research-article

Segmentation of Trajectories on Nonmonotone Criteria

Authors:

Marc Van Kreveld,

Maarten Löffler,

Frank StaalsAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 12, Issue 2

Article No.: 26, Pages 1 - 28

https://doi.org/10.1145/2660772

Published: 08 December 2015 Publication History

Abstract

In the trajectory segmentation problem, we are given a polygonal trajectory with n vertices that we have to subdivide into a minimum number of disjoint segments (subtrajectories) that all satisfy a given criterion. The problem is known to be solvable efficiently for monotone criteria: criteria with the property that if they hold on a certain segment, they also hold on every subsegment of that segment. To the best of our knowledge, no theoretical results are known for nonmonotone criteria.

We present a broader study of the segmentation problem, and suggest a general framework for solving it, based on the start-stop diagram: a 2-dimensional diagram that represents all valid and invalid segments of a given trajectory. This yields two subproblems: (1) computing the start-stop diagram, and (2) finding the optimal segmentation for a given diagram. We show that (2) is NP-hard in general. However, we identify properties of the start-stop diagram that make the problem tractable and give a polynomial-time algorithm for this case.

We study two concrete nonmonotone criteria that arise in practical applications in more detail. Both are based on a given univariate attribute function f over the domain of the trajectory. We say a segment satisfies an outlier-tolerant criterion if the value of f lies within a certain range for at least a given percentage of the length of the segment. We say a segment satisfies a standard deviation criterion if the standard deviation of f over the length of the segment lies below a given threshold. We show that both criteria satisfy the properties that make the segmentation problem tractable. In particular, we compute an optimal segmentation of a trajectory based on the outlier-tolerant criterion in O(n²log n + kn²) time and on the standard deviation criterion in O(kn²) time, where n is the number of vertices of the input trajectory and k is the number of segments in an optimal solution.

References

[1]

H. Alt and M. Godau. 1995. Computing the Fréchet distance between two polygonal curves. Int. Journal of Computer and Geometry Applications 5 (1995), 75--91.

[2]

A. Anagnostopoulos, M. Vlachos, M. Hadjieleftheriou, E. J. Keogh, and P. S. Yu. 2006. Global distance-based segmentation of trajectories. In Proc. 12th Conference on Knowledge Discovery and Data Mining. 34--43.

Digital Library

[3]

P. Bovet and S. Benhamou. 1988. Spatial analysis of animals’ movements using a correlated random walk model. Journal of Theoretical Biology 131, 4 (1988), 419--433. 10.1016/S0022-5193(88)80038-9

[4]

M. Buchin, A. Driemel, M. J. van Kreveld, and V. Sacristan. 2011. Segmenting trajectories: A framework and algorithms using spatiotemporal criteria. Journal of Spatial Information Science 3, 1 (2011), 33--63.

[5]

C. Calenge, S. Dray, and M. Royer-Carenzi. 2009. The concept of animals’ trajectories from a data analysis perspective. Ecological Informatics 4, 1 (2009), 34--41.

[6]

W. S. Chan and F. Chin. 1992. Approximation of polygonal curves with minimum number of line segments. In Proc. Algorithms and Computation (LNCS 650). 378--387.

Digital Library

[7]

P. Chundi and D. J. Rosenkrantz. 2009. Segmentation of time series data. In Encyclopedia of Data Warehousing and Mining, John Wang (Ed.). IGI Global, 1753--1758.

[8]

S. Dasgupta, C. Papadimitriou, and U. Vazirani. 2008. Algorithms. McGraw-Hill.

Digital Library

[9]

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. 1997. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin.

[10]

S. Dodge, R. Weibel, and E. Forootan. 2009. Revealing the physics of movement: Comparing the similarity of movement characteristics of different types of moving objects. Computers, Environment and Urban Systems 33, 6 (2009), 419--434.

[11]

David H. Douglas and Thomas K. Peucker. 1911. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Cartographica: The International Journal for Geographic Information and Geovisualization 10, 2 (1911), 112--122.

