Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-14T04:21:25.977Z Has data issue: false hasContentIssue false

Topological reconstruction of compact supports of dependent stationary random variables

Published online by Cambridge University Press:  02 April 2024

Sadok Kallel*
Affiliation:
American University of Sharjah
Sana Louhichi*
Affiliation:
Université Grenoble Alpes
*
*Postal address: American University of Sharjah, UAE, and Laboratoire Painlevé, Université de Lille, France. Email address: skallel@aus.edu
**Postal address: Université Grenoble Alpes, CNRS, Grenoble INP, LJK 38000 Grenoble, France. Email address: sana.louhichi@univ-grenoble-alpes.fr

Abstract

In this paper we extend results on reconstruction of probabilistic supports of independent and identically distributed random variables to supports of dependent stationary ${\mathbb R}^d$-valued random variables. All supports are assumed to be compact of positive reach in Euclidean space. Our main results involve the study of the convergence in the Hausdorff sense of a cloud of stationary dependent random vectors to their common support. A novel topological reconstruction result is stated, and a number of illustrative examples are presented. The example of the Möbius Markov chain on the circle is treated at the end with simulations.

Type
Original Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aamari, E. and Levrard, C. (2019). Nonasymptotic rates for manifold, tangent space and curvature estimation, Ann. Statist. 47, 177204.CrossRefGoogle Scholar
Amézquita, E. J. et al. (2020). The shape of things to come: topological data analysis and biology, from molecules to organisms. Dev. Dynamics 249, 816833.CrossRefGoogle ScholarPubMed
Attali, D., Lieutier, A. and Salinas, D. (2013). Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes. Comput. Geom. 46, 448465.CrossRefGoogle Scholar
Barb, S. (2009). Topics in geometric analysis with applications to partial differential equations. Doctoral Thesis, University of Missouri-Columbia.Google Scholar
Bradley, R. C. (1983). Absolute regularity and functions of Markov chains. Stoch. Process. Appl. 14, 6777.CrossRefGoogle Scholar
Bradley, R. C. (2007). Introduction to Strong Mixing Conditions, Vol. 1, 2, 3. Kendrick Press, Heber City, UT.Google Scholar
Chazal, F., Glisse, M., Labruyère, C. and Michel, B. (2015). Convergence rates for persistence diagram estimation in topological data analysis. J. Machine Learning Res. 16, 36033635.Google Scholar
Chazal, F. and Oudot, S. Y. (2008). Towards persistence-based reconstruction in Euclidean spaces. In Scg ’08: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, Association for Computing Machinery, New York, pp. 232–241.CrossRefGoogle Scholar
Cisewski-Kehe, J. et al. (2018). Investigating the cosmic web with topological data analysis. In American Astronomical Society Meeting Abstracts 231, id. 213.07.Google Scholar
Cuevas, A. (2009). Set estimation: another bridge between statistics and geometry. Bol. Estadist. Investig. Oper. 25, 7185.Google Scholar
Cuevas, A. and Rodrguez-Casal, A. (2004). On boundary estimation. Adv. Appl. Prob. 36, 340354.CrossRefGoogle Scholar
Dedecker, J. et al. (2007). Weak Dependence: With Examples and Applications. Springer, New York.CrossRefGoogle Scholar
Divol, V. (2021). Minimax adaptive estimation in manifold inference. Electron. J. Statist. 15, 58885932.CrossRefGoogle Scholar
Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stoch. Process. Appl. 84, 313342.CrossRefGoogle Scholar
Ellis, J. C. (2012). On the geometry of sets of positive reach. Doctoral Thesis, University of Georgia.Google Scholar
Fasy, B. T. et al. (2014). Confidence sets for persistence diagrams. Ann. Statist. 42, 23012339.CrossRefGoogle Scholar
Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93, 418491.CrossRefGoogle Scholar
Fefferman, C., Mitter, S. and Narayanan, H. (2016). Testing the manifold hypothesis. J. Amer. Math. Soc. 29, 9831049.CrossRefGoogle Scholar
Fu, J. H. G. (1989). Curvature measures and generalized Morse theory. J. Differential Geom. 30, 619642.CrossRefGoogle Scholar
Iniesta, R. et al. (2022). Topological Data Analysis and its usefulness for precision medicine studies. Statist. Operat. Res. Trans. 46, 115136.Google Scholar
Goldie, C. M. and Maller, R. A. (2001). Stability of perpetuities. Ann. Prob. 28, 11951218.Google Scholar
Hoef, L. V., Adams, H., King, E. J. and Ebert-Uphoff, I. (2023). A primer on topological data analysis to support image analysis tasks in environmental science. Artificial Intellig. Earth Systems 2, 118.Google Scholar
Hatcher, A. (2002.) Algebraic Topology. Cambridge University Press.Google Scholar
Hörmander, L. (1994). Notions of Convexity. Birkhäuser, Boston.Google Scholar
Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data. Ann. Statist. 38, 18451884.CrossRefGoogle Scholar
Kato, S. (2010). A Markov process for circular data. J. R. Statist. Soc. B. [Statist. Methodology] 72, 655672.CrossRefGoogle Scholar
Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207248.CrossRefGoogle Scholar
Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Prob. 2, 355386.CrossRefGoogle Scholar
Kim, J. et al. (2020). Homotopy reconstruction via the Cech complex and the Vietoris–Rips complex. In 36th International Symposium on Computational Geometry, Dagstuhl Publishing, Wadern, article no. 54.Google Scholar
Lee, J. M. (2013). Introduction to Smooth Manifolds. Springer, New York.Google Scholar
MathOverflow (2017). Tubular neighborhood theorem for $C^1$ submanifold. Available at https://mathoverflow.net/questions/286512/tubular-neighborhood-theorem-for-c1-submanifold.Google Scholar
Moreno, R., Koppal, S. and de Muinck, E. (2013). Robust estimation of distance between sets of points. Pattern Recognition Lett. 34, 21922198.CrossRefGoogle Scholar
Niyogi, P., Smale, S. and Weinberger, S. (2008). Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39, 419441.CrossRefGoogle Scholar
Rio, E. (2013). Inequalities and limit theorems for weakly dependent sequences. Available at https://cel.hal.science/cel-00867106v2.Google Scholar
Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA 42, 4347.CrossRefGoogle Scholar
Scholtes, S. (2013). On hypersurfaces of positive reach, alternating Steiner formulae and Hadwiger’s Problem. Preprint. Available at https://arxiv.org/abs/1304.4179.Google Scholar
Singh, Y. et al. (2023). Topological data analysis in medical imaging: current state of the art. Insights Imaging 14, article no. 58.CrossRefGoogle ScholarPubMed
Thale, C. (2008). 50 years sets with positive reach—a survey. Surveys Math. Appl. 3, 123165.Google Scholar
Wang, Y. and Wang, B. (2020). Topological inference of manifolds with boundary. Comput. Geom. 88, article no. 101606.CrossRefGoogle Scholar
Yu, B. (1994). Rates of convergence for empirical processes of stationary mixing sequences. Ann. Prob. 22, 94116.CrossRefGoogle Scholar