Robust functional PCA for relative data

Technical Summary: Robust Functional PCA for Relative Data

Problem Statement
Functional Data Analysis (FDA) has traditionally focused on absolute magnitude, yet many applied settings involve "relative" functional data, such as probability density functions (PDFs), age-specific fertility rates, or spectral curves. In these contexts, the information of interest lies in the distributional pattern or shape rather than the absolute scale. Standard FDA techniques often fail in this domain because they do not respect the compositional nature of the data (non-negativity and unit integral constraints).

While recent work has established the Bayes space framework to model such data as infinite-dimensional compositions, existing dimension reduction tools within this space, specifically Simplicial Functional PCA (SFPCA), rely on sample covariance operators. These standard estimators are highly sensitive to outliers and anomalous observations. In relative data, outliers often manifest as structural shape anomalies (e.g., unexpected redistribution of mass) rather than simple magnitude deviations, rendering traditional robust methods (designed for Euclidean spaces) or standard SFPCA ineffective. There is a lack of robust methodologies capable of handling these anomalies while preserving the geometric structure of the Bayes space.

Methodology
The paper proposes Robust Density Principal Component Analysis (RDPCA), a method designed to estimate functional principal components (FPCs) for density data in the presence of outliers. The methodology proceeds through the following theoretical and algorithmic steps:

1. Bayes Space Framework: The data $X$ is modeled as an element of a Hilbert space $B^2(I)$ of positive densities with unit integral. The analysis utilizes the centered logratio (clr) transformation, which maps the Bayes space isometrically to a subspace of $L^2(I)$ (functions integrating to zero), allowing the use of standard Hilbert space tools while respecting the relative nature of the data.

2. Regularized Mahalanobis Distance (RDMD):

   • The core innovation is the extension of the Mahalanobis distance to Bayes spaces. In infinite-dimensional settings, the covariance operator is not invertible. To address this, the authors define a regularized standardization of a density $X$ by solving a Tikhonov-type optimization problem:
     $$X^{\alpha, L}_{st} = \arg \min_{Y \in B^2(I)} \left\{ \|C^{1/2}_{B^2,X} Y \ominus X\|^2_{B} + \alpha \|LY\|^2_{B} \right\}$$
     where $C_{B^2,X}$ is the covariance operator, $\alpha > 0$ is a regularization parameter, and $L$ is a regularization operator (e.g., an orthogonal projector onto the complement of the first $k$ eigenfunctions).
   • The Regularized Density Mahalanobis Distance (RDMD) is then defined as the squared norm of the difference between the standardized densities: $M^2_{\alpha,k,B^2}(X, Y) = \|X^{\alpha,k}_{st} \ominus Y^{\alpha,k}_{st}\|^2_{B^2}$.
   • This distance generalizes previous functional Mahalanobis concepts (truncated and $\alpha$-Mahalanobis distances) and accounts for the specific constraints of density data.

3. Robust Covariance Estimation:

   • Using the RDMD as a measure of centrality, the method employs a trimmed covariance estimator. An optimal subset $H_0$ of the data is identified by minimizing the average squared RDMD of the observations in the subset.
   • The robust mean and covariance are calculated using only this central subset. A scaling factor is applied to ensure consistency under Gaussian assumptions, though the method does not strictly require Gaussianity for the estimation itself.

4. Algorithm (RDPCA):

   • The procedure is iterative. It alternates between estimating the robust mean/covariance on the current subset and re-evaluating the subset based on the updated RDMDs until convergence.
   • The method supports both discretized data (approximated via integral sums) and functional representations (projected onto a basis, such as compositional splines). Proposition 4.1 demonstrates that when data is expressed in a fixed orthonormal basis, the RDMD can be computed exactly using matrix operations on the basis coefficients.

Key Results
The performance of RDPCA was evaluated through simulation studies and real-world applications:

• Simulations:

  • In the presence of outliers (introduced via Huber's $\epsilon$-contamination model), RDPCA significantly outperformed standard SFPCA in estimating both the covariance operator and the principal components.
  • Metrics included Integrated Square Error (ISE) for covariance estimation and mean cosine similarity for eigenfunctions. RDPCA maintained high accuracy even with contamination rates up to 20%.
  • Interestingly, RDPCA also showed superior performance to SFPCA in non-contaminated settings, particularly when the underlying score distribution was heavy-tailed (e.g., multivariate $t$-distribution).
  • The method successfully identified outliers based on the limiting distribution of the RDMD, whereas non-robust methods failed to distinguish atypical curves.

• Real-Data Applications:

  • EPXMA Spectra: Applied to glass vessel spectra, RDPCA effectively distinguished between different glass types and identified outlying spectra that standard methods missed. The robust correlation function revealed structural differences obscured by the bias of non-robust methods.
  • Fertility Data: Analyzed age-specific fertility rates (ASFR) across countries and years. The method identified countries with unusual fertility patterns (e.g., Japan in 1956) and captured the evolution of fertility dynamics (shifting peaks from early 20s to around 30 years) more robustly than standard approaches.

Significance and Claims
The paper claims to provide the first robust framework for PCA specifically tailored to relative functional data within the Bayes space. Its primary contributions are:

1. Theoretical Extension: It extends the concept of regularized Mahalanobis distance to Bayes spaces, addressing the non-invertibility of the covariance operator in infinite dimensions while respecting the compositional geometry of density data.
2. Robustness to Structural Anomalies: Unlike traditional FDA methods that treat outliers as magnitude deviations, RDPCA is designed to detect and downweight structural shape anomalies inherent in relative data.
3. Practical Utility: The method offers a computationally feasible algorithm (Algorithm 1) that works with both discretized and basis-expanded data, providing a viable tool for analyzing density functions in the presence of noise and outliers.

The authors emphasize that the goal is not merely outlier detection but the construction of robust PCA methods that yield accurate covariance and principal component estimates. They note that while the current work focuses on univariate densities, the framework opens avenues for future extensions to multivariate densities and random surfaces, though these remain outside the current scope.