arXiv subjects

Nathan Willey

Publications and source records attributed to Nathan Willey.

3 recordsLinked to original sources

arXiv2026

Visualizing Local Maxima of the Ohio overdose epidemic with Vineyards

Understanding how spatial patterns evolve over time is a complex task that often arises in the analysis of public health data. In this work, we investigate the use of vineyards from topological data analysis (TDA) in this setting by applying them to time series data related to the overdose epidemic in the state of Ohio. We begin by proposing statistical tests that can be used in order to evaluate whether vineyards are a reasonable technique to study a spatiotemporal dataset. We then apply these tests to the data of drug overdose deaths in Ohio and, finding the data suitable, perform a subsequent analysis using vineyards to visualize the evolution of local maxima of death rates throughout the Ohio overdose epidemic. We conclude by developing statistical methods to quantify the significance and uncertainty of vineyard features and by exploring how vineyard-derived summaries can be used for forecasting.

math.AT↗

arXiv2026

Approximation theorems in bilipschitz invariant theory

Bilipschitz invariant theory concerns low-distortion embeddings of orbit spaces into Euclidean space. To date, embeddings with the smallest-possible distortion are known for only a few cases, to include: (a) planar rotations, (b) real phase retrieval, and (c) finite reflection groups. Here, we prove that for all three of these cases, the smallest possible distortion is nearly achieved by a composition of a "max filter bank" with a linear transformation. Our proof amounts to a two-step process: first, we show it suffices to demonstrate a certain inclusion of Lipschitz function spaces, and second, we prove that inclusion, using fundamentally different approaches for the three cases. We also show that these cases interact differently with a few related function spaces, which suggests that a unified treatment would be nontrivial.

math.FA↗

arXiv2025

Tracking the Spatiotemporal Spread of the Ohio Overdose Epidemic with Topological Data Analysis

In recent years, techniques from Topological Data Analysis (TDA) have proven effective at capturing spatial features of multidimensional data. However, applying TDA to spatiotemporal data remains relatively underexplored. In this work, we extend previous studies of disease spread by using the Mapper algorithm to analyze the Ohio drug overdose epidemic from 2007 to 2024. We introduce a novel method for constructing covers in Mapper graphs of spatiotemporal data that respects geographic structure and highlights the time-dependent variables. Finally, we generate a Mapper visualization of regional demographics to examine how these factors relate to overdose deaths. Our approach effectively reveals temporal trends, overdose hotspots, and time-lagged patterns in relation to both geography and community demographics.

stat.AP↗