
- Previous Article
- FoDS Home
- This Issue
-
Next Article
On adaptive estimation for dynamic Bernoulli bandits
EmT: Locating empty territories of homology group generators in a dataset
Department of Statistics and Data Science, Yale University, New Haven, CT 06511, USA |
Persistent homology is a tool within topological data analysis to detect different dimensional holes in a dataset. The boundaries of the empty territories (i.e., holes) are not well-defined and each has multiple representations. The proposed method, Empty Territory (EmT), provides representations of different dimensional holes with a specified level of complexity of the territory boundary. EmT is designed for the setting where persistent homology uses a Vietoris-Rips complex filtration, and works as a post-analysis to refine the hole representation of the persistent homology algorithm. In particular, EmT uses alpha shapes to obtain a special class of representations that captures the empty territories with a complexity determined by the size of the alpha balls. With a fixed complexity, EmT returns the representation that contains the most points within the special class of representations. This method is limited to finding 1D holes in 2D data and 2D holes in 3D data, and is illustrated on simulation datasets of a homogeneous Poisson point process in 2D and a uniform sampling in 3D. Furthermore, the method is applied to a 2D cell tower location geography dataset and 3D Sloan Digital Sky Survey (SDSS) galaxy dataset, where it works well in capturing the empty territories.
References:
[1] |
S. Asaeedi, F. Didehvar and A. Mohades,
α-concave hull, a generalization of convex hull, Theoretical Computer Science, 702 (2017), 48-59.
doi: 10.1016/j.tcs.2017.08.014. |
[2] |
P. Bendich, J. S. Marron, E. Miller, A. Pieloch and S. Skwerer,
Persistent homology analysis of brain artery trees, The annals of applied statistics, 10 (2016), 198-218.
doi: 10.1214/15-AOAS886. |
[3] |
G. Carlsson, A. Zomorodian, A. Collins and L. J. Guibas,
Persistence barcodes for shapes, International Journal of Shape Modeling, 11 (2005), 149-187.
|
[4] |
F. Chazal, B. Fasy, F. Lecci, B. Michel, A. Rinaldo, A. Rinaldo and L. Wasserman, Robust topological inference: Distance to a measure and kernel distance, The Journal of Machine Learning Research, 18 (2017), Paper No. 159, 40 pp. |
[5] |
S. N. Chiu, D. Stoyan, W. S. Kendall and J. Mecke, Stochastic Geometry and Its Applications, John Wiley & Sons, 2013.
doi: 10.1002/9781118658222. |
[6] |
D. Cohen-Steiner, H. Edelsbrunner and J. Harer,
Stability of persistence diagrams, Discrete & Computational Geometry, 37 (2007), 103-120.
doi: 10.1007/s00454-006-1276-5. |
[7] |
D. Kahle and H. Wickham,
ggmap: Spatial Visualization with ggplot2, The R Journal, 5 (2013), 144-161.
|
[8] |
V. De Silva and G. E. Carlsson,
Topological estimation using witness complexes., SPBG, 4 (2004), 157-166.
|
[9] |
Edelsbrunner, Letscher and Zomorodian, Topological persistence and simplification, Discrete & Computational Geometry, 28 (2002), 511–533, URL https://doi.org/10.1007/s00454-002-2885-2. |
[10] |
H. Edelsbrunner and J. Harer,
Persistent homology-a survey, Contemporary Mathematics, 453 (2008), 257-282.
doi: 10.1090/conm/453/08802. |
[11] |
H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, American Mathematical Soc., 2010. |
[12] |
H. Edelsbrunner, D. Kirkpatrick and R. Seidel,
On the shape of a set of points in the plane, IEEE Transactions on information theory, 29 (1983), 551-559.
doi: 10.1109/TIT.1983.1056714. |
[13] |
H. Edelsbrunner and D. Morozov, Persistent homology: Theory and practice, European Congress of Mathematics, 31–50, Eur. Math. Soc., Zürich, 2013. |
[14] |
H. Edelsbrunner and E. P. Mücke,
Three-dimensional alpha shapes, ACM Transactions on Graphics (TOG), 13 (1994), 43-72.
|
[15] |
S. Emrani, T. Gentimis and H. Krim,
Persistent homology of delay embeddings and its application to wheeze detection, IEEE Signal Processing Letters, 21 (2014), 459-463.
|
[16] |
B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan and A. Singh et al.,
Confidence sets for persistence diagrams, The Annals of Statistics, 42 (2014), 2301-2339.
