1932

Abstract

Topological data analysis (TDA) can broadly be described as a collection of data analysis methods that find structure in data. These methods include clustering, manifold estimation, nonlinear dimension reduction, mode estimation, ridge estimation and persistent homology. This paper reviews some of these methods.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-statistics-031017-100045
2018-03-07
2024-04-18
Loading full text...

Full text loading...

/deliver/fulltext/statistics/5/1/annurev-statistics-031017-100045.html?itemId=/content/journals/10.1146/annurev-statistics-031017-100045&mimeType=html&fmt=ahah

Literature Cited

  1. Adcock A, Rubin D, Carlsson G. 2014. Classification of hepatic lesions using the matching metric. Comput. Vis. Image Underst. 121:36–42 [Google Scholar]
  2. Adler RJ, Taylor JE. 2009. Random Fields and Geometry New York: Springer
  3. Arai M, Brandt V, Dabaghian Y. 2014. The effects of theta precession on spatial learning and simplicial complex dynamics in a topological model of the hippocampal spatial map. PLOS Comput. Biol. 10:6e1003651 [Google Scholar]
  4. Arias-Castro E, Chen G, Lerman G. 2011. Spectral clustering based on local linear approximations. Electron. J. Stat. 5:1537–87 [Google Scholar]
  5. Arias-Castro E, Mason D, Pelletier B. 2015. On the estimation of the gradient lines of a density and the consistency of the mean-shift algorithm. J. Mach. Learn. Res. 17:1–28 [Google Scholar]
  6. Azizyan M, Chen YC, Singh A, Wasserman L. 2015. Risk bounds for mode clustering. arXiv1505.00482v1 [math.ST]
  7. Babichev A, Dabaghian Y. 2016. Persistent memories in transient networks. arXiv1602.00681 [qbio.NC]
  8. Balakrishnan S, Narayanan S, Rinaldo A, Singh A, Wasserman L. 2013. Cluster trees on manifolds. Advances in Neural Information Processing Systems 26 (NIPS 2013), CJC Burges, L Bottou, M Welling, Z Ghahramani, KQ Weinberger 2679–87 Red Hook, NY: Curran [Google Scholar]
  9. Basso E, Arai M, Dabaghian Y. 2016. Gamma synchronization of the hippocampal spatial map—topological model. arXiv1603.06248 [qbio.NC]
  10. Belkin M, Niyogi P. 2001. Laplacian eigenmaps and spectral techniques for embedding and clustering. Advances in Neural Information Processing Systems 14 (NIPS 2011) TG Dietterich, S Becker, Z Ghahramani 585–91 Cambridge, MA: MIT Press [Google Scholar]
  11. Belkin M, Niyogi P. 2003. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15:1373–96 [Google Scholar]
  12. Bendich P, Cohen-Steiner D, Edelsbrunner H, Harer J, Morozov D. 2007. Inferring local homology from sampled stratified spaces. 48th Annu. IEEE Symp. Found. Comput. Sci. 2007 (FOCS '07)536–46 New York: IEEE [Google Scholar]
  13. Bendich P, Edelsbrunner H, Kerber M. 2010. Computing robustness and persistence for images. IEEE Trans. Visualization Comput. Gr. 16:1251–60 [Google Scholar]
  14. Bendich P, Marron JS, Miller E, Pieloch A, Skwerer S. 2016. Persistent homology analysis of brain artery trees. Ann. Appl. Stat. 10:198–218 [Google Scholar]
  15. Bobrowski O, Mukherjee S, Taylor JE. 2014. Topological consistency via kernel estimation. arXiv1407.5272 [math.ST]
  16. Bonis T, Ovsjanikov M, Oudot S, Chazal F. 2016. Persistence-based pooling for shape pose recognition. Computational Topology in Image Context A Bac, JL Mari 19–29 New York: Springer [Google Scholar]
  17. Brown J, Gedeon T. 2012. Structure of the afferent terminals in terminal ganglion of a cricket and persistent homology. PLOS ONE 7:5e37278 [Google Scholar]
  18. Bubenik P. 2015. Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16:77–102 [Google Scholar]
  19. Cadre B. 2006. Kernel estimation of density level sets. J. Multivar. Anal. 97:999–1023 [Google Scholar]
