Mathematical Modeling
Vol. 3 (2019), Issue 4, pg(s) 109-112

THEORETICAL FOUNDATIONS AND SPECIFICITY OF MATHEMATICAL MODELLING

APPLICATION OF PERSISTENT HOMOLOGY ON BIO-MEDICAL DATA – A CASE STUDY

  • 1 Faculty of Computer Science and Engineering, ”Ss. Cyril and Methodius” University, Skopje, Macedonia

Abstract

In this paper we introduce, analyze and apply persistent homology, one of the main algorithms of TDA, on some real data sets from the bio-medical field. Topological data analysis (TDA) is a field which is a synergy between mathematics, data science and computer science. The main goal of TDA is studying the shape of data using topological techniques. TDA proposes new algorithms that deal with these problems based on tools or concepts from algebraic topology and pure mathematics. We analyze the results and give a topological characterization of the dataset and propose to use them in future work.

Keywords

References

  1. H. Edelsbrunner, “Persistent homology: theory and practice”, 2014.
  2. Gunnar Carlsson, “Topology and data”. Bulletin of the American Mathematical Society. 46 (2), 2009, pp. 255–308.
  3. J. R. Munkres, Topology. vol. 2. Upper Saddle River: Prentice Hall, 2000
  4. llen Hatcher, Algebraic topology. Cambridge University Press, 2002
  5. G. Carlsson, A. Zomorodian, A. Collins, L. Guibas, J. (2005-12-01). "Persistence barcodes for shapes". International Journal of Shape Modeling.
  6. Brittany Terese Fasy, Jisu Kim, Fabrizio Lecci, and Clément Maria. “Introduction to the r package tda. “, arXiv preprint arXiv:1411.1830, 2014.
  7. https://s3.amazonaws.com/cdn.ayasdi.com/wpcontent/uploads/2018/11/12131418/TDA-Based-Approaches-to-Deep-Learning.pdf
  8. https://en.wikipedia.org/wiki/Fluoroscopy
  9. Ulrich Bauer and Michael Lesnick.” Induced matchings of barcodes and the algebraic stability of persistence. In Proceedings of the thirtieth annual symposium on Computational geometry”, p. 355, 2014.
  10. Reaven, G. M. and Miller, R. G. (1979). An attempt to define the nature of chemical diabetes using a multidimensional analysis. Diabetologia, 16, 17-24.
  11. A. J. Zomorodian, Topology for Computing, Cambridge, 2005
  12. J. Nicponski and J.-H. Jung, Topological data analysis of vascular disease: A theoretical framework, BioRxiv, (2019), p. 637090.
  13. D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, Stability of persistence diagrams, Discrete & Computational Geometry, (2007), pp. 103–120
  14. A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete & Computational Geometry, 33 (2005), pp. 249–274

Article full text

Download PDF