On the Effect of Trajectory Compression in Spatiotemporal Querying

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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
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Talk Outline

• 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

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Talk Outline

• 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

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Problem Statement (1)

• 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.

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Problem Statement (2)

• 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

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Talk Outline

• 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

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Compressing Trajectories SED

• 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

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Compressing Trajectories TD-TR algorithm

• 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

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Compressing Trajectories OPW-TR algorithm

• 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

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Talk Outline

• 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

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Related work on Error Estimation

• 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

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Talk Outline

• 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

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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

• 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

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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

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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 dx and dy values
• Finally by integrating the area Ai,j over all the
  timestamps inside the unit space we obtain
  AvgPi,P / AvgPi,N

W
W
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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

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Talk Outline

• 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

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Evaluating the Effect of Compression in ST
Querying

• The evaluation of this formula is a costly
  operation O(n?m) its calculation requires to
  process the entire original dataset along with
  its compressed counterpart
• However, any compression algorithm evaluating
  SED, need also to calculate dxi,k dyi,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

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Talk Outline

• 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

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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

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Experimental Study On the performance

• Scaling the value of the TD-TR threshold
• The algorithms execution time reduces as the
  value of the TD-TR threshold increases
• The overhead introduced in the algorithms
  execution, is typically small (bellow 7)
• In absolute times, the overhead introduced never
  exceeds 0.2 milliseconds per trajectory

Trucks dataset
Synthetic dataset
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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
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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
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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

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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

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On the Effect of Trajectory Compression in
Spatiotemporal Querying
Thank you!