The Math Behind TDA: A Primer on Persistent Homology

Table of Links

I. Introduction

II. Methodology

III. TDA Approach to analyzing multiple time series

IV. Data Analyzed

V. Results and Discussion

A. Obtaining point cloud from stock price time-series

B. EE due to the 2008 Financial crisis

C. EE due to COVID-19 pandemic

D. Impact of COVID-19 on different Indian sectors

VI. Conclusion

VII. Acknowledgments and References

II. METHODOLOGY

A. Topological Data Analysis

Topological data analysis (TDA) allows us to extract robust topological features from noisy datasets. One of the advantages of using TDA is its robustness, i.e., the output doesn’t change under small perturbations[29]. In our work, we use Persistent homology which is a powerful tool in TDA, that provides a quantitative representation of how the shape and connectivity of data evolve as the resolution (ε) increases[41].

Sections II A 1– II A 4 contain the definitions and notations from persistent homology used in the paper.

1. Vietoris–Rips Complexes

2. Persistence Diagrams

To develop a better understanding of Vietoris–Rips complexes and persistence diagrams through visualization, let us consider the following set of four data points, A = {[2, 2], [2, 6], [6, 2], [6, 6]} in the two-dimensional Euclidean space.

3. Wasserstein Distance

A key advantage of persistence homology is its robustness under small perturbations. If the underlying data changes slightly, the corresponding persistence diagram only moves a small Wasserstein distance from the diagram of the original data[29,44]. This property makes persistence homology a valuable tool for analyzing complex systems.

4. Persistence Landscapes and their Norms