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Exploring uses of persistent homology for statistical analysis of landmark-based shape data. (English) Zbl 1203.62116

Summary: A method for the use of persistent homology in the statistical analysis of landmark-based shape data is given. Three-dimensional landmark configurations are used as input for separate filtrations, persistent homology is performed, and persistence diagrams are obtained. Groups of configurations are compared using distances between persistence diagrams combined with dimensionality reduction methods. A three-dimensional landmark-based data set is used from a longitudinal orthodontic study, and the persistent homology method is able to distinguish clinically relevant treatment effects. Comparisons are made with the traditional landmark-based statistical shape analysis methods of I. L. Dryden and K. V. Mardia [Statistical shape analysis. Chichester: Wiley (1998; Zbl 0901.62072)], and Euclidean distance matrix analysis.

MSC:

62H99 Multivariate analysis
62P10 Applications of statistics to biology and medical sciences; meta analysis
55U10 Simplicial sets and complexes in algebraic topology
62H11 Directional data; spatial statistics

Citations:

Zbl 0901.62072

Software:

LFM-Pro
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Full Text: DOI

References:

[1] Borg, I.; Croenen, J. F., Modern Multidimensional Scaling (2005), Springer
[2] P. Bubenik, G. Carlsson, P. Kim, Z. Luo, Statistical topology via morse theory, persistence, and nonparmetric estimation, Contemp. Math. (in press).; P. Bubenik, G. Carlsson, P. Kim, Z. Luo, Statistical topology via morse theory, persistence, and nonparmetric estimation, Contemp. Math. (in press). · Zbl 1196.62041
[3] Bubenik, P.; Kim, P., A statistical approach to persistent homology, Homology Homotopy Appl., 9, 2, 337-362 (2007) · Zbl 1136.55004
[4] Carlsson, G., Topology and data, Bull. Amer. Math. Soc., 46, 2, 255-308 (2009) · Zbl 1172.62002
[5] Chung, M.; Bubenik, P.; Kim, P., Persistence diagrams of cortical surface data, (LNCS: Proceedings of IPMI 2009, vol. 5636 (2009)), 386-397
[6] Cohen-Steiner, D.; Edelsbrunner, H.; Harer, J.; Mileyko, Y., Lipschitz functions have \(L_p\)-stable persistence, Found. Comput. Math, 10, 2, 127-139 (2010) · Zbl 1192.55007
[7] Dequéant, M.-L.; Ahnert, S.; Edelsbrunner, H.; Fink, T. M.A.; Glynn, E. F., Comparison of pattern detection methods in microarray time series of the segmentation clock, PLoS ONE, 3, 8, e2856 (2008)
[8] V. de Silva, et al. Plex, online, 2000-2003. Available at: http://comptop.stanford.edu/programs/plex/; V. de Silva, et al. Plex, online, 2000-2003. Available at: http://comptop.stanford.edu/programs/plex/
[9] Dryden, I. L.; Mardia, K. V., Statistical Shape Analysis (1998), John Wiley and Sons · Zbl 0901.62072
[10] Edelsbrunner, H.; Harer, J., Persistent homology—a survey, Contemp. Math., 453, 257-282 (2008) · Zbl 1145.55007
[11] Edelsbrunner, H.; Letscher, D.; Zomorodian, A., Topological persistence and simplification, Discrete Comput. Geom., 28, 4, 511-533 (2002) · Zbl 1011.68152
[12] Ghrist, R., Barcodes: the persistent topology of data, Bull. Amer. Math. Soc., 45, 1, 61-75 (2007) · Zbl 1391.55005
[13] Ghrist, R.; de Silva, V., Homological sensor networks, Notic. Amer. Math. Soc., 54, 1, 10-17 (2007) · Zbl 1142.94006
[14] Hatcher, A., Algebraic Topology (2001), Cambridge University Press
[15] Heo, G.; Small, C., Form representations and means for landmarks: a survey and comparative study, Comput. Vis. Image Underst., 102, 2, 188-203 (2006)
[16] Huckemann, S.; Hotz, T.; Munk, A., Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric lie group actions, Statist. Sinica, 20, 1, 1-100 (2010) · Zbl 1180.62087
[17] Lele, S. R.; Richtsmeier, J. T., An Invariant Approach to Statistical Analysis of Shapes (2001), Chapman and Hall/CRC Press · Zbl 0983.62030
[18] Lovasz, L.; Plummer, M. D., Matching Theory (1986), Elsevier Science Publishers B.V. · Zbl 0618.05001
[19] Sacan, A.; Ozturk, O.; Ferhatosmanoglu, H.; Wang, Y., LFM-Pro: a tool for detecting significant local structural sites in proteins, Bioinform., 23, 6, 709-716 (2007)
[20] Tenenbaum, J. B.; de Silva, V.; Langford, J. C., A global geometric framework for nonlinear dimensionality reduction, Science, 290, 5500, 2319-2323 (2000)
[21] Zomorodian, A., Topology for Computing (2005), Cambridge University Press · Zbl 1065.68001
[22] Zomorodian, A.; Carlsson, G., Computing persistent homology, Discrete Comput. Geom., 33, 2, 249-274 (2005) · Zbl 1069.55003
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