doi: 10.1214/14-AOS1252. |
[17] |
A. Galton, Pareto-optimality of cognitively preferred polygonal hulls for dot patterns, in Spatial Cognition VI. Learning, Reasoning, and Talking about Space (eds. C. Freksa, N. S. Newcombe, P. Gärdenfors and S. Wölfl), Springer Berlin Heidelberg, Berlin, Heidelberg, 45 (2008), 409–425. |
[18] |
R. Ghrist,
Barcodes: the persistent topology of data, Bulletin of the American Mathematical Society, 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[19] |
J. A. Hartigan and M. A. Wong,
Algorithm as 136: A k-means clustering algorithm, Journal of the Royal Statistical Society. Series C (Applied Statistics), 28 (1979), 100-108.
|
[20] |
Y. Hoffman, O. Metuki, G. Yepes, S. Gottlöber, J. E. Forero-Romero, N. I. Libeskind and A. Knebe,
A kinematic classification of the cosmic web, Monthly Notices of the Royal Astronomical Society, 425 (2012), 2049-2057.
|
[21] |
V. Icke, R. Weygaert et al., The galaxy distribution as a voronoi foam, Quarterly Journal of the Royal Astronomical Society, 32 (1991), 85. |
[22] |
F.-S. Kitaura and R. E. Angulo,
Linearization with cosmological perturbation theory, Monthly Notices of the Royal Astronomical Society, 425 (2012), 2443-2454.
|
[23] |
S. Lloyd,
Least squares quantization in pcm, IEEE Transactions on Information Theory, 28 (1982), 129-137.
doi: 10.1109/TIT.1982.1056489. |
[24] |
J. MacQueen et al., Some methods for classification and analysis of multivariate observations, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, Oakland, CA, USA, 1967,281–297. |
[25] |
A. Moreira and M. Y. Santos, Concave hull: A k-nearest neighbours approach for the computation of the region occupied by a set of points. |
[26] |
M. C. Neyrinck,
ZOBOV: A parameter-free void-finding algorithm, Monthly Notices of the Royal Astronomical Society, 386 (2008), 2101-2109.
|
[27] |
P. Peebles, The void phenomenon, The Astrophysical Journal, 557 (2001), 495. |
[28] |
A. Pisani, P. Sutter, N. Hamaus, E. Alizadeh, R. Biswas, B. D. Wandelt and C. M. Hirata, Counting voids to probe dark energy, Physical Review D, 92 (2015), 083531. |
[29] |
R. R. Rojas, M. S. Vogeley, F. Hoyle and J. Brinkmann, Photometric properties of void galaxies in the sloan digital sky survey, The Astrophysical Journal, 617 (2004), 50. |
[30] |
T. Sousbie,
The persistent cosmic web and its filamentary structure–i. theory and implementation, Monthly Notices of the Royal Astronomical Society, 414 (2011), 350-383.
|
[31] |
M. A. Strauss, D. H. Weinberg, R. H. Lupton, V. K. Narayanan, J. Annis, M. Bernardi, M. Blanton, S. Burles, A. Connolly, J. Dalcanton et al., Spectroscopic target selection in the Sloan Digital Sky Survey: The main galaxy sample, The Astronomical Journal, 124 (2002), 1810. |
[32] |
P. Sutter, G. Lavaux, B. D. Wandelt and D. H. Weinberg, A public void catalog from the sdss dr7 galaxy redshift surveys based on the watershed transform, The Astrophysical Journal, 761 (2012), 44. |
[33] |
R. van de Weygaert,
Fragmenting the universe. 3: The constructions and statistics of 3-d voronoi tessellations, Astronomy and Astrophysics, 283 (1994), 361-406.
|
[34] |
L. Vietoris,
Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Mathematische Annalen, 97 (1927), 454-472.
doi: 10.1007/BF01447877. |
[35] |
H. Wagner, C. Chen and E. Vuçini, Efficient computation of persistent homology for cubical data, in Topological Methods in Data Analysis and Visualization II, Springer, 2012, 91–106.
doi: 10.1007/978-3-642-23175-9_7. |
[36] |
L. Wasserman,
Topological data analysis, Annual Review of Statistics and Its Application, 5 (2018), 501-535.
doi: 10.1146/annurev-statistics-031017-100045. |
[37] |
K. Xia and G.-W. Wei,
Persistent homology analysis of protein structure, flexibility, and folding, International Journal for Numerical Methods in Biomedical Engineering, 30 (2014), 814-844.
