Topological Data Analysis in Python

## Introduction

mogutda contains Python codes that demonstrate the numerical calculation of algebraic topology in an application to topological data analysis (TDA). Its core code is the numerical methods concerning implicial complex, and the estimation of homology and Betti numbers.

Topological data analysis aims at studying the shapes of the data, and draw some insights from them. A lot of machine learning algorithms deal with distances, which are extremely useful, but they miss the information the data may carry from their geometry.

## History

The codes in this package were developed as a demonstration of a few posts of my blog. It was not designed to be a Python package but a pedagogical collection of codes. (See: PyTDA.) However, the codes and the blog posts have been unexpectedly popular. Therefore, I modularized the code into the package mogu. (or corresponding repository: MoguNumerics) However, mogu is simply a collection of unrelated numerical routines with a lot of dependencies, but the part of TDA can be quite independent.

In order to provide other researchers and developers an independent package, which is compact (without unecessary alternative packages to load), and efficient, I decided to modularize the codes of TDA separately, and name this package mogutda.

## Prerequisite

It runs under Python 3.5, 3.6, 3.7, and 3.8.

Release 0.1.5 can work under numpy>0.16.0, but previous release will constitute error under the new numpy.

## Simple Tutorial: Simplicial Complex

You can install by:

pip install -U mogutda

To establish a simplicial complex for a torus, type

import numpy as np
from mogutda import SimplicialComplex

torus_sc = [(1,2,4), (4,2,5), (2,3,5), (3,5,6), (5,6,1), (1,6,2), (6,7,2), (7,3,2),
(1,3,4), (3,4,6), (4,6,7), (4,5,7), (5,7,1), (7,3,1)]
torus_c = SimplicialComplex(simplices=torus_sc)

To retrieve its Betti numbers, type:

print(torus_c.betti_number(0))   # print 1
print(torus_c.betti_number(1))   # print 2
print(torus_c.betti_number(2))   # print 1

## Demo Codes and Blog Entries

Codes in this repository are demo codes for a few entries of my blog, Everything about Data Analytics, and the corresponding entries are:

## Wolfram Demonstration

Richard Hennigan put a nice Wolfram Demonstration online explaining what the simplicial complexes are, and how homologies are defined:

## News

• 09/09/2022: mogutda 0.3.4 released.
• 07/15/2021: mogutda 0.3.3 released.
• 04/10/2021: mogutda 0.3.2 released.
• 11/28/2020: mogutda 0.3.1 released.
• 08/16/2020: mogutda 0.3.0 released.
• 04/28/2020: mogutda 0.2.1 released.
• 01/16/2020: mogutda 0.2.0 released.
• 02/20/2019: mogutda 0.1.5 released.
• 12/31/2018: mogutda 0.1.4 released.
• 07/18/2018: mogutda 0.1.3 released.
• 07/02/2018: mogutda 0.1.2 released.
• 06/13/2018: mogutda 0.1.1 released.
• 06/11/2018: mogutda 0.1.0 released.

## Other TDA Packages

It is recommended that for real application, you should use the following packages for efficiency, because my codes serve the pedagogical purpose only.

## Contributions

If you want to contribute, feel free to fork the repository, and submit a pull request whenever you are ready.

If you spot any bugs or issues, go to the Issue page.

I may not approve pull request immediately if your suggested change is big. If you want to incorporate something big, please discuss with me first.

## References

• Afra J. Zomorodian. Topology for Computing (New York, NY: Cambridge University Press, 2009). [Amazon]
• Afra J. Zomorodian. "Topological Data Analysis," Proceedings of Symposia in Applied Mathematics (2011). [link]
• Afra Zomorodian, Gunnar Carlsson, “Computing Persistent Homology,” Discrete Comput. Geom. 33, 249-274 (2005). [pdf]
• Gunnar Carlsson, “Topology and Data”, Bull. Amer. Math. Soc. 46, 255-308 (2009). [link]
• Jeffrey Ray, Marcello Trovati, "A Survey of Topological Data Analysis (TDA) Methods Implemented in Python," in Advances in Intelligent Networking and Collaborative Systems, Springer (2018).
• P. Y. Lum, G. Singh, A. Lehman, T. Ishkanov, M. Vejdemo-Johansson, M. Alagappan, J. Carlsson, G. Carlsson, “Extracting insights from the shape of complex data using topology”, Sci. Rep. 3, 1236 (2013). [link]
• Robert Ghrist, “Barcodes: The persistent topology of data,” Bull. Amer. Math. Soc. 45, 1-15 (2008). [pdf]