Tukey's Depth for Object Data

TL;DR

This paper introduces a new data depth measure for non-Euclidean object data, extending Tukey's depth to general metric spaces, enabling robust analysis and ranking of complex data types.

Contribution

The paper develops the metric halfspace depth, a novel extension of Tukey's depth for non-Euclidean data, with theoretical properties, an efficient algorithm, and practical applications.

Findings

Depth median shows high robustness as a location descriptor.

Algorithm effectively adapts to intrinsic data geometry.

Applications reveal group differences and outliers in complex data.

Abstract

We develop a novel exploratory tool for non-Euclidean object data based on data depth, extending the celebrated Tukey's depth for Euclidean data. The proposed metric halfspace depth, applicable to data objects in a general metric space, assigns to data points depth values that characterize the centrality of these points with respect to the distribution and provides an interpretable center-outward ranking. Desirable theoretical properties that generalize standard depth properties postulated for Euclidean data are established for the metric halfspace depth. The depth median, defined as the deepest point, is shown to have high robustness as a location descriptor both in theory and in simulation. We propose an efficient algorithm to approximate the metric halfspace depth and illustrate its ability to adapt to the intrinsic data geometry. The metric halfspace depth was applied to an Alzheimer's disease study, revealing group differences in the brain connectivity, modeled as covariance matrices, for subjects in different stages of dementia. Based on phylogenetic trees of 7 pathogenic parasites, our proposed metric halfspace depth was also used to construct a meaningful consensus estimate of the evolutionary history and to identify potential outlier trees.