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.

翻译：针对存在个体相位变化的多元函数数据，本文开发了一种鲁棒的基于深度的方法，用于估计存在跨分量时间扭曲时的主模式函数。具体而言，我们考虑潜在变形模型（Carroll and Müller, 2023），其中多元函数变量的不同分量也是一个共同模板函数的时间扭曲版本。我们并不专注于特定的函数深度度量，而是讨论了在不同模型假设下，深度函数能够为中心模式提供一致估计所需满足的必要条件。通过模拟实验，我们评估了该方法的性能及其对非典型观测值和模型假设违反情况的鲁棒性，并在两个实际数据集上展示了其应用。