  20. Carreira-Perpiñán MA. 2010. The elastic embedding algorithm for dimensionality reduction. Proceedings of the 27th International Conference on International Conference on Machine Learning (ICML 2010) ed. J Fürnkranz, T Joachims, pp. 167–74. Madison, WI: Omnipress
  21. Carrière M, Oudot SY, Ovsjanikov M. 2015. Stable topological signatures for points on 3D shapes. Comput. Gr. Forum 34:1–12 [Google Scholar]
  22. Carstens C, Horadam K. 2013. Persistent homology of collaboration networks. Math. Probl. Eng. 2013:815035 [Google Scholar]
  23. Cassidy B, Rae C, Solo V. 2015. Brain activity: conditional dissimilarity and persistent homology. 2015 IEEE 12th Int. Symp. Biomed. Imaging1356–59 New York: IEEE [Google Scholar]
  24. Chacón JE. 2012. Clusters and water flows: a novel approach to modal clustering through Morse theory. arXiv1212.1384 [math.ST]
  25. Chacón JE. 2015. A population background for nonparametric density-based clustering. Stat. Sci. 30:518–32 [Google Scholar]
  26. Chacón JE, Duong T. 2013. Data-driven density derivative estimation, with applications to nonparametric clustering and bump hunting. Electron. J. Stat. 7:499–532 [Google Scholar]
  27. Chaudhuri K, Dasgupta S. 2010. Rates of convergence for the cluster tree. Advances in Neural Information Processing Systems 23 (NIPS 2010) JD Lafferty, CKI Williams, J Shawe-Taylor, RS Zemel, A Culotta 343–51 Red Hook, NY: Curran [Google Scholar]
  28. Chaudhuri P, Marron JS. 1999. Sizer for exploration of structures in curves. J. Am. Stat. Assoc. 94:807–23 [Google Scholar]
  29. Chaudhuri P, Marron JS. 2000. Scale space view of curve estimation. Ann. Stat. 28:408–28 [Google Scholar]
  30. Chazal F, Cohen-Steiner D, Lieutier A. 2009. A sampling theory for compact sets in Euclidean space. Discrete Comput. Geometry 41:461–79 [Google Scholar]
  31. Chazal F, Cohen-Steiner D, Mérigot Q. 2011. Geometric inference for probability measures. Found. Comput. Math. 11:733–51 [Google Scholar]
  32. Chazal F, Fasy BT, Lecci F, Michel B, Rinaldo A, Wasserman L. 2014.a Robust topological inference: Distance to a measure and kernel distance. arXiv1412.7197 [math.ST]
  33. Chazal F, Glisse M, Labruère C, Michel B. 2014.b Convergence rates for persistence diagram estimation in topological data analysis. Proc. Mach. Learn. Res. 32:163–71 [Google Scholar]
  34. Chazal F, Guibas LJ, Oudot SY, Skraba P. 2013. Persistence-based clustering in Riemannian manifolds. J. ACM 60:41 [Google Scholar]
  35. Chazal F, Massart P, Michel B. 2015. Rates of convergence for robust geometric inference. arXiv1505.07602 [math.ST]
  36. Chen YC, Genovese CR, Wasserman L. 2015.a Asymptotic theory for density ridges. Ann. Stat. 43:1896–928 [Google Scholar]
  37. Chen YC, Genovese CR, Wasserman L. 2015.b Density level sets: asymptotics, inference, and visualization. arXiv1504.05438 [stat.ME]
  38. Chen YC, Ho S, Freeman PE, Genovese CR, Wasserman L. 2015.c Cosmic web reconstruction through density ridges: method and algorithm. MNRAS 454:1140–56 [Google Scholar]
  39. Chen YC, Ho S, Tenneti A, Mandelbaum R, Croft R. et al. 2015.d Investigating galaxy-filament alignments in hydrodynamic simulations using density ridges. MNRAS 454:3341–50 [Google Scholar]
  40. Chen YC, Kim J, Balakrishnan S, Rinaldo A, Wasserman L. 2016. Statistical inference for cluster trees. arXiv1605.06416 [math.ST]
  41. Chen Z, Gomperts SN, Yamamoto J, Wilson MA. 2014. Neural representation of spatial topology in the rodent hippocampus. Neural Comput 26:1–39 [Google Scholar]
  42. Cheng Y. 1995. Mean shift, mode seeking, and clustering. IEEE Trans. Pattern Anal. Mach. Intell. 17:790–99 [Google Scholar]
