The Signed Translation Transformed Depth: Order, Quantiles, Spread, Skewness, and Quantile Regression for Multivariate Functional Data

Overview

Multivariate functional data play a central role in modern research across fields such as neuroscience, medicine, and environmental science. Key challenges in analyzing such data include extracting distributional characteristics, performing supervised and unsupervised learning, and building nonlinear functional regression models. Central to addressing these challenges is defining a suitable notion of order that captures the distributional properties of functional data. While functional depth has been widely used, its interpretation lacks symmetry with respect to the deepest curve, which limits its effectiveness in representing distributional aspects. To address this, we introduce the Signed Translation transformed Depth (STtD), a novel and interpretable notion of order for multivariate functional data. With this new notion in place, we define functional quantiles and descriptive tools such as probability bands, interquartile range, spread, and skewness for functional data analysis. Furthermore, building on STtD, we propose a flexible framework for multivariate functional quantile regression using vine copulas, allowing interaction between scalar and functional predictors. This research primarily advances the understanding of distributional analysis and nonlinear regression for functional data while offering tools that are both interpretable and adaptable to complex structures in high-dimensional settings.