Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this talk, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions p is comparable to, or even larger than the sample size n. Aided by the Hilbert–Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding estimator by incorporating the variance effects of individual entries of the sample covariance function into functional thresholding. To handle the practical scenario where curves are partially observed with errors, we also develop a nonparametric smoothing approach to obtain the smoothed adaptive functional thresholding estimator and its binned implementation to accelerate the computation. We investigate the theoretical properties of our proposals when p grows exponentially with n under both fully and partially observed functional scenarios. Finally, we demonstrate that the proposed adaptive functional thresholding estimators significantly outperform the competitors through extensive simulations and the functional connectivity analysis of a neuroimaging dataset.
|About the speaker||
Dr. Xinghao Qiao is an Associate Professor in the Department of Statistics of the London School of Economics and Political Science. He obtained PhD in Business Statistics from Marshall School of Business at the University of Southern California. His research focuses on functional and longitudinal data analysis, high dimensional statistical inference, time series analysis and statistical machine learning with applications in Business, Neuroimaging Analysis and Environmental Sciences.