Functional Convex Averaging and Synchronization for
Time-Warped Random Curves
Monday, 4 October 2004
Functional data for dynamic phenomena may be time-warped. It is then often inadequate to use commonly employed sample statistics such as the cross-sectional mean or the cross-sectional sample variance. If one observes trajectories that exhibit random transformations of the time scale, one may consider subjecting each observed curve to a time transformation in an attempt to reverse the warping of the time scale. For this purpose, 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 to develop a functional convex calculus. Observed random functions in warped time space are represented by a bivariate random function in synchronized time space. This leads to the definition of a functional convex average. We discuss various implementations of functional convex averaging and derive a functional limit theorem and asymptotic confidence intervals for functional convex means. In a second part, we discuss the implementation of a simple time-shift warping method and its application to gene expression profiles. This talk is based on joint work with Xueli Liu and Xiaoyan Leng.
Refreshments will be served at 3:45 p.m. in Room 008 of the Statistics Building