  43. Choi H, Kim YK, Kang H, Lee H, Im HJ. et al. 2014. Abnormal metabolic connectivity in the pilocarpine-induced epilepsy rat model: a multiscale network analysis based on persistent homology. NeuroImage 99:226–36 [Google Scholar]
  44. Chung MK, Bubenik P, Kim PT. 2009. Persistence diagrams of cortical surface data. Inf. Proc. Med. Imaging 21:386–97 [Google Scholar]
  45. Coifman RR, Lafon S. 2006. Diffusion maps. Appl. Comput. Harmon. Anal. 21:5–30 [Google Scholar]
  46. Comaniciu D, Meer P. 2002. Mean shift: a robust approach toward feature space analysis. IEEE Trans. Pattern Anal. Mach. Intell. 24:603–19 [Google Scholar]
  47. Costa JA, Hero AO. 2004. Geodesic entropic graphs for dimension and entropy estimation in manifold learning. IEEE Trans. Signal Proc. 52:2210–21 [Google Scholar]
  48. Cuevas A. 2009. Set estimation: another bridge between statistics and geometry. Bol. Estad. Investig. Oper. 25:71–85 [Google Scholar]
  49. Cuevas A, Febrero M, Fraiman R. 2001. Cluster analysis: a further approach based on density estimation. Comput. Stat. Data Anal. 36:441–59 [Google Scholar]
  50. Curto C. 2016. What can topology tell us about the neural code. Bull. Am. Math. Soc. 54:63–78 [Google Scholar]
  51. Curto C, Gross E, Jeffries J, Morrison K, Omar M. et al. 2015. What makes a neural code convex?. arXiv1508.00150 [q-bio.NC]
  52. Curto C, Itskov V. 2008. Cell groups reveal structure of stimulus space. PLOS Comput. Biol. 4:e1000205 [Google Scholar]
  53. Curto C, Itskov V, Veliz-Cuba A, Youngs N. 2013. The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes. Bull. Math. Biol. 75:1571–611 [Google Scholar]
  54. Curto C, Youngs N. 2015. Neural ring homomorphisms and maps between neural codes. arXiv1511.00255 [math.NC]
  55. Dabaghian Y. 2015. Geometry of spatial memory replay. arXiv1508.06579 [q-bio.NC]
  56. Dabaghian Y, Brandt VL, Frank LM. 2014. Reconceiving the hippocampal map as a topological template. eLife 3:e03476 [Google Scholar]
  57. Dabaghian Y, Cohn AG, Frank L. 2011. Topological coding in the hippocampus. Computational Modeling and Simulation of Intellect: Current State and Future Perspectives Y Dabaghian, AG Cohn, L Frank 290–320 Hershey, PA: IGI Global [Google Scholar]
  58. Dabaghian Y, Mémoli F, Frank L, Carlsson G. 2012. A topological paradigm for hippocampal spatial map formation using persistent homology. PLOS Comput. Biol. 8:e1002581 [Google Scholar]
  59. De'ath G. 1999. Extended dissimilarity: a method of robust estimation of ecological distances from high beta diversity data. Plant Ecol 144:191–99 [Google Scholar]
  60. Devroye L, Wise GL. 1980. Detection of abnormal behavior via nonparametric estimation of the support. SIAM J. Appl. Math. 38:480–88 [Google Scholar]
  61. Dlotko P, Hess K, Levi R, Nolte M, Reimann M. et al. 2016. Topological analysis of the connectome of digital reconstructions of neural microcircuits. arXiv1601.01580 [q-bio.NC]
  62. Eberly D. 1996. Ridges in Image and Data Analysis New York: Springer
  63. Edelsbrunner H, Harer J. 2008. Persistent homology—a survey. Contemp. Math. 453:257–82 [Google Scholar]
  64. Edelsbrunner H, Harer J. 2010. Computational Topology: An Introduction Washington, DC: Am. Math. Soc.
  65. Edelsbrunner H, Letscher D, Zomorodian A. 2002. Topological persistence and simplification. Discrete Comput. Geom. 28:511–33 [Google Scholar]
  66. Eldridge J, Wang Y, Belkin M. 2015. Beyond Hartigan consistency: Merge distortion metric for hierarchical clustering. arXiv1506.06422 [stat.ML]
  67. Ellis SP, Klein A. 2014. Describing high-order statistical dependence using concurrence topology, with application to functional MRI brain data. Homol. Homotopy Appl. 16:245–64 [Google Scholar]
