1b9ffa5e-04ef-478d-91cc-86a039657eac20221128040555010wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051919202291920222210.37394/23206.2023.22https://wseas.com/journals/mathematics/2023.phpS. Z.RidaMathematics Department, Faculty of Science, South Valley University, Qena, EGYPTAlaa HassanNoreldeenMathematics Department, Faculty of Science, Aswan University, Aswan, EGYPTFaten R.KararMathematics Department, Faculty of Science, Aswan University, Aswan, EGYPTStatistical topology inference is a branch of algebraic topology that analyzes the geometric structure's global topological properties underlying a point cloud dataset. There is an increasing need to analyze massive data sets and screen large databases to address real-world problems. A central challenge to modern applied mathematics is the need to generate tools to simplify the data in high dimensional order to extract the important features or the relationships while performing the analysis. A growing field of study at the intersection of algebraic topology, computational geometry, and statistics is topological data analysis (TDA) inference. This study applies TDA tools to test hypothesis between two high-dimensional data sets. Hypothesis testing is one of the most important topics of statistical topology inference. A proposed test was created, which was built on the nearest-neighbor function. Three tests such as (Hypothesis testing based on persistent homology, hypothesis testing based on persistent landscapes, and hypothesis testing based on density estimation) based on TDA, are discussed. Moreover, a modification of these tests was proposed. Monte Carlo simulation was conducted to compare the power of the previous tests. We displayed the use of TDA tools in hypothesis testing. It was proposed that this test might be established based on the nearest neighbor distance function. Furthermore, a suggested modification for the present tests based on TDA was introduced. Finally, the tests specified in the vignette were enabled by two empirical applications within the biology field. We demonstrated the efficacy of the above tests on the heart disease dataset from Statlog and the Wisconsin breast cancer dataset.112820221231202222343https://wseas.com/journals/mathematics/2023/a065106-1711.pdf10.37394/23206.2023.22.3https://wseas.com/journals/mathematics/2023/a065106-1711.pdf10.1007/s10208-020-09482-9Bubenik, Peter, and Nikola Milićević, "Homological algebra for persistence modules", Foundations of Computational Mathematics 21.5 (2021): 1233-1278. https://doi.org/10.1007/s10208-020-09482-9. 10.1214/14-aos1252Fasy, Brittany Terese, et al. "Confidence sets for persistence diagrams", The Annals of Statistics (2014): 2301-2339. https://doi.org/10.1214/14- AOS1252. 10.1016/j.joems.2017.07.001Alaa, H. N., and S. A. Mohamed. "On the topological data analysis extensions and comparisons", Journal of the Egyptian Mathematical Society 25.4 (2017): 406-413. https://doi.org/10.1016/j.joems.2017.07.001. 10.2172/1172555Fasy, Brittany Terese, et al. "Introduction to the R package TDA", arXiv preprint arXiv:1411.1830 (2014). https://doi.org/10.48550/arXiv.1411.1830. 10.1109/sfcs.2000.892133Edelsbrunner, Herbert, David Letscher, and Afra Zomorodian. "Topological persistence and simplification", Proceedings 41st annual symposium on foundations of computer science. IEEE, 2000. DOI:10.1109/SFCS.2000.892133. Bubenik, Peter. "Statistical topological data analysis using persistence landscapes", J. Mach. Learn. Res.16.1 (2015): 77-102. https://doi.org/10.48550/arXiv.1207.6437 Balchin, Scott, and Etienne Pillin. "Comparing Metrics on Arbitrary Spaces using Topological Data Analysis", arXiv preprint arXiv:1503.04619 (2015). https://doi.org/10.48550/arXiv.1503.04619 10.1090/s0273-0979-09-01249-xCarlsson, Gunnar. "Topology and data", Bulletin of the American Mathematical Society 46.2 (2009): 255-308. DOI:10.1090/S0273- 0979-09-01249-X. 10.2172/1172555Emmett, Kevin, et al. "Parametric inference using persistence diagrams: A case study in population genetics", arXiv preprint arXiv:1406.4582 (2014). https://doi.org/10.48550/arXiv.1406.4582 10.1016/j.jmva.2010.04.016Gamble, Jennifer, and Giseon Heo. "Exploring uses of persistent homology for statistical analysis of landmark-based shape data", Journal of Multivariate Analysis 101.9 (2010): 2184-2199. https://doi.org/10.1016/j.jmva.2010.04.016. 10.1016/j.topol.2020.107523Kim, Wonse, et al. "Investigation of flash crash via topological data analysis", Topology and its Applications 301 (2021): 107523. https://doi.org/10.1016/j.topol.2020.107523. 10.1016/j.physd.2016.04.015Dłotko, Paweł, and Thomas Wanner. "Topological microstructure analysis using persistence landscapes", Physica D: Nonlinear Phenomena 334 (2016): 60-81. https://doi.org/10.1016/j.physd.2016.04.015. Sizemore, Ann E., et al. "The importance of the whole: topological data analysis for the network neuroscientist", Network Neuroscience 3.3 (2019): 656-673. https://doi.org/10.1162/netn_a_00073. Artamonov, Oleg. "Topological methods for the representation and analysis of exploration data in oil industry", Diss. Technische Universität Kaiserslautern, 2010. urn:nbn:de:hbz:386-kluedo-25456. Curran, James, and Maintainer James M. Curran. "Package 'Hotelling'”, R package version (2017): 1-0. https://github.com/jmcurran/Hotelling. 10.1007/s41468-017-0008-7Robinson, Andrew, and Katharine Turner. “Hypothesis testing for topological data analysis”, Journal of Applied and Computational Topology 1.2 (2017): 241-261. https://doi.org/10.1007/s41468-017-0008-7 Edelsbrunner, Herbert, and John L. Harer. “Computational topology: an introduction”, American Mathematical Society, 2010. https://doi.org/10.1007/978-3-540-33259-6_7.