Minimum regularized covariance trace estimator and outlier detection for functional data

TL;DR

This paper introduces the MRCT estimator, a robust covariance estimation method for functional data that effectively detects outliers, handles high-dimensional and sparse data, and automates regularization parameter selection.

Contribution

The paper presents the MRCT estimator, a novel robust covariance estimator for functional data that operates without preprocessing and adapts to sparsity, with automated regularization.

Findings

Demonstrates effective outlier detection and covariance estimation in simulations.

Performs favorably compared to existing functional outlier detection methods.

Handles high-dimensional and sparsely observed data efficiently.

Abstract

In this paper, we propose the Minimum Regularized Covariance Trace (MRCT) estimator, a novel method for robust covariance estimation and functional outlier detection. The MRCT estimator employs a subset-based approach that prioritizes subsets exhibiting greater centrality based on the generalization of the Mahalanobis distance, resulting in a fast-MCD type algorithm. Notably, the MRCT estimator handles high-dimensional data sets without the need for preprocessing or dimension reduction techniques, due to the internal smoothening whose amount is determined by the regularization parameter $\alpha > 0$. The selection of the regularization parameter $\alpha$ is automated. The proposed method adapts seamlessly to sparsely observed data by working directly with the finite matrix of basis coefficients. An extensive simulation study demonstrates the efficacy of the MRCT estimator in terms of robust covariance estimation and automated outlier detection, emphasizing the balance between noise exclusion and signal preservation achieved through appropriate selection of $\alpha$. The method converges fast in practice and performs favorably when compared to other functional outlier detection methods.