Directional Outlyingness for Multivariate Functional Data

TL;DR

This paper introduces a new concept called directional outlyingness for multivariate functional data, decomposing outlyingness into magnitude and shape, and develops visualization and outlier detection tools that outperform existing methods.

Contribution

It generalizes classical depth to directional outlyingness, providing a novel decomposition, visualization, and outlier detection framework for multivariate functional data.

Findings

Decomposition into magnitude and shape outlyingness.

Outlier detection method outperforms competing approaches.

Practical applications demonstrated on weather and ECG data.

Abstract

The direction of outlyingness is crucial to describing the centrality of multivariate functional data. Motivated by this idea, we generalize classical depth to directional outlyingness for functional data. We investigate theoretical properties of functional directional outlyingness and find that the total outlyingness can be naturally decomposed into two parts: magnitude outlyingness and shape outlyingness which represent the centrality of a curve for magnitude and shape, respectively. Using this decomposition, we provide a visualization tool for the centrality of curves. Furthermore, we design an outlier detection procedure based on functional directional outlyingness. This criterion applies to both univariate and multivariate curves and simulation studies show that it outperforms competing methods. Weather and electrocardiogram data demonstrate the practical application of our proposed framework.