A Stochastic Process Model for Time Warping Functions

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.