Applications of Non-Euclidean Metrics for the Statistical Analysis of Multivariate Data

In many modern statistical applications, the relationships of interest among measured features may be described and interpreted through the notions of distance and similarity resulting from an appropriately chosen metric. While non-Euclidean metrics naturally arise in many modeling contexts, the choice of metric and relative utility of alternative metrics in answering specific scientific questions is often not explicitly considered. However, approaching statistical analysis and model design from the perspectives of distance, similarity, and feature space geometry can inform the development of new methodologies, guide analysis, and aid the interpretation of inferential results in many scientific settings. In this work, we propose novel statistical methods utilizing non-Euclidean distance and similarity metrics for the analysis of multivariate data, motivated by challenges in the analysis of cognitive and neurophysiological data. Our primary contributions include: the adaptive Mantel test, a metric-based test for association of two high-dimensional feature sets, with applications to genetic heritability testing of electroencephalogram data; shape-theoretic inference methods for multi-group samples using the Riemannian shape metric, for the study of a newly-discovered neurological cell type in relation to two known cell types; and a latent factor Gaussian process model for time-varying covariance processes incorporating the log-Euclidean metric for positive definite matrices, for the analysis of dynamic functional connectivity of local field potential recordings from a rat hippocampus during a working-memory experiment