Analyzing distributions using a systematic programmable approach as persistent homology

Abstract

Persistent homology is a tool in mathematics used for analyzing data topologically. This analysis is made possible through one of its components called a filtered simplicial complex, which is a sequence of nested simplicial complexes. With this, we can obtain our persistent π0 (gaps) which is the number of connected components, and also apply homology on the filtration to obtain the persistent homology. This thesis focuses on the comparison of three fundamental probability distributions which are the Normal distribution, Uniform distribution, and Exponential distribution, using a Python code to derive a filtration, which we analyze to obtain persistent homology. The comparison enables us to categorize any random data into a distinct distribution after analyzing its behavior with the Python code. Firstly, the three distributions are compared statistically, after which a Python program is developed which is divided into four different sections. The first and second sections input parameters and generate the distinct probability distribution data respectively, while the third and fourth sections analyze the data generated from each distribution to output relevant information which includes the persistent π0 and the connected components. The connected components are further analyzed to form a filtration, from which we obtain our persistent gaps and persistent homology. In conclusion, this project provides worthwhile insights into the behavior and characteristics of the Normal, Uniform, and Exponential distributions through the spectacles of topology. These findings create the building blocks for analyzing and classifying random data and also for better understanding and analyzing various real-world phenomena governed by these distributions to enhance future research in diverse fields.