Projection depth for functional data: Theoretical properties

2025-12-23

AI Summary

Functional data in infinite-dimensional Hilbert spaces pose challenges for classical projection depth, including degeneracy, reliance on moment conditions, and inconsistent convergence of sample depth. To address these, we propose Regularized Projection Depth (RPD), which imposes regularization constraints on projection directions to circumvent degeneracy arising from the non-compactness of direction sets in infinite dimensions. RPD is the first functional depth measure satisfying all six fundamental depth axioms—affine invariance (here, translation and scale invariance), monotonicity relative to deepest points, vanishing at infinity, continuity, quasi-concavity, and maximality at center—without any moment assumptions. We establish uniform consistency of its sample version. Practically, RPD yields a highly robust functional median estimator that avoids rank ties and significantly enhances detection of shape outliers. Thus, RPD provides a theoretically rigorous and practically superior tool for robust inference with functional data.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We introduce a novel projection depth for data lying in a general Hilbert space, called the regularized projection depth, with a focus on functional data. By regularizing projection directions, the proposed depth does not suffer from the degeneracy issue that may arise when the classical projection depth is naively defined on an infinite-dimensional space. Compared to existing functional depth notions, the regularized projection depth has several advantages: (i) it requires no moment assumptions on the underlying distribution, (ii) it satisfies many desirable depth properties including invariance, monotonicity, and vanishing at infinity, (iii) its sample version uniformly converges under mild conditions, and (iv) it generates a highly robust median. Furthermore, the proposed depth is statistically useful as it (v) does not produce ties in the induced ranks and (vi) effectively detects shape outlying functions. This paper focuses mainly on the theoretical properties of the regularized projection depth.

Problem

• Introduces regularized projection depth for functional data in Hilbert spaces
• Addresses degeneracy issues of classical projection depth in infinite dimensions
• Ensures robust statistical properties without moment assumptions

Innovation

• Regularized projection depth for functional data
• Avoids degeneracy by regularizing projection directions
• No moment assumptions and robust median generation