Testing to distinguish measures on metric spaces
This work addresses a fundamental statistical problem for researchers dealing with metric spaces, such as in computational biology, but it appears incremental as it builds on existing theorems and tests.
The paper tackles the problem of distinguishing between two distributions on a metric space using finite data, and it achieves this by employing a reconstruction theorem and a two-sample Kolmogorov-Smirnov test, with a real-world application to phylogenetic trees and flu data.
We study the problem of distinguishing between two distributions on a metric space; i.e., given metric measure spaces $({\mathbb X}, d, μ_1)$ and $({\mathbb X}, d, μ_2)$, we are interested in the problem of determining from finite data whether or not $μ_1$ is $μ_2$. The key is to use pairwise distances between observations and, employing a reconstruction theorem of Gromov, we can perform such a test using a two sample Kolmogorov--Smirnov test. A real analysis using phylogenetic trees and flu data is presented.