Statistical learning with dependent functional data

We propose new methods inspired by elastic data analysis to for spatially-dependent functional data. We first propose a new classification algorithm of fMRI data based on the notion of amplitude and phase variability separation and a new between-voxel similarity measure based on the spatial dependence structure of the fMRI data. The approach is based on support vector machine (SVM) with a new kernel. The new kernel varies by the functional structure of interest in the brain, and it also captures the amplitude variability of the activation in the region of interest (ROI). This classification algorithm is illustrated via simulated datasets as well as a real dataset on schizophrenia, and its performance is compared with existing SVM based algorithms. Second, we propose a new Bayesian registration model that incorporates the spatial dependence structure in the priors. We propose a Bayesian registration model with a geometrically-inspired conditional autoregressive prior on warping functions. The CAR prior allows for warping functions to be dependent a priori, with dependencies characterized by a graph representing spatial structure of the data. The posterior is sampled via a MCMC algorithm that is robust to the discretization of the observed functions. This graph-based Bayesian registration algorithm is shown to be robust to outliers in a simulation study. Its performance is compared with an existing Bayesian registration algorithm.