor
construction
Deal with missing
values (𝐶𝐴𝑇𝐷)
Conclusion
• A very fundamental but challenging task
– Data sparsity
– Trade off between length of a path and support of trajectories
– Efficiency and scalability
• Our method
– Context-Aware Tensor Decomposition
– Optimal Concatenation
• Results
– Effectiveness
• 12,384 query paths and 114 in the field study
• Relative error ratio: 19% and 11.6%
– Efficiency
• Partition a city into grids
• Suffix-tree-based pattern mining and index
• Infer the travel time of a path in 2.3ms
Download data and codes
Search for “Urban Computing”
Thanks!
Yu Zheng
[email protected]
Homepage
Experiments
Download data here
• Datasets
– Taxi Trajectories:
•
•
•
•
•
Generated by 32,670 taxicabs in Beijing
From Sept. 1 to Oct. 31, 2013.
673+ million GPS points
Total length: over 26 million km.
Sampling rate: 96 seconds per point.
Max. Num. of Trajectories (Support)
10.0k
8.0k
6.0k
4.0k
2.0k
0.0
0
– Road networks:
– POIs:
• Th273,165 POIs of Beijing
• 195 tier two categories.
• Chose the top 10 categories that occur around
road segments
20
30
40
50
Length of a Path
1-2
3-4
5-6
7-8
>8
50k
Num. of Road Segments
• 148,110 nodes and 196,307 edges.
• Covers a 40×50km spatial range
• Total length of road segments: 21,985km.
10
40k
30k
20k
10k
0
0
4
8
12
Time of Day
16
20