Abstract

For real-valued functional data, it is well known that failing to distinguish between amplitude variation and phase variation can contaminate subsequent statistical analysis. To address this, time warping methods have been extensively investigated. However, much less is known about handling phase variation in random processes that take values in a general metric space which by default does not have a linear structure. In this paper, we introduce a latent deformation model designed for multivariate random processes, where each component resides in a shared metric space and accounts for both amplitude and phase variations. We provide a thorough analysis of the asymptotic rates of convergence for estimates of subject-specific and component-specific warping functions, as well as the latent template trajectory. The finite-sample performance of our method is assessed through simulation studies, and its practical application is demonstrated by an analysis of yearly gender-specific age-at-death distributions from different countries, which provides valuable insights into comparisons of absolute longevity and the pace of longevity improvement across genders and countries.