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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 ADBIS, October

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Frentzos and Theodoridis, ADBIS 2007 On 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

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Frentzos and Theodoridis, ADBIS 2007 On 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

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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 4 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)

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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 5 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)

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Frentzos and Theodoridis, ADBIS 2007 On 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

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Frentzos and Theodoridis, ADBIS 2007 On 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 P i (x i, y i, t i ) being under examination, and the point of the line (P s, P e ) where the moving object would lie, supposed it was moving on this line, at time instance t i determined by the point under examination Compressing Trajectories: SED P s (x s,y s,t s ) P e (x e,y e,t e ) P i (x i,y i,t i ) SED(P,P)

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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 8 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

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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 9 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

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Frentzos and Theodoridis, ADBIS 2007 On 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

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Frentzos and Theodoridis, ADBIS 2007 On 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

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Frentzos and Theodoridis, ADBIS 2007 On 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

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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 13 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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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 14 Estimating the Effect of Compression in ST Querying: Analysis (1) We first calculate AvgP i,P / AvgP i,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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Frentzos and Theodoridis, ADBIS 2007 On 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 T i to be retrieved as false positive / negative regarding a timeslice query window at timestamp t j The quantity of timeslice query windows that may retrieve a compressed trajectory as false positive / negative at timestamp t j can be extracted geometrically We distinguish among 4 cases, regarding the signs of δx and δy values Finally by integrating the area A i,j over all the timestamps inside the unit space we obtain AvgP i,P / AvgP i,N δy i, j >0 δx i, j <0 [0,1] [0,1], t j A i,j W

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Frentzos and Theodoridis, ADBIS 2007 On 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

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Frentzos and Theodoridis, ADBIS 2007 On 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

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Frentzos and Theodoridis, ADBIS 2007 On 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(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 δx i,k δy i,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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Frentzos and Theodoridis, ADBIS 2007 On 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

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Frentzos and Theodoridis, ADBIS 2007 On 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 randomly distributed timeslice queries

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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 21 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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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 22 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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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 23 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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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 24 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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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 25 Acknowledgements Research partially supported by: GEOPKDD (Geographic Privacy-aware Knowledge Discovery and Delivery) project funded by the European Community under FP contract

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Frentzos and Theodoridis, ADBIS 2007 On the Effect of Trajectory Compression in Spatiotemporal Querying 26 Thank you! On the Effect of Trajectory Compression in Spatiotemporal Querying

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