Data-driven density derivative estimation, with applications to nonparametric clustering and bump hunting

TL;DR

This paper introduces fully automatic, data-driven bandwidth selectors for multivariate density derivative estimation, enabling improved nonparametric clustering and bump hunting with theoretical and practical validation.

Contribution

It presents the first fully automatic bandwidth selectors for multivariate density derivatives, integrating recent matrix analytic advances for tractable estimation.

Findings

New data-driven bandwidth selectors outperform existing methods.

Enhanced clustering algorithms based on mean shift outperform traditional mixture models.

Application to real data demonstrates improved bump detection.

Abstract

Important information concerning a multivariate data set, such as clusters and modal regions, is contained in the derivatives of the probability density function. Despite this importance, nonparametric estimation of higher order derivatives of the density functions have received only relatively scant attention. Kernel estimators of density functions are widely used as they exhibit excellent theoretical and practical properties, though their generalization to density derivatives has progressed more slowly due to the mathematical intractabilities encountered in the crucial problem of bandwidth (or smoothing parameter) selection. This paper presents the first fully automatic, data-based bandwidth selectors for multivariate kernel density derivative estimators. This is achieved by synthesizing recent advances in matrix analytic theory which allow mathematically and computationally tractable representations of higher order derivatives of multivariate vector valued functions. The theoretical asymptotic properties as well as the finite sample behaviour of the proposed selectors are studied. {In addition, we explore in detail the applications of the new data-driven methods for two other statistical problems: clustering and bump hunting. The introduced techniques are combined with the mean shift algorithm to develop novel automatic, nonparametric clustering procedures which are shown to outperform mixture-model cluster analysis and other recent nonparametric approaches in practice. Furthermore, the advantage of the use of smoothing parameters designed for density derivative estimation for feature significance analysis for bump hunting is illustrated with a real data example.