doi: 10.1002/cnm.2655. |
[38] |
X. Xu, J. Cisewski-Kehe, S. B. Green and D. Nagai, Finding cosmic voids and filament loops using topological data analysis, Astronomy and Computing. |
[39] |
X. Zhu, Persistent homology: An introduction and a new text representation for natural language processing., in IJCAI, 2013, 1953–1959. |
[40] |
A. Zomorodian,
Fast construction of the Vietoris-Rips complex, Computers & Graphics, 34 (2010), 263-271.
|
[41] |
A. Zomorodian and G. Carlsson,
Computing persistent homology, Discrete & Computational Geometry, 33 (2005), 249-274.
doi: 10.1007/s00454-004-1146-y. |
show all references
References:
[1] |
S. Asaeedi, F. Didehvar and A. Mohades,
α-concave hull, a generalization of convex hull, Theoretical Computer Science, 702 (2017), 48-59.
doi: 10.1016/j.tcs.2017.08.014. |
[2] |
P. Bendich, J. S. Marron, E. Miller, A. Pieloch and S. Skwerer,
Persistent homology analysis of brain artery trees, The annals of applied statistics, 10 (2016), 198-218.
doi: 10.1214/15-AOAS886. |
[3] |
G. Carlsson, A. Zomorodian, A. Collins and L. J. Guibas,
Persistence barcodes for shapes, International Journal of Shape Modeling, 11 (2005), 149-187.
|
[4] |
F. Chazal, B. Fasy, F. Lecci, B. Michel, A. Rinaldo, A. Rinaldo and L. Wasserman, Robust topological inference: Distance to a measure and kernel distance, The Journal of Machine Learning Research, 18 (2017), Paper No. 159, 40 pp. |
[5] |
S. N. Chiu, D. Stoyan, W. S. Kendall and J. Mecke, Stochastic Geometry and Its Applications, John Wiley & Sons, 2013.
doi: 10.1002/9781118658222. |
[6] |
D. Cohen-Steiner, H. Edelsbrunner and J. Harer,
Stability of persistence diagrams, Discrete & Computational Geometry, 37 (2007), 103-120.
doi: 10.1007/s00454-006-1276-5. |
[7] |
D. Kahle and H. Wickham,
ggmap: Spatial Visualization with ggplot2, The R Journal, 5 (2013), 144-161.
|
[8] |
V. De Silva and G. E. Carlsson,
Topological estimation using witness complexes., SPBG, 4 (2004), 157-166.
|
[9] |
Edelsbrunner, Letscher and Zomorodian, Topological persistence and simplification, Discrete & Computational Geometry, 28 (2002), 511–533, URL https://doi.org/10.1007/s00454-002-2885-2. |
[10] |
H. Edelsbrunner and J. Harer,
Persistent homology-a survey, Contemporary Mathematics, 453 (2008), 257-282.
doi: 10.1090/conm/453/08802. |
[11] |
H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, American Mathematical Soc., 2010. |
[12] |
H. Edelsbrunner, D. Kirkpatrick and R. Seidel,
On the shape of a set of points in the plane, IEEE Transactions on information theory, 29 (1983), 551-559.
doi: 10.1109/TIT.1983.1056714. |
[13] |
H. Edelsbrunner and D. Morozov, Persistent homology: Theory and practice, European Congress of Mathematics, 31–50, Eur. Math. Soc., Zürich, 2013. |
[14] |
H. Edelsbrunner and E. P. Mücke,
Three-dimensional alpha shapes, ACM Transactions on Graphics (TOG), 13 (1994), 43-72.
|
[15] |
S. Emrani, T. Gentimis and H. Krim,
Persistent homology of delay embeddings and its application to wheeze detection, IEEE Signal Processing Letters, 21 (2014), 459-463.
|
[16] |
B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan and A. Singh et al.,
Confidence sets for persistence diagrams, The Annals of Statistics, 42 (2014), 2301-2339.
doi: 10.1214/14-AOS1252. |
[17] |
A. Galton, Pareto-optimality of cognitively preferred polygonal hulls for dot patterns, in Spatial Cognition VI. Learning, Reasoning, and Talking about Space (eds. C. Freksa, N. S. Newcombe, P. Gärdenfors and S. Wölfl), Springer Berlin Heidelberg, Berlin, Heidelberg, 45 (2008), 409–425. |
[18] |
R. Ghrist,
Barcodes: the persistent topology of data, Bulletin of the American Mathematical Society, 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[19] |
J. A. Hartigan and M. A. Wong,
Algorithm as 136: A k-means clustering algorithm, Journal of the Royal Statistical Society. Series C (Applied Statistics), 28 (1979), 100-108.
|
[20] |
Y. Hoffman, O. Metuki, G. Yepes, S. Gottlöber, J. E. Forero-Romero, N. I. Libeskind and A. Knebe,
A kinematic classification of the cosmic web, Monthly Notices of the Royal Astronomical Society, 425 (2012), 2049-2057.
