Homogeneity Tests of Covariance and Change-Points Identification for High-Dimensional Functional Data

Abstract

We consider inference problems for high-dimensional (HD) 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 detecting and identifying change points among covariance matrices from HD functional data. The second aim is to provide computationally efficient and tuning-free tools with a guaranteed stochastic error control. The change point detection procedure is developed in the form of testing the homogeneity of covariance matrices. The weak convergence of the stochastic process formed by the test statistics is established under the "large p, large T and small n" setting. Under a mild set of conditions, our change point identification estimator is proven to be consistent for change points in any location of a sequence. Its rate of convergence depends 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 such as fMRI data with a large number of HD repeated measurements. 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.