The promises of persistent homology, machine learning, and deep neural networks in topological data analysis of democracy survival

This paper presents a new approach to survival analysis using topological data analysis (TDA) within Bayesian statistics combined with machine learning algorithms suitable to time-to-event data. The paper brings into the analysis aspects of topological invariance through what is known as persistence homology. TDA demonstrates the existence and statistical significance of a kind of unmeasured heterogeneity originating from the topology of the data as a whole. Combined with machine learning tools persistence homology provides us with new tools to construct a rich set of ways to analyze data and build predictive models that are optimized using inherent topological invariants such as one-dimensional loops as regularization. Specifically, this paper incorporates persistent homology effects in different ways in the analysis of survival data through the technique of functional principal component analysis (FPCA): first, by using topological invariants converted into FPCA factors that shape Bayesian statistical analysis of time-to-event data; second, by using FPCA measures of topological invariants in regularizing the process of optimizing the data and the posterior distributions of the Bayesian estimation; three, by using FPCA factors of measures of topological invariants in machine learning algorithms and deep neural networks suitable for analyzing survival data as a way of going beyond usual parametric and semi-parametric models of survival analysis. The approach is illustrated through a running example of multi-frailty survival analysis of democracies in the period of 1950–2010.