Persistent Homology for Statistical Analysis of Random Fields

Abstract

In order to understand and predict the behaviour of the climate in the future, simulations of climate models are often performed. The Community Earth System ModelLarge Ensemble provides 40 massive simulations of climate variables for the period 1920-2100. The volume of the data produced is huge and scientists have applied compression methods to reduce storage costs. This involves treating simulated output with different initial conditions as being independent and perhaps identically distributed realisations of a stochastic model. It is of particular interest to investigate if there are any potential systematic differences between different realisations of the climate model. This is the focus of this thesis, where we concentrate on precipitation, pressure, temperature and wind data for the year 2020, over the latitudinal bands from 62 degrees south to 70 degrees north. We employ persistent homology to detect ensemble members that have unusual properties compared to the others. Persistent homology is used to track the evolution of topological features over different levels of the topological space. This approach is further coupled with survival analysis methods, such as the Nelson-Aalen estimator, to detect differences between random fields, to evaluate the fitting performance of spatial models and for testing the standard assumption of Gaussian random fields. Motivated by our topological results, a second part of the thesis is focused on modelling the filamentarity structure of galaxy or other point process data. The large-scale structure of the universe is formed of a network of filaments, or clusters of galaxies, and empty areas which are called voids. From a statistical point of view the locations of galaxies in the universe forms a spatial point pattern. We develop a testing procedure to identify increased filamentarity compared to a Poisson point process, and we introduce a filament model based on a Poisson process of parent positions and correlated random walks for offspring. A Bayesian approach is used for estimation.