Bump hunting through density curvature features

TL;DR

This paper introduces a novel approach to bump hunting using curvature functionals of the density, providing theoretical guarantees and practical applications in sports analytics.

Contribution

It defines an abstract bump concept based on density curvature, proposes multivariate implementation, and offers asymptotic consistency results with real-world data applications.

Findings

Curvature-based bumps effectively identify meaningful data regions.

The methodology provides consistent boundary estimation with theoretical guarantees.

Applications in sports analytics demonstrate insightful visualizations.

Abstract

Bump hunting deals with finding in sample spaces meaningful data subsets known as bumps. These have traditionally been conceived as modal or concave regions in the graph of the underlying density function. We define an abstract bump construct based on curvature functionals of the probability density. Then, we explore several alternative characterizations involving derivatives up to second order. In particular, a suitable implementation of Good and Gaskins' original concave bumps is proposed in the multivariate case. Moreover, we bring to exploratory data analysis concepts like the mean curvature and the Laplacian that have produced good results in applied domains. Our methodology addresses the approximation of the curvature functional with a plug-in kernel density estimator. We provide theoretical results that assure the asymptotic consistency of bump boundaries in the Hausdorff distance with affordable convergence rates. We also present asymptotically valid and consistent confidence regions bounding curvature bumps. The theory is illustrated through several use cases in sports analytics with datasets from the NBA, MLB and NFL. We conclude that the different curvature instances effectively combine to generate insightful visualizations.