Depth-based estimation for multivariate functional data with phase variability

In the context of multivariate functional data with individual phase variation, we develop a robust depth-based approach to estimate the main pattern function when cross-component time warping is also present. In particular, we consider the latent deformation model (Carroll and Müller, 2023) in which the different components of a multivariate functional variable are also time-distorted versions of a common template function. Rather than focusing on a particular functional depth measure, we discuss the necessary conditions on a depth function to be able to provide a consistent estimation of the central pattern, considering different model assumptions. We evaluate the method performance and its robustness against atypical observations and violations of the model assumptions through simulations, and illustrate its use on two real data sets.

💡 Research Summary

The paper addresses the challenging problem of estimating a common underlying pattern when multivariate functional observations are subject to two sources of phase variation: (i) individual warping that is shared across all components of a subject, and (ii) component‑specific warping that distorts each coordinate of the multivariate process. The authors adopt the latent deformation model (LDM) introduced by Carroll and Müller (2023), which assumes that each observed curve (X_{ij}(t)) can be written as
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