  68. Fasy BT, Kim J, Lecci F, Maria C. 2014.a Introduction to the R package TDA. arXiv:1411.1830 [cs.MS].
  69. Fasy BT, Lecci F, Rinaldo A, Wasserman L, Balakrishnan S. et al. 2014.b Confidence sets for persistence diagrams. Ann. Stat. 42:2301–39 [Google Scholar]
  70. Genovese CR, Perone-Pacifico M, Verdinelli I, Wasserman L. 2012.a Manifold estimation and singular deconvolution under Hausdorff loss. Ann. Stat. 40:941–63 [Google Scholar]
  71. Genovese CR, Perone-Pacifico M, Verdinelli I, Wasserman L. 2012.b Minimax manifold estimation. J. Mach. Learn. Res. 13:1263–91 [Google Scholar]
  72. Genovese CR, Perone-Pacifico M, Verdinelli I, Wasserman L. 2016. Non-parametric inference for density modes. J. R. Stat. Soc. B 78:99–126 [Google Scholar]
  73. Genovese CR, Perone-Pacifico M, Verdinelli I, Wasserman L. 2014. Nonparametric ridge estimation. Ann. Stat. 42:1511–45 [Google Scholar]
  74. Giusti C, Ghrist R, Bassett DS. 2016. Two's company, three (or more) is a simplex: algebraic-topological tools for understanding higher-order structure in neural data. J. Comput. Neurosci. 41: [Google Scholar]
  75. Giusti C, Itskov V. 2013. A no-go theorem for one-layer feedforward networks. Neural Comput 26:2527–40 [Google Scholar]
  76. Giusti C, Pastalkova E, Curto C, Itskov V. 2015. Clique topology reveals intrinsic geometric structure in neural correlations. PNAS 112:13455–60 [Google Scholar]
  77. Godtliebsen F, Marron J, Chaudhuri P. 2002. Significance in scale space for bivariate density estimation. J. Comput. Gr. Stat. 11:1–21 [Google Scholar]
  78. Guibas L, Morozov D, Mérigot Q. 2013. Witnessed k-distance. Discrete Comput. Geom. 49:22–45 [Google Scholar]
  79. Hartigan JA. 1975. Clustering Algorithms New York: Wiley
  80. Hartigan JA. 1981. Consistency of single linkage for high-density clusters. J. Am. Stat. Assoc. 76:388–94 [Google Scholar]
  81. Hatcher A. 2000. Algebraic Topology Cambridge, UK: Cambridge Univ. Press
  82. Hein M, Audibert JY. 2005. Intrinsic dimensionality estimation of submanifolds in Rd. Proc. 22nd Int. Conf. Mach. Learn (ICML 2005)289–96 New York: ACM [Google Scholar]
  83. Hoffman K, Babichev A, Dabaghian Y. 2016. Topological mapping of space in bat hippocampus. arXiv1601.04253 [q-bio.NC]
  84. Jeffs RA, Omar M, Suaysom N, Wachtel A, Youngs N. 2015. Sparse neural codes and convexity. arXiv1511.00283 [math.CO]
  85. Kanari L, Dłotko P, Scolamiero M, Levi R, Shillcock J. et al. 2016. Quantifying topological invariants of neuronal morphologies. arXiv1603.08432 [q-bio.NC]
  86. Kégl B. 2002. Intrinsic dimension estimation using packing numbers. Proceedings of the 15th International Conference on Neural Information Processing Systems (NIPS 2002) S Becker, S Thrun, K Obermayer 697–704 Cambridge, MA: MIT Press [Google Scholar]
  87. Kendall DG. 1984. Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16:81–121 [Google Scholar]
  88. Khalid A, Kim BS, Chung MK, Ye JC, Jeon D. 2014. Tracing the evolution of multi-scale functional networks in a mouse model of depression using persistent brain network homology. Neuroimage 101:351–63 [Google Scholar]
  89. Kim E, Kang H, Lee H, Lee HJ, Suh MW. et al. 2014. Morphological brain network assessed using graph theory and network filtration in deaf adults. Hear. Res. 315:88–98 [Google Scholar]
  90. Kim J, Rinaldo A, Wasserman L. 2016. Minimax rates for estimating the dimension of a manifold. arXiv1605.01011 [math.ST]
  91. Koltchinskii VI. 2000. Empirical geometry of multivariate data: a deconvolution approach. Ann. Stat. 28:591–629 [Google Scholar]
  92. Kovacev-Nikolic V, Bubenik P, Nikolić D, Heo G. 2016. Using persistent homology and dynamical distances to analyze protein binding. Stat. Appl. Genet. Mol. Biol. 15:19–38 [Google Scholar]
