Higher-order patterns reveal sequential multistep state transitions, which are usually superior to origin-destination analyses that depict only first-order geospatial movement patterns. Conventional methods for higher-order movement modeling first construct a directed acyclic graph (DAG) of movements and then extract higher-order patterns from the DAG. However, DAG-based methods rely heavily on identifying movement keypoints, which are challenging for sparse movements and fail to consider the temporal variants critical for movements in urban environments. To overcome these limitations, we propose HoLens, a novel approach for modeling and visualizing higher-order movement patterns in the context of an urban environment. HoLens mainly makes twofold contributions: First, we designed an auto-adaptive movement aggregation algorithm that self-organizes movements hierarchically by considering spatial proximity, contextual information, and temporal variability. Second, we developed an interactive visual analytics interface comprising well-established visualization techniques, including the H-Flow for visualizing the higher-order patterns on the map and the higher-order state sequence chart for representing the higher-order state transitions. Two real-world case studies demonstrate that the method can adaptively aggregate data and exhibit the process of exploring higher-order patterns using HoLens. We also demonstrate the feasibility, usability, and effectiveness of our approach through expert interviews with three domain experts.
Xu, J.; Wickramarathne, T. L.; Chawla, N. V. Representing higher-order dependencies in networks. Science Advances Vol. 2, No. 5, e1600028, 2016.
Shannon, C. E. A mathematical theory of communication. The Bell System Technical Journal Vol. 27, No. 3, 379–423, 1948.
Grundy, E.; Jones, M. W.; Laramee, R. S.; Wilson, R. P.; Shepard, E. L. C. Visualisation of sensor data from animal movement. Computer Graphics Forum Vol. 28, No. 3, 815–822, 2009.
Nekovee, M.; Moreno, Y.; Bianconi, G.; Marsili, M. Theory of rumour spreading in complex social networks. Physica A: Statistical Mechanics and Its Applications Vol. 374, No. 1, 457–470, 2007.
Kareiva, P. M.; Shigesada, N. Analyzing insect movement as a correlated random walk. Oecologia Vol. 56, No. 2, 234–238, 1983.
Andrienko, N.; Andrienko, G. State transition graphs for semantic analysis of movement behaviours. Information Visualization Vol. 17, No. 1, 41–65, 2018.
Andrienko, N.; Andrienko, G.; Stange, H.; Liebig, T.; Hecker, D. Visual analytics for understanding spatial situations from episodic movement data. KI - Künstliche Intelligenz Vol. 26, No. 3, 241–251, 2012.
Gaffney, S. J.; Robertson, A. W.; Smyth, P.; Camargo, S. J.; Ghil, M. Probabilistic clustering of extratropical cyclones using regression mixture models. Climate Dynamics Vol. 29, No. 4, 423–440, 2007.
Adrienko, N.; Adrienko, G. Spatial generalization and aggregation of massive movement data. IEEE Transactions on Visualization and Computer Graphics Vol. 17, No. 2, 205–219, 2011.
Zhou, Z.; Meng, L.; Tang, C.; Zhao, Y.; Guo, Z.; Hu, M.; Chen, W. Visual abstraction of large scale geospatial origin-destination movement data. IEEE Transactions on Visualization and Computer Graphics Vol. 25, No. 1, 43–53, 2019.
Blaas, J.; Botha, C.; Grundy, E.; Jones, M.; Laramee, R.; Post, F. Smooth graphs for visual exploration of higher-order state transitions. IEEE Transactions on Visualization and Computer Graphics Vol. 15, No. 6, 969–976, 2009.
Rosvall, M.; Esquivel, A. V.; Lancichinetti, A.; West, J. D.; Lambiotte, R. Memory in network flows and its effects on spreading dynamics and community detection. Nature Communications Vol. 5, Article No. 4630, 2014.
Zeng, W.; Fu, C. W.; Müller Arisona, S.; Schubiger, S.; Burkhard, R.; Ma, K. L. Visualizing the relationship between human mobility and points of interest. IEEE Transactions on Intelligent Transportation Systems Vol. 18, No. 8, 2271–2284, 2017.
Dodge, S.; Weibel, R.; Lautenschütz, A. K. Towards a taxonomy of movement patterns. Information Visualization Vol. 7, Nos. 3–4, 240–252, 2008.
