Tukey's Depth for Object Data.

Area of Science:

• * Statistics
• * Data Science
• * Computational Biology

Background:

• * Traditional data depth methods are limited to Euclidean spaces.
• * Analyzing non-Euclidean data, such as covariance matrices and phylogenetic trees, requires advanced techniques.
• * Centrality measures are crucial for understanding data distributions and identifying key features.

Purpose of the Study:

• * To introduce metric halfspace depth, a novel data depth measure for general metric spaces.
• * To establish theoretical properties generalizing Tukey's depth for non-Euclidean data.
• * To demonstrate the utility of metric halfspace depth in real-world applications.

Main Methods:

• * Development of the metric halfspace depth concept for arbitrary metric spaces.
• * Theoretical analysis establishing generalization of standard depth properties.
• * Implementation of an efficient algorithm for approximating metric halfspace depth.
• * Application to Alzheimer's disease brain connectivity data (covariance matrices).
• * Application to phylogenetic trees of pathogenic parasites.

Main Results:

• * Metric halfspace depth provides interpretable center-outward rankings for non-Euclidean data.
• * The depth median demonstrates robust location descriptor properties.
• * The method adapts to intrinsic data geometry, outperforming standard approaches.
• * Significant group differences in brain connectivity were identified in Alzheimer's patients.
• * A meaningful consensus evolutionary history was constructed, and outlier trees identified.

Conclusions:

• * Metric halfspace depth is a versatile and powerful tool for exploratory data analysis in non-Euclidean spaces.
• * The method offers robust centrality estimation and valuable insights in diverse scientific fields.
• * Metric halfspace depth advances the analysis of complex data structures like covariance matrices and phylogenetic trees.