  93. Lee H, Chung MK, Kang H, Kim BN, Lee DS. 2011. Discriminative persistent homology of brain networks. Proc. 2011 IEEE Int. Symp. Biomed. Imaging Nano Macro, Chicago, IL841–44 New York: IEEE [Google Scholar]
  94. Lee JA, Verleysen M. 2007. Nonlinear Dimensionality Reduction New York: Springer
  95. Levina E, Bickel PJ. 2004. Maximum likelihood estimation of intrinsic dimension. Advances in Neural Information Processing Systems 17 (NIPS 2004) LK Saul, Y Weiss, L Bottou 777–784 Cambridge, MA: MIT Press [Google Scholar]
  96. Li C, Ovsjanikov M, Chazal F. 2014. Persistence-based structural recognition. Proc. IEEE Conf. Comput. Vis. Pattern Recognit2003–10 New York: IEEE [Google Scholar]
  97. Li J, Ray S, Lindsay B. 2007. A nonparametric statistical approach to clustering via mode identification. J. Mach. Learn. Res. 8:1687–723 [Google Scholar]
  98. Lienkaemper C, Shiu A, Woodstock Z. 2015. Obstructions to convexity in neural codes. arXiv1509.03328 [q-bio.NC]
  99. Little AV, Maggioni M, Rosasco L. 2011. Multiscale geometric methods for estimating intrinsic dimension. Proc. SampTA 4:2 [Google Scholar]
  100. Lombardi G, Rozza A, Ceruti C, Casiraghi E, Campadelli P. 2011. Minimum neighbor distance estimators of intrinsic dimension. Machine Learning and Knowledge Discovery in Databases D Funopulos, T Hofmann, D Malerba, M Vazirgiannis 374–89 New York: Springer [Google Scholar]
  101. Maaten L, Hinton G. 2008. Visualizing data using t-SNE. J. Mach. Learn. Res. 9:2579–605 [Google Scholar]
  102. Manin YI. 2015. Neural codes and homotopy types: mathematical models of place field recognition. arXiv1501.00897 [math.HO]
  103. Markov A. 1958. Insolubility of the problem of homeomorphy. Proc. Int. Congr. Math. 121:300–306 [Google Scholar]
  104. Masulli P, Villa AE. 2015. The topology of the directed clique complex as a network invariant. arXiv1510.00660 [q-bio.NC]
  105. Milnor J. 2016. Morse Theory Princeton, NJ: Princeton Univ. Press
  106. Müller DW, Sawitzki G. 1991. Excess mass estimates and tests for multimodality. J. Am. Stat. Assoc. 86:738–46 [Google Scholar]
  107. Niyogi P, Smale S, Weinberger S. 2008. Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geometry 39:419–41 [Google Scholar]
  108. Niyogi P, Smale S, Weinberger S. 2011. A topological view of unsupervised learning from noisy data. SIAM J. Comput. 40:646–63 [Google Scholar]
  109. Offroy M, Duponchel L. 2016. Topological data analysis: a promising big data exploration tool in biology, analytical chemistry and physical chemistry. Anal. Chim. Acta 910:1–11 [Google Scholar]
  110. Ozertem U, Erdogmus D. 2011. Locally defined principal curves and surfaces. J. Mach. Learn. Res. 12:1249–86 [Google Scholar]
  111. Patrangenaru V, Ellingson L. 2015. Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis Boca Raton, FL: Chapman and Hall/CRC Press
  112. Petri G, Expert P, Turkheimer F, Carhart-Harris R, Nutt D. et al. 2014. Homological scaffolds of brain functional networks. J. R. Soc. Interface 11:20140873 [Google Scholar]
  113. Phillips JM, Wang B, Zheng Y. 2015. Geometric inference on kernel density estimates. 31st International Symposium on Computational Geometry (SoCG 2015) L Arge, J Pach 857–71 Saarbrücken, Ger.: Schloss Dagstuhl [Google Scholar]
  114. Pirino V, Riccomagno E, Martinoia S, Massobrio P. 2014. A topological study of repetitive co-activation networks in in vitro cortical assemblies. Phys. Biol. 12:016007 [Google Scholar]
  115. Polonik W. 1995. Measuring mass concentrations and estimating density contour clusters—an excess mass approach. Ann. Stat. 23:855–81 [Google Scholar]
  116. Richardson E, Werman M. 2014. Efficient classification using the Euler characteristic. Pattern Recognit. Lett. 49:99–106 [Google Scholar]
  117. Rinaldo A, Wasserman L. 2010. Generalized density clustering. Ann. Stat. 38:2678–722 [Google Scholar]
