  Multivariate functional data present theoretical and practical complications
which are not found in univariate functional data. One of these is a situation
where the component functions of multivariate functional data are positive and
are subject to mutual time warping. That is, the component processes exhibit a
common shape but are subject to systematic phase variation across their domains
in addition to subject-specific time warping, where each subject has its own
internal clock. This motivates a novel model for multivariate functional data
that connects such mutual time warping to a latent deformation-based framework
by exploiting a novel time warping separability assumption. This separability
assumption allows for meaningful interpretation and dimension reduction. The
resulting Latent Deformation Model is shown to be well suited to represent
commonly encountered functional vector data. The proposed approach combines a
random amplitude factor for each component with population based registration
across the components of a multivariate functional data vector and includes a
latent population function, which corresponds to a common underlying
trajectory. We propose estimators for all components of the model, enabling
implementation of the proposed data-based representation for multivariate
functional data and downstream analyses such as Fr\'echet 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 and multivariate environmental pollution data.
