Functional convex averaging and synchronization for time-warped random curves

Data that can be best described as a sample of curves are now fairly common in science and engineering. When the dynamics of development, growth, or response over time are at issue, subjects or experimental units may experience events at different temporal paces. For functional data where trajectories may be individually time-transformed, it is usually inadequate to use commonly used sample statistics, such as the cross-sectional mean or median or the cross-sectional sample variance. If one observes time-warped curve data (i.e., random curves or random trajectories that exhibit random transformations of the time scale), then the usual L-2 norm and metric typically are inadequate. One may then consider subjecting each observed curve to a time transformation in an attempt to reverse the warping of the time scale before further statistical analysis. Dynamic time warping, alignment, curve registration, and landmark-based methods have been put forward with the goal of finding adequate empirical time transformations. Previous analyses of warping typically have not been based on a model in which individual observed curves are viewed as realizations of a stochastic process. We propose a functional convex synchronization model, under the premise that each observed curve is the realization of a stochastic process. Monotonicity constraints on time evolution provide the motivation for a functional convex calculus with the goal of obtaining sample statistics such as a functional mean. Observed random functions in warped time space are represented by a bivariate random function in synchronized time space, consisting of a stochastic monotone time transformation function and an unrestricted random amplitude function. Our theory assumes a monotone time warping transformation that maps synchronized time to warped (i.e., observed) time. This leads to the definition of a functional convex average or longitudinal average, in contrast to the conventional cross-sectional average. We discuss various implementations of functional convex averaging and derive a functional limit theorem and asymptotic confidence intervals for functional convex means. The results are illustrated with a novel time-warping transformation and extend to commonly used warping and registration methods, such as landmark registration. The methods are applied to simulated data and the Berkeley growth data.