Abstract: Existing functional principal component analysis (FPCA) methods are restricted to data with a single or finite number of random functions (much smaller than the sample size n). In this work, we focus on high-dimensional functional processes where the number of random functions p is comparable to, or even much larger than n. Such data are ubiquitous in various fields, such as neuroimaging analysis, and cannot be modeled properly by existing methods. We propose a new algorithm, called sparse FPCA, that models principal eigenfunctions effectively under sensible sparsity regimes. The sparsity structure motivates a thresholding rule that is easy to compute by exploiting the relationship between univariate orthonormal basis expansions and the multivariate Karhunen–Loève representation. We investigate the theoretical properties of the resulting estimators, and illustrate the performance using simulated and real-data examples.

Key words and phrases: Basis expansion, multivariate Karhunen-Loève expansion, sparsity regime.