|
[21] |
V. Icke, R. Weygaert et al., The galaxy distribution as a voronoi foam, Quarterly Journal of the Royal Astronomical Society, 32 (1991), 85. |
[22] |
F.-S. Kitaura and R. E. Angulo,
Linearization with cosmological perturbation theory, Monthly Notices of the Royal Astronomical Society, 425 (2012), 2443-2454.
|
[23] |
S. Lloyd,
Least squares quantization in pcm, IEEE Transactions on Information Theory, 28 (1982), 129-137.
doi: 10.1109/TIT.1982.1056489. |
[24] |
J. MacQueen et al., Some methods for classification and analysis of multivariate observations, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, Oakland, CA, USA, 1967,281–297. |
[25] |
A. Moreira and M. Y. Santos, Concave hull: A k-nearest neighbours approach for the computation of the region occupied by a set of points. |
[26] |
M. C. Neyrinck,
ZOBOV: A parameter-free void-finding algorithm, Monthly Notices of the Royal Astronomical Society, 386 (2008), 2101-2109.
|
[27] |
P. Peebles, The void phenomenon, The Astrophysical Journal, 557 (2001), 495. |
[28] |
A. Pisani, P. Sutter, N. Hamaus, E. Alizadeh, R. Biswas, B. D. Wandelt and C. M. Hirata, Counting voids to probe dark energy, Physical Review D, 92 (2015), 083531. |
[29] |
R. R. Rojas, M. S. Vogeley, F. Hoyle and J. Brinkmann, Photometric properties of void galaxies in the sloan digital sky survey, The Astrophysical Journal, 617 (2004), 50. |
[30] |
T. Sousbie,
The persistent cosmic web and its filamentary structure–i. theory and implementation, Monthly Notices of the Royal Astronomical Society, 414 (2011), 350-383.
|
[31] |
M. A. Strauss, D. H. Weinberg, R. H. Lupton, V. K. Narayanan, J. Annis, M. Bernardi, M. Blanton, S. Burles, A. Connolly, J. Dalcanton et al., Spectroscopic target selection in the Sloan Digital Sky Survey: The main galaxy sample, The Astronomical Journal, 124 (2002), 1810. |
[32] |
P. Sutter, G. Lavaux, B. D. Wandelt and D. H. Weinberg, A public void catalog from the sdss dr7 galaxy redshift surveys based on the watershed transform, The Astrophysical Journal, 761 (2012), 44. |
[33] |
R. van de Weygaert,
Fragmenting the universe. 3: The constructions and statistics of 3-d voronoi tessellations, Astronomy and Astrophysics, 283 (1994), 361-406.
|
[34] |
L. Vietoris,
Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Mathematische Annalen, 97 (1927), 454-472.
doi: 10.1007/BF01447877. |
[35] |
H. Wagner, C. Chen and E. Vuçini, Efficient computation of persistent homology for cubical data, in Topological Methods in Data Analysis and Visualization II, Springer, 2012, 91–106.
doi: 10.1007/978-3-642-23175-9_7. |
[36] |
L. Wasserman,
Topological data analysis, Annual Review of Statistics and Its Application, 5 (2018), 501-535.
doi: 10.1146/annurev-statistics-031017-100045. |
[37] |
K. Xia and G.-W. Wei,
Persistent homology analysis of protein structure, flexibility, and folding, International Journal for Numerical Methods in Biomedical Engineering, 30 (2014), 814-844.
doi: 10.1002/cnm.2655. |
[38] |
X. Xu, J. Cisewski-Kehe, S. B. Green and D. Nagai, Finding cosmic voids and filament loops using topological data analysis, Astronomy and Computing. |
[39] |
X. Zhu, Persistent homology: An introduction and a new text representation for natural language processing., in IJCAI, 2013, 1953–1959. |
[40] |
A. Zomorodian,
Fast construction of the Vietoris-Rips complex, Computers & Graphics, 34 (2010), 263-271.
|
[41] |
A. Zomorodian and G. Carlsson,
Computing persistent homology, Discrete & Computational Geometry, 33 (2005), 249-274.
doi: 10.1007/s00454-004-1146-y. |












1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
original | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
k=60 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 |
k=120 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
k=180 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |
k=240 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
k=300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
k=360 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
original | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
k=60 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 |
k=120 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
k=180 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |
k=240 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
k=300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
k=360 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
[1] |
Hassan Abdallah, Adam Regalski, Mohammad Behzad Kang, Maria Berishaj, Nkechi Nnadi, Asadur Chowdury, Vaibhav A. Diwadkar, Andrew Salch. Statistical inference for persistent homology applied to simulated fMRI time series data. Foundations of Data Science, 2022 doi: 10.3934/fods.2022014 |
[2] |
George Siopsis. Quantum topological data analysis with continuous variables. Foundations of Data Science, 2019, 1 (4) : 419-431. doi: 10.3934/fods.2019017 |
[3] |
Tyrus Berry, Timothy Sauer. Consistent manifold representation for topological data analysis. Foundations of Data Science, 2019, 1 (1) : 1-38. doi: 10.3934/fods.2019001 |
[4] |
Erik Carlsson, John Gunnar Carlsson, Shannon Sweitzer. Applying topological data analysis to local search problems. Foundations of Data Science, 2022 doi: 10.3934/fods.2022006 |
[5] |
Jelena Grbić, Jie Wu, Kelin Xia, Guo-Wei Wei. Aspects of topological approaches for data science. Foundations of Data Science, 2022, 4 (2) : 165-216. doi: 10.3934/fods.2022002 |
[6] |
Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of borehole seismic data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022026 |
[7] |
Pooja Bansal, Aparna Mehra. Integrated dynamic interval data envelopment analysis in the presence of integer and negative data. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1339-1363. doi: 10.3934/jimo.2021023 |
[8] |
Jochen Abhau, Oswin Aichholzer, Sebastian Colutto, Bernhard Kornberger, Otmar Scherzer. Shape spaces via medial axis transforms for segmentation of complex geometry in 3D voxel data. Inverse Problems and Imaging, 2013, 7 (1) : 1-25. doi: 10.3934/ipi.2013.7.1 |
[9] |
Karol Mikula, Jozef Urbán, Michal Kollár, Martin Ambroz, Ivan Jarolímek, Jozef Šibík, Mária Šibíková. An automated segmentation of NATURA 2000 habitats from Sentinel-2 optical data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1017-1032. doi: 10.3934/dcdss.2020348 |
[10] |
Esther Klann, Ronny Ramlau, Wolfgang Ring. A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. Inverse Problems and Imaging, 2011, 5 (1) : 137-166. doi: 10.3934/ipi.2011.5.137 |
[11] |
Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2&3) : 139-161. doi: 10.3934/bdia.2016001 |
[12] |
Pankaj Sharma, David Baglee, Jaime Campos, Erkki Jantunen. Big data collection and analysis for manufacturing organisations. Big Data & Information Analytics, 2017, 2 (2) : 127-139. doi: 10.3934/bdia.2017002 |
[13] |
Runqin Hao, Guanwen Zhang, Dong Li, Jie Zhang. Data modeling analysis on removal efficiency of hexavalent chromium. Mathematical Foundations of Computing, 2019, 2 (3) : 203-213. doi: 10.3934/mfc.2019014 |
[14] |
Habibe Zare Haghighi, Sajad Adeli, Farhad Hosseinzadeh Lotfi, Gholam Reza Jahanshahloo. Revenue congestion: An application of data envelopment analysis. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1311-1322. doi: 10.3934/jimo.2016.12.1311 |
[15] |
Mahdi Mahdiloo, Abdollah Noorizadeh, Reza Farzipoor Saen. Developing a new data envelopment analysis model for customer value analysis. Journal of Industrial and Management Optimization, 2011, 7 (3) : 531-558. doi: 10.3934/jimo.2011.7.531 |
[16] |
Jiang Xie, Junfu Xu, Celine Nie, Qing Nie. Machine learning of swimming data via wisdom of crowd and regression analysis. Mathematical Biosciences & Engineering, 2017, 14 (2) : 511-527. doi: 10.3934/mbe.2017031 |
[17] |
Mohammad Afzalinejad, Zahra Abbasi. A slacks-based model for dynamic data envelopment analysis. Journal of Industrial and Management Optimization, 2019, 15 (1) : 275-291. doi: 10.3934/jimo.2018043 |
[18] |
Andreas Chirstmann, Qiang Wu, Ding-Xuan Zhou. Preface to the special issue on analysis in machine learning and data science. Communications on Pure and Applied Analysis, 2020, 19 (8) : i-iii. doi: 10.3934/cpaa.2020171 |
[19] |
Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026 |
[20] |
Jingmei Zhou, Xiangmo Zhao, Xin Cheng, Zhigang Xu. Visualization analysis of traffic congestion based on floating car data. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1423-1433. doi: 10.3934/dcdss.2015.8.1423 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]