Analytical and statistical properties of local depth functions motivated by clustering applications

TL;DR

This paper studies local depth functions for multivariate data, establishing their theoretical properties, developing a clustering algorithm based on these properties, and demonstrating its effectiveness through experiments.

Contribution

It provides a rigorous analysis of local general depth functions, introduces a new clustering algorithm, and proves its consistency with applications to mode and level set estimation.

Findings

Scaled local depth functions converge to the density function

Sample local depth functions converge to a Gaussian process

Proposed clustering algorithm is consistent and effective

Abstract

Local general depth ($LGD$) functions are used for describing the local geometric features and mode(s) in multivariate distributions. In this paper, we undertake a rigorous systematic study of $LGD$ and establish several analytical and statistical properties. First, we show that, when the underlying probability distribution is absolutely continuous with density $f(\cdot)$, the scaled version of $LGD$ (referred to as $\tau$-approximation) converges, uniformly and in $L^d(\mathbb{R}^p)$ to $f(\cdot)$ when $\tau$ converges to zero. Second, we establish that, as the sample size diverges to infinity the centered and scaled sample $LGD$ converge in distribution to a centered Gaussian process uniformly in the space of bounded functions on $\mathcal{H}_G$, a class of functions yielding $LGD$. Third, using the sample version of the $\tau$-approximation ($S \tau A$) and the gradient system analysis, we develop a new clustering algorithm. The validity of this algorithm requires several results concerning the uniform finite difference approximation of the gradient system associated with $S \tau A$. For this reason, we establish \emph{Bernstein}-type inequality for deviations between the centered and scaled sample $LGD$, which is also of independent interest. Finally, invoking the above results, we establish consistency of the clustering algorithm. Applications of the proposed methods to mode estimation and upper level set estimation are also provided. Finite sample performance of the methodology are evaluated using numerical experiments and data analysis.