Improved Modeling of Persistence Diagram

TL;DR

This paper introduces a modified RST model for better statistical analysis of persistence diagrams in topological data analysis, enhancing the modeling of data shape and outlier detection.

Contribution

The paper proposes a modification to the RST model, improving its goodness of fit for persistence diagrams in topological data analysis.

Findings

Modified RST outperforms original in simulation studies

Improved goodness of fit demonstrated via MCMC sampling

Enhances outlier detection in high-dimensional data

Abstract

High-dimensional reduction methods are powerful tools for describing the main patterns in big data. One of these methods is the topological data analysis (TDA), which modeling the shape of the data in terms of topological properties. This method specifically translates the original data into two-dimensional system, which is graphically represented via the 'persistence diagram'. The outliers points on this diagram present the data pattern, whereas the other points behave as a random noise. In order to determine which points are significant outliers, replications of the original data set are needed. Once only one original data is available, replications can be created by fitting a model for the points on the persistence diagram, and then using the MCMC methods. One of such model is the RST (Replicating Statistical Topology). In this paper we suggest a modification of the RST model. Using a simulation study, we show that the modified RST improves the performance of the RST in terms of goodness of fit. We use the MCMC Metropolis-Hastings algorithm for sampling according to the fitted model.