A stochastic process representation 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. However, a principled, linear, generative representation on time warping functions is still under-explored. This is highly challenging because the warping functions are non-linear in the conventional L2 space. To address this problem, a new linear warping space is defined and a stochastic process representation is proposed to characterize time warping functions. The key is to define an inner-product structure on the time warping space, followed by a transformation which maps the warping functions into a sub-space of the L2 space. With certain constraints on the warping functions, this transformation is an isometric isomorphism. In the transformed space, the L2 basis in the Hilbert space is adopted for representation, which can be easily utilized to generate time warping functions by using different types of stochastic process. The effectiveness of this representation is demonstrated through its use as a new penalty in the penalized function registration, accompanied by an efficient gradient method to minimize the cost function. The new penalized method is illustrated through simulations that properly characterize nonuniform and correlated constraints in the time domain. Furthermore, this representation is utilized to develop a boxplot for warping functions, which can estimate templates and identify warping outliers. Finally, this representation is applied to a Covid-19 dataset to construct boxplots and identify states with outlying growth patterns.