  Time warping function provides a mathematical representation to measure phase
variability in functional data. Recent studies have developed various
approaches to estimate optimal warping between functions and provide
non-Euclidean models. However, a principled, linear, generative model on time
warping functions is still under-explored. This is a highly challenging problem
because the space of warping functions is non-linear with the conventional
Euclidean metric. To address this problem, we propose a stochastic process
model for time warping functions, where the key is to define a linear,
inner-product structure on the time warping space and then transform the
warping functions into a sub-space of the $\mathbb L^2$ Euclidean space. With
certain constraints on the warping functions, this transformation is an
isometric isomorphism. In the transformed space, we adopt the $\mathbb L^2$
basis in the Hilbert space for representation. This new framework can easily
build generative model on time warping by using different types of stochastic
process. It can also be used to conduct statistical inferences such as
functional PCA, functional ANOVA, and functional regressions. Furthermore, we
demonstrate the effectiveness of this new framework by using it as a new prior
in the Bayesian registration, and propose an efficient gradient method to
address the important maximum a posteriori estimation. We illustrate the new
Bayesian method using simulations which properly characterize nonuniform and
correlated constraints in the time domain. Finally, we apply the new framework
to the famous Berkeley growth data and obtain reasonable results on modeling,
resampling, group comparison, and classification analysis.
