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On the Effect of Trajectory Compression in Spatio-temporal Querying. Elias Frentzos , and Yannis Theodoridis Data Management Group, University of Piraeus http://isl.cs.unipi.gr/db. ADBIS, October 2 2007. Talk Outline. Problem Statement Background Compressing Trajectories - PowerPoint PPT Presentation
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On the Effect of Trajectory Compression in Spatio-
temporal Querying
Elias Frentzos, and Yannis TheodoridisData Management Group, University of Piraeushttp://isl.cs.unipi.gr/db
ADBIS, October 2 2007
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
2
Problem Statement Background
Compressing Trajectories Related work on Error Estimation
Estimating the Effect of Compression ST Querying
Evaluating the Effect of Compression ST Querying Experimental Results
On the performance On the quality
Conclusions and Future Work
Talk Outline
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
3
Problem Statement Background
Compressing Trajectories Related work on Error Estimation
Estimating the Effect of Compression ST Querying
Evaluating the Effect of Compression ST Querying Experimental Results
On the performance On the quality
Conclusions and Future Work
Talk Outline
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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Trajectory is the data obtained from moving point objects and can be seen as a string in the 3D space
Trajectory compression is a very promising field since moving objects recording their position in time produce large amounts of frequently redundant data
Existing work on trajectory compression is mainly driven by research advances in the fields of line generalization and time series compression.
Our interest is in lossy compression techniques which eliminate some repeated or unnecessary information under well-defined error bounds.
Problem Statement (1)
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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The objectives for trajectory compression are: To obtain a lasting reduction in data size; To obtain a data series that still allows various
computations at acceptable (low) complexity; To obtain a data series with known, small margins of
error, which are preferably parametrically adjustable. Our goal is to calculate the mean error introduced
in query results over compressed trajectory data, which is by no means a trivial task We argue that this mean error can be used for
deciding whether the compressed data are suitable for the user needs
We restrict our discussion in a special type of spatiotemporal query, the timeslice queries
Problem Statement (2)
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
6
Problem Statement Background
Compressing Trajectories Related work on Error Estimation
Estimating the Effect of Compression ST Querying
Evaluating the Effect of Compression ST Querying Experimental Results
On the performance On the quality
Conclusions and Future Work
Talk Outline
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
7
Methods exploiting line simplification algorithms for compressing a trajectory are based on the so called Synchronous Euclidean Distance (SED)
SED is the distance between the sampled point Pi (xi , yi , ti ) being under examination, and the point of the line (Ps, Pe) where the moving object would lie, supposed it was moving on this line, at time instance ti determined by the point under examination
Compressing Trajectories: SED
Ps(xs,ys,ts)
Pe(xe,ye,te)
Pi(xi,yi,ti)
Pi’(xi’,yi’,ti)
SED(P,P’)
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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The TD-TR algorithm (Meratnia and By, EDBT 2004) is a spatiotemporal extension of the quite famous Top – Down Douglas – Peucker algorithm which was originally used in cartography
The algorithm tries (and achieves) to preserve directional trends in the approximated line using a distance threshold
The TD-TR algorithm uses SED instead of the perpendicular distance It is a batch algorithm since it requires the full line at its start
Compressing Trajectories: TD-TR algorithm
A
B
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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Opening window (OW) algorithms anchor the start point of a potential segment, and then attempt to approximate the subsequent data series with increasingly longer segments.
The algorithm also achieves to preserve directional trends in the approximated line using a distance threshold
The OPW-TR algorithm (Meratnia and By, EDBT 2004) also uses SED instead of the perpendicular distance
It can be used as an online algorithm
Compressing Trajectories: OPW-TR algorithm
A
B
C
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
10
Problem Statement Background
Compressing Trajectories Related work on Error Estimation
Estimating the Effect of Compression ST Querying
Evaluating the Effect of Compression ST Querying Experimental Results
On the performance On the quality
Conclusions and Future Work
Talk Outline
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
11
The only relative work estimates the average value of the Synchronous Euclidean Distance (SED), also termed as Synchronous Error, between an original trajectory and its approximation.
