Homogeneity Tests of Covariance for High-Dimensional Functional Data with Applications to Event Segmentation
Abstract: We consider inference problems for high-dimensional functional data with a dense number (T) of repeated measurements taken for a large number of p variables from a small number of n experimental units. The spatial and temporal dependence, high dimensionality, and the dense number of repeated measurements all make theoretical studies and computation challenging. This paper has two aims; our first aim is to solve the theoretical and computational challenges in testing equivalence among covariance matrices from high-dimensional functional data. The second aim is to provide computationally efficient and tuning-free tools with a guaranteed stochastic error control. The weak convergence of the stochastic process formed by the test statistics is established under the “large p, large T and small n'' setting. If the null is rejected, we further show that the locations of the change points can be estimated consistently. Its rate of convergence is shown to depend on the data dimension, sample size, number of repeated measurements, and signal-to-noise ratio. We also show that our proposed computation algorithms can significantly reduce the computation time and are applicable to real-world data with a large number of high-dimensional repeated measurements (e.g. fMRI data). Simulation results demonstrate both finite sample performance and computational effectiveness of our proposed procedures. We observe that the empirical size of the test is well controlled at the nominal level, and the locations of multiple change points can accurately be identified. An application to fMRI data demonstrates that our proposed methods can identify event boundaries in the preface of the movie Sherlock. Our proposed procedures are implemented in an R package TechPhD. This is a joint work with Shawn Santo (Duke).