Time-warping and convex synchronization for random curves, with applications to biological functional data.

摘要

In biological, chemical or physical experiments, data that can be best described as a sample of curves are now fairly common. When the dynamics of development, growth or response over time are at issue, subjects or experimental units may experience events at a different temporal pace. For functional data where trajectories may be individually time-transformed, it is usually inadequate to use commonly employed sample statistics for curves such as cross-sectional mean or median, or cross-sectional sample variance. Previous analysis of time warping which was motivated from speech recognition has typically not been based on a model where individual observed curves are viewed as realization 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. A model has been developed where observed random functions are assumed to have a randomly warped time scale. They are described by a latent bivariate random process that consists of a time transformation function and an amplitude function. Estimates and algorithms have been developed within the framework of this model. Important applications are the definition of a longitudinal sample mean (as opposed to the cross-sectional sample mean), the estimation of time warping functions for individual subjects and groups of subjects, and a synchronized mean update algorithm to find mode functions for samples of time-warped stochastic processes. We discuss various examples of functional convex averaging. The methods are illustrated with simulated data and mRNA gene expression microarray data. Theoretical motivation is provided by an asymptotic functional limit theorem, which can be used to construct asymptotic confidence bands for the functional convex mean function.