Slingsby, A.; van Loon, E. Exploratory visual analysis for animal movement ecology. Computer Graphics Forum Vol. 35, No. 3, 471–480, 2016.
Chen, S.; Yuan, X.; Wang, Z.; Guo, C.; Liang, J.; Wang, Z.; Zhang, X.; Zhang, J. Interactive visual discovering of movement patterns from sparsely sampled geo-tagged social media data. IEEE Transactions on Visualization and Computer Graphics Vol. 22, No. 1, 270–279, 2016.
Chen, W.; Guo, F.; Wang, F. Y. A survey of traffic data visualization. IEEE Transactions on Intelligent Transportation Systems Vol. 16, No. 6, 2970–2984, 2015.
Andrienko, G.; Andrienko, N.; Dykes, J.; Fabrikant, S. I.; Wachowicz, M. Geovisualization of dynamics, movement and change: Key issues and developing approaches in visualization research. Information Visualization Vol. 7, Nos. 3–4, 173–180, 2008.
Adrienko, N.; Adrienko, G. Spatial generalization and aggregation of massive movement data. IEEE Transactions on Visualization and Computer Graphics Vol. 17, No. 2, 205–219, 2011.
Guo, D.; Zhu, X. Origin-destination flow data smoothing and mapping. IEEE Transactions on Visualization and Computer Graphics Vol. 20, No. 12, 2043–2052, 2014.
Scheepens, R.; Willems, N.; van de Wetering, H.; Andrienko, G.; Andrienko, N.; van Wijk, J. J. Composite density maps for multivariate trajectories. IEEE Transactions on Visualization and Computer Graphics Vol. 17, No. 12, 2518–2527, 2011.
Feng, Z.; Li, H.; Zeng, W.; Yang, S. H.; Qu, H. Topology density map for urban data visualization and analysis. IEEE Transactions on Visualization and Computer Graphics Vol. 27, No. 2, 828–838, 2021.
Laube, P.; Purves, R. S. An approach to evaluating motion pattern detection techniques in spatio-temporal data. Computers, Environment and Urban Systems Vol. 30, No. 3, 347–374, 2006.
Zeng, W.; Fu, C. W.; Müller Arisona, S.; Schubiger, S.; Burkhard, R.; Ma, K. L. A visual analytics design for studying rhythm patterns from human daily movement data. Visual Informatics Vol. 1, No. 2, 81–91, 2017.
Deng, Z.; Weng, D.; Liu, S.; Tian, Y.; Xu, M.; Wu, Y. A survey of urban visual analytics: Advances and future directions. Computational Visual Media Vol. 9, No. 1, 3–39, 2023.
Huang, X.; Zhao, Y.; Ma, C.; Yang, J.; Ye, X.; Zhang, C. TrajGraph: A graph-based visual analytics approach to studying urban network centralities using taxi trajectory data. IEEE Transactions on Visualization and Computer Graphics Vol. 22, No. 1, 160–169, 2016.
Zeng, W.; Shen, Q.; Jiang, Y.; Telea, A. Route-aware edge bundling for visualizing origin-destination trails in urban traffic. Computer Graphics Forum Vol. 38, No. 3, 581–593, 2019.
Lu, W. L.; Wang, Y. S.; Lin, W. C. Chess evolution visualization. IEEE Transactions on Visualization and Computer Graphics Vol. 20, No. 5, 702–713, 2014.
Krstajic, M.; Bertini, E.; Keim, D. CloudLines: Compact display of event episodes in multiple time-series. IEEE Transactions on Visualization and Computer Graphics Vol. 17, No. 12, 2432–2439, 2011.
Schmidt, M. The sankey diagram in energy and material flow management. Journal of Industrial Ecology Vol. 12, No. 1, 82–94, 2008.
Ham, F. V.; van de Wetering, H.; van Wijk, J. J. Interactive visualization of state transition systems. IEEE Transactions on Visualization and Computer Graphics Vol. 8, No. 4, 319–329, 2002.
Monroe, M.; Lan, R.; Lee, H.; Plaisant, C.; Shneiderman, B. Temporal event sequence simplification. IEEE Transactions on Visualization and Computer Graphics Vol. 19, No. 12, 2227–2236, 2013.