There is no obvious way on how to use it in order to determine the error introduced in query results
Related work on Error Estimation
11
,1
( , ) ( )k
k
tn
p qk t
AvgE p q E t dt
11 2
2, 2
2 4 2( ) arcsinh
4 8 4
kk
kk
tt
p q
tt
at b b ac at bE t dt at bt c
a a a ac b
t1
tn
t
q
p
x
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
12
Problem Statement Background
Compressing Trajectories Related work on Error Estimation
Estimating the Effect of Compression in ST Querying
Evaluating the Effect of Compression in ST Querying
Experimental Results On the performance On the quality
Conclusions and Future Work
Talk Outline
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
13
t y
Q1
Q2
x
1 2
Q3
3 4
Q5
Q4
t1
t4
t2
t6
t3
Estimating the Effect of Compression in ST Querying: Preliminaries
Our goal is to provide closed-form formulas that estimate the number of false hits introduced in query results over compressed trajectory datasets
Among the query types executed against trajectory datasets, we focus on a special type or range query, the so-called timeslice query
Two types of errors are introduced in query results when executing a timeslice query over a trajectory dataset
t
Q1
x
1 false negatives are the
trajectories which originally qualified the query but their compressed counterparts were not retrieved
false positives are the compressed trajectories retrieved by the query while their original counterparts are not qualifying it
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
14
Query Window
Wj
a
b
†2, jp
p2,j
δy2(tj)
δx2(tj)
x
y t=tj
δy1(tj) δx1(tj)
Estimating the Effect of Compression in ST Querying: Analysis (1)
We first calculate AvgPi,P / AvgPi,N, which is the average probability of a single compressed trajectory to be retrieved as false positive / negative, regarding all possible timeslice query windows with sides a b
We then sum-up these average probabilities of all dataset trajectories in order to produce the global average probability
The error introduced in the position of a trajectory can be calculated as a function of time
,1
n
P a b i P a bi
E R AvgP R
,1
n
N a b i N a bi
E R AvgP R
, 1 ,, ,
, 1 ,
( ) i k i ki i k i k
i k i k
x xx t x t t
t t
, 1 ,, ,
, 1 ,
( ) i k i ki i k i k
i k i k
y yy t y t t
t t
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
15
W
Estimating the Effect of Compression in ST Querying: Analysis (2)
We calculate the average probability of a compressed trajectory Ti to be retrieved as false positive / negative regarding a timeslice query window at timestamp tj
The quantity of timeslice query windows that may retrieve a compressed trajectory as false positive / negative at timestamp tj can be extracted geometrically
We distinguish among 4 cases, regarding the signs of δx and δy values
Finally by integrating the area Ai,j over all the timestamps inside the unit space we obtain AvgPi,P / AvgPi,N
δyi,j>0
δxi,j<0
[0,1][0,1], tj
Ai,j , , ,i j i j i jA a b a x b y W
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
16
Estimating the Effect of Compression in ST Querying: Analysis (3)
Summing up the average probabilities of all trajectories and performing the necessary calculations, we obtain:
where
1, , 1 , , 1, 1 ,
1 1 (1 ) (1 ) 2 2 6
i
N a b P a b
mni k i k i k i ki k i k
i k
E R E R
b x x a y yt t e
a b
, , , 1 , 1 , , 1 , 1 ,2 2i k i k i k i k i k i k i k i ke x y x y x y x y
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
17
Problem Statement Background
Compressing Trajectories Related work on Error Estimation
Estimating the Effect of Compression in ST Querying
Evaluating the Effect of Compression in ST Querying
Experimental Results On the performance On the quality
Conclusions and Future Work
Talk Outline
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
18
Evaluating the Effect of Compression in ST Querying
The evaluation of this formula is a costly operation O(nm); its calculation requires to process the entire original dataset along with its compressed counterpart
However, any compression algorithm evaluating SED, need also to calculate δxi,k δyi,k in every timestamp
As a consequence, the evaluation of the average error in the query results, can be integrated in the compressions algorithm, introducing only a small overhead on its execution
1, , 1 , , 1, 1 ,
1 1 (1 ) (1 ) 2 2 6
i
N a b P a b
mni k i k i k i ki k i k
i k
E R E R
b x x a y yt t e
a b
2 2
i i iSED t x t y t
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
19
Problem Statement Background
Compressing Trajectories Related work on Error Estimation
Estimating the Effect of Compression in ST Querying
Evaluating the Effect of Compression in ST Querying
Experimental Results On the performance On the quality
Conclusions and Future Work
Talk Outline
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
20
Experimental Study: Settings
Datasets One real trajectory dataset of a fleet of trucks (273
trajectories, 112K entries) A synthetic dataset of 2000 trajectories generated using
network-based data generator and the San Joaquin road network
Implementation We implemented the TD-TR algorithm and compressed
the real and synthetic datasets varying its threshold
Experiments Average overhead introduced in the TD-TR algorithm Average number of false positives and false negatives in
10000 randomly distributed timeslice queries
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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Experimental Study: On the performance Scaling the value of the
TD-TR threshold The algorithm’s execution
time reduces as the value of the TD-TR threshold increases
The overhead introduced in the algorithm’s execution, is typically small (bellow 7%)
In absolute times, the overhead introduced never exceeds 0.2 milliseconds per trajectory
Trucks dataset
Synthetic dataset
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.001 0.005 0.01 0.015 0.02TD-TR threshold
Ex
ecu
tio
n t
ime
(mse
c)
Model calculations included
Model calculations excluded
0
0.2
0.4
0.6
0.8
1
1.2
0.001 0.005 0.01 0.015 0.02TD-TR threshold
Ex
ecu
tio
n t
ime
(mse
c) Model calculations included
Model calculations excluded
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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Experimental Study: On the quality (1) Scaling the value of the TD-
TR threshold The average number of false hits
(negatives and positives) is linear with the value of the TD-TR compression threshold
The average error in the estimation for the synthetic dataset is around 6%, varying between 0.2% and 14%
In the trucks dataset the average error increases around 10.6%, mainly due to the error introduced in small values of TD-TR threshold
Trucks dataset
Synthetic dataset
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.001 0.005 0.01 0.015 0.02TD-TR threshold
Av
erag
e Fa
lse
Hit
s
False Negatives
False Positives
Estimation
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.001 0.005 0.01 0.015 0.02TD-TR threshold
Av
erag
e Fa
lse
Hit
s
False Negatives
False Positives
Estimation
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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Experimental Study: On the quality (2) Scaling the query size
The average number of false hits (negatives and positives) is sub-linear with the size of the query
The average error in the estimation for the synthetic dataset is around 2.9%, varying between 0.2% and 8.7%
In the trucks dataset the average error increases around 7.5%
Trucks dataset
Synthetic dataset
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.05 0.1 0.15 0.2 0.25 0.3Query size (a = b )
Av
erag
e Fa
lse
Hit
s
False Negatives
False Positives
Estimation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.05 0.1 0.15 0.2 0.25 0.3Query size (a = b )
Av
erag
e Fa
lse
Hit
s
False Negatives
False Positives
Estimation
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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Summary and Future Work
We provided a closed formula of the average number of false negatives and false positives covering the case of uniformly distributed query windows and arbitrarily distributed trajectory data
Through an experimental study we demonstrated the efficiency of the proposed model We illustrated the applicability of our model under real-life
requirements – it turns out that the estimation of the model parameters introduce only a small overhead in the trajectory compression algorithm
We presented the accuracy of our estimations, with an average error being around 6%.
Future work: Extension of our model in nearest neighbor and general range
queries Applicability of our model in the case of spatiotemporal warehouses
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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Acknowledgements
Research partially supported by: GEOPKDD (“Geographic Privacy-aware Knowledge
Discovery and Delivery”) project funded by the European Community under FP6-014915 contract
Frentzos and Theodoridis, ADBIS 2007On the Effect of Trajectory Compression in Spatiotemporal Querying
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Thank you!
On the Effect of Trajectory Compression in Spatiotemporal Querying