[12]

K. S. Fu and J. K. Mui. 1981. A survey on image segmentation. Pattern Recognition 13, 1 (1981), 3--16.

[13]

L. J. Guibas, J. Hershberger, J. S. B. Mitchell, and J. Snoeyink. 1993. Approximating polygons and subdivisions with minimum link paths. International Journal of Computer and Geometry Applications 3, 4 (1993), 383--415.

[14]

E. Gurarie, R. D. Andrews, and K. L. Laidre. 2009. A novel method for identifying behavioural changes in animal movement data. Ecology Letters 12, 5 (2009), 395--408.

[15]

H. Imai and M. Iri. 1988. Polygonal approximation of curve-formulations and algorithms. Computational Morphology: A Computational Geometric Approach to the Analysis of Form (1988), 71--86.

[16]

X. Li, X. Li, D. Tang, and X. Xu. 2010. Deriving features of traffic flow around an intersection from trajectories of vehicles. In Proc. 18th International Conference on Geoinformatics. IEEE, 1--5.

[17]

R. Mann, A. D. Jepson, and T. El-Maraghi. 2002. Trajectory segmentation using dynamic programming. In Proc. 16th International Conference on Pattern Recognition (ICPR’02), Vol. 1. 331--334.

Digital Library

[18]

R. Nathan, W. M. Getz, E. Revilla, M. Holyoak, R. Kadmon, D. Saltz, and P. E. Smouse. 2008. A movement ecology paradigm for unifying organismal movement research. Proceedings of the National Academy of Sciences 105 (2008), 19052--19059.

[19]

U. Ramer. 1972. An iterative procedure for the polygonal approximation of plane curves. Computer Graphics and Image Processing 1, 3 (1972), 244--256.

[20]

A. Stohl. 1998. Computation, accuracy and applications of trajectories -- A review and bibliography. Atmospheric Environment 32, 6 (1998), 947--966.

[21]

E. Terzi and P. Tsaparas. 2006. Efficient algorithms for sequence segmentation. In Proc. 6th SIAM International Conference on Data Mining. 314--325.

[22]

B. van Moorter, D. R. Visscher, C. L. Jerde, J. L. Frair, and E. H. Merrill. 2010. Identifying movement states from location data using cluster analysis. Journal of Wildlife Management 74, 3 (2010), 588--594.

[23]

X. Zhou, S. Shekhar, P. Mohan, S. Liess, and P. K. Snyder. 2011. Discovering interesting sub-paths in spatiotemporal datasets: A summary of results. In Proc. 19th ACM SIGSPATIAL Conference on Advances in Geographic Information Systems. 44--53.

Digital Library

Cited By

Forsch AFunke SHaunert JStorandt S(2023)Efficient Mining of Volunteered Trajectory DatasetsVolunteered Geographic Information10.1007/978-3-031-35374-1_3(43-77)Online publication date: 9-Dec-2023
https://doi.org/10.1007/978-3-031-35374-1_3
Ziaei SBidgoli MRudi A(2022)Recommending Popular Locations Based on Collected Trajectories2022 12th International Conference on Computer and Knowledge Engineering (ICCKE)10.1109/ICCKE57176.2022.9960068(181-189)Online publication date: 17-Nov-2022
https://doi.org/10.1109/ICCKE57176.2022.9960068
Damiani MHachem FQuadri CRossini MGaito S(2020)On Location Relevance and Diversity in Human Mobility DataACM Transactions on Spatial Algorithms and Systems10.1145/34234047:2(1-38)Online publication date: 27-Oct-2020
https://dl.acm.org/doi/10.1145/3423404
Show More Cited By

Index Terms

Segmentation of Trajectories on Nonmonotone Criteria
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Recommendations

A framework for trajectory segmentation by stable criteria
SIGSPATIAL '14: Proceedings of the 22nd ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems

We present an algorithmic framework for criteria-based segmentation of trajectories that can efficiently process a large class of criteria. Criteria-based segmentation is the problem of subdividing a trajectory into a small number of parts such that ...
An algorithmic framework for segmenting trajectories based on spatio-temporal criteria
GIS '10: Proceedings of the 18th SIGSPATIAL International Conference on Advances in Geographic Information Systems