Liu, Z.; Kerr, B.; Dontcheva, M.; Grover, J.; Hoffman, M.; Wilson, A. CoreFlow: Extracting and visualizing branching patterns from event sequences. Computer Graphics Forum Vol. 36, No. 3, 527–538, 2017.
Zeng, W.; Fu, C. W.; Arisona, S. M.; Qu, H. Visualizing interchange patterns in massive movement data. Computer Graphics Forum Vol. 32, No. 3pt3, 271–280, 2013.
Slingsby, A.; van Loon, E. Exploratory visual analysis for animal movement ecology. Computer Graphics Forum Vol. 35, No. 3, 471–480, 2016.
Ware, C.; Arsenault, R.; Plumlee, M.; Wiley, D. Visualizing the underwater behavior of humpback whales. IEEE Computer Graphics and Applications Vol. 26, No. 4, 14–18, 2006.
Lu, Y.; Steptoe, M.; Burke, S.; Wang, H.; Tsai, J. Y.; Davulcu, H.; Montgomery, D.; Corman, S. R.; Maciejewski, R. Exploring evolving media discourse through event cueing. IEEE Transactions on Visualization and Computer Graphics Vol. 22, No. 1, 220–229, 2016.
Wongsuphasawat, K.; Gotz, D. Exploring flow, factors, and outcomes of temporal event sequences with the outflow visualization. IEEE Transactions on Visualization and Computer Graphics Vol. 18, No. 12, 2659–2668, 2012.
Gotz, D.; Stavropoulos, H. DecisionFlow: Visual analytics for high-dimensional temporal event sequence data. IEEE Transactions on Visualization and Computer Graphics Vol. 20, No. 12, 1783–1792, 2014.
Perer, A.; Sun, J. MatrixFlow: Temporal network visual analytics to track symptom evolution during disease progression. Annual Symposium Proceedings Vol. 2012, 716–725, 2012.
Schwarz, G. Estimating the dimension of a model. The Annals of Statistics Vol. 6, No. 2, 461–464, 1978.
Akaike, H. A new look at the statistical model identification. IEEE Transactions on Automatic Control Vol. 19, No. 6, 716–723, 1974.
Van der Heyden, M. J.; Diks, C. G. C.; Hoekstra, B. P. T.; DeGoede, J. Testing the order of discrete Markov chains using surrogate data. Physica D: Nonlinear Phenomena Vol. 117, Nos. 1–4, 299–313, 1998.
Kullback, S.; Leibler, R. A. On information and sufficiency. The Annals of Mathematical Statistics Vol. 22, No. 1, 79–86, 1951.
Meulemans, W.; Riche, N. H.; Speckmann, B.; Alper, B.; Dwyer, T. KelpFusion: A hybrid set visualization technique. IEEE Transactions on Visualization and Computer Graphics Vol. 19, No. 11, 1846–1858, 2013.
Baydas, S.; Karakas, B. Defining a curve as a Bezier curve. Journal of Taibah University for Science Vol. 13, No. 1, 522–528, 2019.
Zeng, W.; Fu, C. W.; Müller Arisona, S.; Erath, A.; Qu, H. Visualizing waypoints-constrained origin-destination patterns for massive transportation data. Computer Graphics Forum Vol. 35, No. 8, 95–107, 2016.
Yang, D.; Qu, B.; Yang, J.; Cudré-Mauroux, P. LBSN2Vec: Heterogeneous hypergraph embedding for location-based social networks. IEEE Transactions on Knowledge and Data Engineering Vol. 34, No. 4, 1843–1855, 2022.
Cao, N.; Gotz, D.; Sun, J.; Qu, H. DICON: Interactive visual analysis of multidimensional clusters. IEEE Transactions on Visualization and Computer Graphics Vol. 17, No. 12, 2581–2590, 2011.
Yang, J.; Hubball, D.; Ward, M. O.; Rundensteiner, E. A.; Ribarsky, W. Value and relation display: Interactive visual exploration of large data sets with hundreds of dimensions. IEEE Transactions on Visualization and Computer Graphics Vol. 13, No. 3, 494–507, 2007.