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Structural Analysis of Multivariate Point Clouds Using Simplicial Chains

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Abstract

Topological and geometrical methods constitute common tools for the analysis of high-dimensional scientific data sets. Geometrical methods such as projection algorithms focus on preserving distances in the data set. Topological methods such as contour trees, by contrast, focus on preserving structural and connectivity information. By combining both types of methods, we want to benefit from their individual advantages. To this end, we describe an algorithm that uses persistent homology to analyse the topology of a data set. Persistent homology identifies high-dimensional holes in data sets, describing them as simplicial chains. We localize these chains using geometrical information of the data set, which we obtain from geodesic distances on a neighbourhood graph. The localized chains describe the structure of point clouds. We represent them using an interactive graph, in which each node describes a single chain and its geometrical properties. This graph yields a more intuitive understanding of multivariate point clouds and simplifies comparisons of time-varying data. Our method focuses on detecting and analysing inhomogeneous regions, i.e. holes, in a data set because these regions characterize data in a different manner, thereby leading to new insights. We demonstrate the potential of our method on data sets from particle physics, political science and meteorology.

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