In this paper we address the problem of segmenting a trajectory such that each segment is in some sense homogeneous. We formally define different spatio-temporal criteria under which a trajectory can be homogeneous, including location, heading, speed, ...
Segmentation of trajectories on non-monotone criteria
SODA '13: Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms

In the trajectory segmentation problem we are given a polygonal trajectory with n vertices that we have to subdivide into a minimum number of disjoint segments (subtrajectories) that all satisfy a given criterion. The problem is known to be solvable ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 12, Issue 2

February 2016

385 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/2846106

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2015 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 December 2015

Accepted: 01 July 2014

Revised: 01 March 2014

Received: 01 May 2013

Published in TALG Volume 12, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

EU Cost Action IC0903 (MOVE)
NSF
NSA MSP
Netherlands Organisation for Scientific Research (NWO)

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

12
Total Citations
View Citations
296
Total Downloads

Downloads (Last 12 months)14
Downloads (Last 6 weeks)1

Reflects downloads up to 04 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Forsch AFunke SHaunert JStorandt S(2023)Efficient Mining of Volunteered Trajectory DatasetsVolunteered Geographic Information10.1007/978-3-031-35374-1_3(43-77)Online publication date: 9-Dec-2023
https://doi.org/10.1007/978-3-031-35374-1_3
Ziaei SBidgoli MRudi A(2022)Recommending Popular Locations Based on Collected Trajectories2022 12th International Conference on Computer and Knowledge Engineering (ICCKE)10.1109/ICCKE57176.2022.9960068(181-189)Online publication date: 17-Nov-2022
https://doi.org/10.1109/ICCKE57176.2022.9960068
Damiani MHachem FQuadri CRossini MGaito S(2020)On Location Relevance and Diversity in Human Mobility DataACM Transactions on Spatial Algorithms and Systems10.1145/34234047:2(1-38)Online publication date: 27-Oct-2020
https://dl.acm.org/doi/10.1145/3423404
Rudi A(2020)Where Were the Birds Staying Last Week?New Mathematics and Natural Computation10.1142/S179300572050035016:03(581-592)Online publication date: 30-Nov-2020
https://doi.org/10.1142/S1793005720500350
Damiani MHachem FQuadri CGaito S(2019)Location relevance and diversity in symbolic trajectories with application to telco dataProceedings of the 16th International Symposium on Spatial and Temporal Databases10.1145/3340964.3340980(41-50)Online publication date: 19-Aug-2019
https://dl.acm.org/doi/10.1145/3340964.3340980
Guo SLi XChing WDan RLi WZhang Z(2018)GPS trajectory data segmentation based on probabilistic logicInternational Journal of Approximate Reasoning10.1016/j.ijar.2018.09.008103(227-247)Online publication date: Dec-2018
https://doi.org/10.1016/j.ijar.2018.09.008
Damiani MHachem FIssa HRanc NMoorcroft PCagnacci F(2018)Cluster-based trajectory segmentation with local noiseData Mining and Knowledge Discovery10.1007/s10618-018-0561-232:4(1017-1055)Online publication date: 1-Jul-2018
https://dl.acm.org/doi/10.1007/s10618-018-0561-2
Alewijnse SBuchin KBuchin MSijben SWestenberg M(2018)Model-Based Segmentation and Classification of TrajectoriesAlgorithmica10.1007/s00453-017-0329-x80:8(2422-2452)Online publication date: 1-Aug-2018
https://dl.acm.org/doi/10.1007/s00453-017-0329-x
Tang LRen CLiu ZLi Q(2017)A Road Map Refinement Method Using Delaunay Triangulation for Big Trace DataISPRS International Journal of Geo-Information10.3390/ijgi60200456:2(45)Online publication date: 15-Feb-2017
https://doi.org/10.3390/ijgi6020045
Moosavi SOmidvar-Tehrani BRamnath R(2017)Trajectory Annotation by Discovering Driving PatternsProceedings of the 3rd ACM SIGSPATIAL Workshop on Smart Cities and Urban Analytics10.1145/3152178.3152184(1-4)Online publication date: 7-Nov-2017
https://dl.acm.org/doi/10.1145/3152178.3152184
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents