Tropical Statistics: New Frontiers in Max-Linear Data Analysis

Objective

Classical statistics assumes data lies in a Euclidean space, but many problems involve non-Euclidean spaces such as manifolds or polyhedral complexes. Tropical linear spaces, which are polyhedral complexes defined by max-linear equations, are a promising but underdeveloped frontier for statistics. Such spaces capture data from a variety of domains, including phylogenetics, extreme-value theory and global economics, where classical methods fail. This proposal develops fundamental statistical measures such as variation, distributions, and models in the tropical setting. The goal of this proposal is to advance our understanding of statistics in the tropical setting through the following four objectives. (O1) concerns the analysis of variation over tropical linear spaces using tropical principal components under the asymmetric tropical distance. Combining two previous approaches, asymmetric tropical PCA has the potential to offer fast, interpretable analysis of variation of max-linear data. (O2) outlines the next steps for fitting Gauss-Laplace distributions over tropical linear spaces, which is an important step in establishing the foundations of tropical statistics. (O3) defines a tropical probabilistic PCA model, which opens new avenues for statistical testing and Bayesian inference for max-linear data. (O4) implements these advancements in a software package in R, in order to facilitate the use of the results in applications. Together, these advances lay the foundations for a rigorous statistical framework for tropical linear spaces. This will provide new insights and practical tools for researchers across phylogenetics, extreme-value theory, and global economics.