UC Davis

Multivariate functional data often present theoretical and practical complications which are not found in their univariate functional counterparts. One of these is a situation where the component functions of multivariate functional data are subject to mutual time warping. That is, the component processes exhibit a similar shape but are subject to systematic phase variation across their time domains. This dissertation addresses this previously unconsidered mode of warping with the introduction of multivariate time warping models which rely on either time-shifting or nonlinear time-distortion frameworks. In the first chapter, we introduce a shift-warping model for multivariate time relations. This model is motivated by the Zürich longitudinal growth dataset, in which the growth trajectories for multiple body parts were observed from birth to adulthood for a sample of children. The proposed method differs from existing registration methods for functional data in a fundamental way. Namely, instead of focusing on the traditional approach to warping, where one aims to recover individual-specific registration, this technique focuses on shift registration across the components of a multivariate functional data vector on a population-wide level. After applying the method to the Zürich data, we find that there exists an archetypal ordering of pubertal growth spurts across modalities: on average, legs tend to experience peak growth velocity approximately a half a year before standing height and arms, which themselves tend to precede the growth spurt of the spine by half a year. Our proposed estimates for these shifts are identifiable, enjoy parametric rates of convergence, and often have intuitive physical interpretations, all in contrast to traditional curve-specific registration approaches. Finite sample properties of these estimators were also investigated in simulation studies.

The second chapter of this dissertation introduces the Latent Transport Model for multivariate functional data. This model widens the class of possible cross-component warps from simple shifts to flexible and nonlinear transport functions. The proposed approach combines a random amplitude factor for each component with population-based registration across the components of a multivariate functional data vector. It also includes a latent population function, which corresponds to a common underlying trajectory as well as a subject-specific warping component. This model allows for meaningful interpretation and is well suited to represent functional vector data. We also propose estimators for all components of the model, which not only lead to a novel representation for multivariate functional data, but can also be used in downstream analyses like Fréchet regression. Rates of convergence are established when curves are fully observed or observed with measurement error. The usefulness of the model, interpretations, and practical aspects are illustrated in simulations and with application to multivariate human growth curves as well as multivariate environmental pollution data.