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 functional observations. However, a principled, generative model on time warping functions is still under-explored. This is a 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 transform the warping function space into an inner-product space with the L2 metric. With certain constraints on the warping functions, the transformation is an isometric isomorphism. In the transformed space, we adopt the L2 basis in Hilbert space for representation. We demonstrate the effectiveness of this new framework using three applications: (1) We build a stochastic process model on observed warping functions with estimated basis functions, and then use the model to resample warping functions. (2) The proposed model is used as a prior term in Bayesian registration, which results in reasonable alignment performance. (3) We apply the modeling framework to the Berkeley growth dataset to conduct resampling and perform statistical testing using functional ANOVA on the transformed data in Euclidean space.