Projection depth for functional data: Practical issues, computation and applications

1. Problem Statement

Functional Data Analysis (FDA) involves analyzing data that are curves, surfaces, or functions, which reside in infinite-dimensional Hilbert spaces. A central challenge in FDA is defining a meaningful ordering structure (ranking) for these data points to distinguish between central (typical) and outlying observations.

While statistical depth functions (extensions of multivariate depth to functional data) provide a solution, existing methods often struggle with:

• Complex Shape Features: Many traditional depths (e.g., integrated, band, or spatial depths) rely heavily on pointwise vertical deviations. They often fail to detect outliers that differ primarily in shape (e.g., phase shifts, frequency changes, or structural deformations) while remaining within the central region of the data cloud.
• Degeneracy in Infinite Dimensions: The classical Projection Depth (Zuo, 2003), which relies on projecting data onto all possible directions, becomes degenerate (yielding infinite outlyingness) in infinite-dimensional spaces, rendering it statistically useless for functional data.
• Practical Implementation: Even when theoretically defined, computing projection depth in high or infinite dimensions is computationally prohibitive.

2. Methodology

The paper investigates and refines the Regularized Projection Depth (RPD), a concept recently proposed by Bočinec et al. (2026), focusing on its practical application, parameter selection, and computation.

A. Regularized Projection Depth (RPD)

The RPD addresses the degeneracy of classical projection depth by restricting the set of projection directions.

• Outlyingness: For a direction $v$, the outlyingness of a point $x$ is defined using the median and Median Absolute Deviation (MAD) of the projected data:
  $$O_v(x; P_X) = \frac{|\langle x, v \rangle - \text{med}[\langle X, v \rangle]|}{\text{MAD}[\langle X, v \rangle]}$$
• Regularization: The set of valid directions $V_\beta$ is restricted to those where the MAD of the projection is at least $\beta$:
  $$V_\beta = \{v \in S : \text{MAD}[\langle X, v \rangle] \ge \beta\}$$
• Depth Definition: The RPD is the infimum of the inverse of $(1 + \text{outlyingness})$ over this restricted set:
  $$D_\beta(x; P_X) = \inf_{v \in V_\beta} (1 + O_v(x; P_X))^{-1}$$
  This ensures the depth is strictly positive and well-defined.

B. Data-Adaptive Parameter Selection

Instead of fixing $\beta$ arbitrarily, the authors propose selecting $\beta$ as the $u$-quantile of the distribution of $\text{MAD}[\langle X, V \rangle]$, where $V$ is a random direction.

• Significance: This transforms $\beta$ into a scalar parameter $u \in [0, 1)$.
  • Small $u$ (e.g., 0.001) implies weak regularization, retaining more directions, making the depth highly sensitive to shape differences.
  • Large $u$ implies stronger regularization, effectively reducing dimensionality and increasing robustness against noise.
• Theoretical Guarantees: The paper proves that this data-adaptive choice preserves key properties of the RPD, including non-degeneracy, consistency of the sample estimator, and a breakdown point of 1/2 for the induced median (under mild conditions).

C. Stochastic Approximation for Computation

Directly computing the infimum over an infinite set of directions is impossible. The authors propose a random projection-based stochastic approximation:

1. Estimate the quantile $\beta$ using $L$ random directions.
2. Sample $M$ directions from the distribution conditioned on the estimated valid set $V_{\hat{\beta}}$.
3. Approximate the depth by minimizing outlyingness over these $M$ directions.

• Convergence: Theorem 5 establishes that this approximation converges almost surely to the true RPD as $M, L \to \infty$.
• Implementation: The method is implemented in an R package (RPD) using C++ via RcppArmadillo for efficiency.

3. Key Contributions

1. Practical Framework for RPD: The paper moves RPD from a theoretical construct to a practical tool by defining a data-driven method for selecting the regularization parameter ($u$-quantile) and establishing its theoretical validity.
2. Stochastic Computation Algorithm: It introduces an efficient, scalable algorithm for computing RPD in functional settings, overcoming the curse of dimensionality through random projections.
3. Comprehensive Empirical Evaluation: The authors conduct extensive simulations across four distinct statistical tasks, comparing RPD against state-of-the-art functional depths (Regularized Halfspace Depth, Integrated Depth, Modified Band Depth, Infimal Depth, Spherical Depth).
4. Shape Sensitivity Discovery: The study demonstrates that RPD is uniquely capable of detecting shape-based anomalies (phase shifts, frequency changes, structural differences) that other depth measures miss because they focus on pointwise magnitude.

4. Results

The paper evaluates RPD in four scenarios:

A. Outlier Detection

• Setup: 500 curves with 50 outliers of various types (magnitude, location, covariance, phase, frequency, shape).
• Findings: RPD (specifically with small $u$, e.g., 0.001) achieves near-perfect detection (average rank of outliers $\approx 0.051$) across all models.
• Comparison: Competing methods (FD, MBD, SD) perform well for magnitude/location outliers but fail significantly on shape outliers (Models D5 and D6), often ranking shape outliers as central. RHD performs better but still lags behind RPD.

B. Supervised Classification

• Setup: Binary classification using DD-plots (Depth-Depth plots) and linear classifiers.
• Findings: RPD enables almost perfect separation of classes even when groups differ only in shape (e.g., central jumps or dimension differences), whereas other depths produce overlapping clouds.
• Efficiency: A simple linear classifier on RPD-based DD-plots suffices for complex shape-differentiated data, whereas other depths require more complex decision boundaries or fail entirely.

C. Hypothesis Testing

• Setup: Two-sample Kruskal-Wallis tests based on depth rankings.
• Findings: RPD-based tests maintain the correct nominal size (Type I error) and exhibit high power, particularly for detecting differences in eigenfunctions (shape) and eigenvalues (smoothness). Other depth-based tests often suffer from inflated Type I errors (oversized tests).

D. Robust Location Estimation

• Setup: Estimating the "deepest" curve (median) under various contamination models (shift, amplitude, phase, shape, peak).
• Findings:
  • For shape detection (outlier tasks), small $u$ is optimal.
  • For robust estimation (median tasks), larger $u$ (e.g., 0.5) improves performance by reducing variance, confirming the theoretical trade-off between sensitivity and robustness.
  • RPD demonstrates superior robustness compared to RHD when contamination involves large shifts, as RPD uses the robust MAD for variance estimation, whereas RHD relies on the non-robust sample covariance.

5. Significance

This paper establishes Regularized Projection Depth (RPD) as a superior tool for functional data analysis, particularly in scenarios where shape features are critical.

• Theoretical Impact: It resolves the degeneracy of projection depth in infinite dimensions and provides a rigorous framework for data-adaptive regularization.
• Practical Impact: It offers a computationally feasible algorithm that outperforms existing methods in detecting subtle structural anomalies (phase, frequency, shape) which are common in real-world functional data (e.g., medical signals, climate data, financial curves).
• Versatility: The method is shown to be effective across the entire spectrum of statistical tasks, from outlier detection to robust estimation, with the flexibility to tune sensitivity via the parameter $u$.

In conclusion, the authors demonstrate that RPD is not just a theoretical extension but a practical, robust, and highly effective method that addresses the specific complexities of ordering and analyzing functional data.