  118. Roweis ST, Saul LK. 2000. Nonlinear dimensionality reduction by locally linear embedding. Science 290:2323–26 [Google Scholar]
  119. Singh G, Memoli F, Ishkhanov T, Sapiro G, Carlsson G, Ringach DL. 2008. Topological analysis of population activity in visual cortex. J. Vis. 8:11 [Google Scholar]
  120. Singh N, Couture HD, Marron JS, Perou C, Niethammer M. 2014. Topological descriptors of histology images. International Workshop on Machine Learning in Medical Imaging G Wu, D Zhang, L Zhou 231–29 New York: Springer [Google Scholar]
  121. Sizemore A, Giusti C, Bassett DS. 2016.a Classification of weighted networks through mesoscale homological features. J. Complex Netw. 5:245–73 [Google Scholar]
  122. Sizemore A, Giusti C, Betzel RF, Bassett DS. 2016.b Closures and cavities in the human connectome. arxiv1608.03520 [q-bio.NC]
  123. Skraba P, Wang B. 2014. Approximating local homology from samples. Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms C Chekuri 174–92 Philadelphia: SIAM [Google Scholar]
  124. Sousbie T. 2011. The persistent cosmic web and its filamentary structure—I. Theory and implementation. MNRAS 414:350–83 [Google Scholar]
  125. Sousbie T, Pichon C, Kawahara H. 2011. The persistent cosmic web and its filamentary structure—II. Illustrations. MNRAS 414:384–403 [Google Scholar]
  126. Spreemann G, Dunn B, Botnan MB, Baas NA. 2015. Using persistent homology to reveal hidden information in neural data. arXiv1510.06629 [q-bio.NC]
  127. Stolz B. 2014. Computational topology in neuroscience Master's Thesis, Univ. Oxford
  128. Taylor JE, Worsley KJ. 2007. Detecting sparse signals in random fields, with an application to brain mapping. J. Am. Stat. Assoc. 102:913–28 [Google Scholar]
  129. Tenenbaum JB, De Silva V, Langford JC. 2000. A global geometric framework for nonlinear dimensionality reduction. Science 290:2319–23 [Google Scholar]
  130. Turner K, Mukherjee S, Boyer DM. 2014. Persistent homology transform for modeling shapes and surfaces. Inf. Inference 3:310–44 [Google Scholar]
  131. Van de Weygaert R, Platen E, Vegter G, Eldering B, Kruithof N. 2010. Alpha shape topology of the cosmic web. 2010 Int. Symp. Voronoi Diagrams Sci. Eng.224–34 New York: IEEE [Google Scholar]
  132. Van de Weygaert R, Pranav P, Jones BJ, Bos E, Vegter G. et al. 2011.a Probing dark energy with alpha shapes and Betti numbers. arXiv1110.5528 [astro-ph.CO]
  133. Van de Weygaert R, Vegter G, Edelsbrunner H, Jones BJ, Pranav P. et al. 2011.b Alpha, Betti and the megaparsec universe: on the topology of the cosmic web. Transactions on Computational Science XIV ML Gavrilova, CJK Tan, MA Mostafavi 60–101 Heidelberg, Ger.: Springer-Verlag [Google Scholar]
  134. Worsley KJ. 1994. Local maxima and the expected Euler characteristic of excursion sets of X2, F and t fields. Adv. Appl. Probability 26:13–42 [Google Scholar]
  135. Worsley KJ. 1995. Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. Appl. Probability 27:943–59 [Google Scholar]
  136. Worsley KJ. 1996. The geometry of random images. Chance 9:27–40 [Google Scholar]
  137. Xia K, Zhao Z, Wei GW. 2015. Multiresolution topological simplification. J. Comput. Biol. 22:887–91 [Google Scholar]
  138. Yoo J, Kim EY, Ahn YM, Ye JC. 2016. Topological persistence vineyard for dynamic functional brain connectivity during resting and gaming stages. J. Neurosci. Methods 267:1–13 [Google Scholar]
  139. Zeeman EC. 1965. The topology of the brain and visual perception. Mathematics and Computer Science in Biology and Medicine AF Bartholomay 240–56 London: H.M. Stationery Off. [Google Scholar]
/content/journals/10.1146/annurev-statistics-031017-100045
Loading
/content/journals/10.1146/annurev-statistics-031017-100045
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error