Event History and Topological Data Analysis

TL;DR

This paper introduces the application of event history analysis methods, including Nelson-Aalen and Cox models, to topological data analysis using persistent homology, enabling statistical comparison of topological features in various data sets.

Contribution

It adapts event history methods to persistent homology, providing new tools for analyzing and comparing topological features in complex data.

Findings

Nelson-Aalen estimator applied to climate and astronomical data.

Cox proportional hazards model used to compare vascular patterns in medical images.

Demonstrates the effectiveness of statistical methods in topological data analysis.

Abstract

Persistent homology is used to track the appearance and disappearance of features as we move through a nested sequence of topological spaces. Equating the nested sequence to a filtration and the appearance and disappearance of features to events, we show that simple event history methods can be used for the analysis of topological data. We propose a version of the well known Nelson-Aalen cumulative hazard estimator for the comparison of topological features of random fields and for testing parametric assumptions. We suggest a Cox proportional hazards approach for the analysis of embedded metric trees. The Nelson-Aalen method is illustrated on globally distributed climate data and on neutral hydrogen distribution in the Milky Way. The Cox method is use to compare vascular patterns in fundus images of the eyes of healthy and diabetic retinopathy patients.