Modeling and replicating statistical topology, and evidence for CMB non-homogeneity

Abstract

Under the banner of `Big Data', the detection and classification of structure in extremely large, high dimensional, data sets, is, one of the central statistical challenges of our times. Among the most intriguing approaches to this challenge is `TDA', or `Topological Data Analysis', one of the primary aims of which is providing non-metric, but topologically informative, pre-analyses of data sets which make later, more quantitative analyses feasible. While TDA rests on strong mathematical foundations from Topology, in applications it has faced challenges due to an inability to handle issues of statistical reliability and robustness and, most importantly, in an inability to make scientific claims with verifiable levels of statistical confidence. We propose a methodology for the parametric representation, estimation, and replication of persistence diagrams, the main diagnostic tool of TDA. The power of the methodology lies in the fact that even if only one persistence diagram is available for analysis -- the typical case for big data applications -- replications can be generated to allow for conventional statistical hypothesis testing. The methodology is conceptually simple and computationally practical, and provides a broadly effective statistical procedure for persistence diagram TDA analysis. We demonstrate the basic ideas on a toy example, and the power of the approach in a novel and revealing analysis of CMB non-homogeneity.

References in corpus (8)

• Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
• Planck 2015 results. XVI. Isotropy and statistics of the CMB
• Hierarchical structures of amorphous solids characterized by persistent homology
• Clique topology reveals intrinsic geometric structure in neural correlations
• The Topology of the Cosmic Web in Terms of Persistent Betti Numbers
• Topological inference for EEG and MEG
• Model-independent analyses of non-Gaussianity in Planck CMB maps using Minkowski Functionals
• DTM-based Filtrations

Cited by in corpus (12)

• Is the Observable Universe Consistent with the Cosmological Principle?
• Topology and Geometry of Gaussian random fields I: on Betti Numbers, Euler characteristic and Minkowski functionals
• The persistence landscape and some of its properties
• Homological Percolation and the Euler Characteristic
• Persistent homology of the cosmic web. I: Hierarchical topology in $\Lambda$CDM cosmologies
• Persistent Homology and Non-Gaussianity
• Topological Learning for Brain Networks
• Tropical Sufficient Statistics for Persistent Homology
• Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics
• Anomalies in the topology of the temperature fluctuations in the cosmic microwave background: An analysis of the $\text{NPIPE}$ and $\text{FFP10}$ data releases
• Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
• Statistical topology of the streamlines of a two-dimensional flow