Event history and 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.

Discussion of ‘Event history and topological data analysis’

“…Other higher-order topological features are simply ignored. However, Garside et al (2021) used somewhat inefficient filtrations in the 2D plane that increase the radius of circles from the root node or points along the tree. Such filtrations produce persistent diagrams that spread points in a 2D plane.…”

“…Then the persistence diagram of the graph filtration is simply ( w (1) , c ), ( w (2) , c ), …, ( w ( q −1) , c ) forming 1D scatter points along the horizontal line y = c , and making various analysis and operations, including matching, significantly simplified ( Songdechakraiwut & Chung, 2020b ). Figure 1 illustrates the graph filtration and corresponding 1D scatter points in persistence diagrams on the binary tree used in Garside et al (2021) . In this example, c = 0.31 is arbitrarily picked to be larger than the maximum edge weight 0.3034.…”

“… Garside et al (2021) proposed modelling the birth and death of cycles as the observed data in event history analysis in a literal sense. Event history analysis has been widely used in diverse areas, including survival analysis in medicine and failure time analysis in engineering, and thus such an approach would open a new direction for research.…”

“…Event history analysis has been widely used in diverse areas, including survival analysis in medicine and failure time analysis in engineering, and thus such an approach would open a new direction for research. From Garside et al (2021) , other event history approaches can be equally applicable to the birth and death of cycles. The Nelson–Aalen method and many other event history methods all bypass the problem of matching births and deaths across different subjects by accumulating events.…”

“…The Nelson–Aalen method and many other event history methods all bypass the problem of matching births and deaths across different subjects by accumulating events. In Garside et al (2021) , functions N x (t) and Y (t) accumulate the indicator variables for the events by summation. Such an approach usually yields the Nelson–Aalen plot type of monotone curves, which make the subsequent analysis stable and easy to perform.…”