Title: Reconstructing gradients from sparse functional data
Authors: Ian McKeague - Columbia University (United States) [presenting]
Abstract: Consider the problem of estimating growth curves $X_j(t), j=1,\ldots , N$ over a time interval $[0,T]$, from data on their integrals over gaps between $n$ observation times. We introduce a new Bayesian approach to this problem taking into account: (1) each trajectory $X_j$ is known to be uniformly bounded, (2) there is measurement error in the data, and (3) pairs of trajectories are unlikely to cross very often. To address (1), we make use of a hyperbolic-tangent transform applied to tied-down Brownian motion, as well as a multivariate normal at observation times, to form a natural bounded process for use as a prior. To address the measurement error issue (2), the log-transformed data are modeled with additive Gaussian noise. To address (3), we use $N$ non-intersecting tied-down Brownian bridges to provide an ensemble prior between the observation times.