People

We propose a two-step procedure to model and predict high-dimensional functional time series, where the number of function-valued time series p is large in relation to the length of time series n. Our first step performs an eigenanalysis of a positive definite matrix, which leads to a one-to-one linear transformation for the original high-dimensional functional time series, and the transformed curve series can be segmented into several groups such that any two subseries from any two different groups are uncorrelated both contemporaneously and serially. Consequently in our second step those groups are handled separately without the information loss on the overall linear dynamic structure. The second step is devoted to establishing a finite-dimensional dynamical structure for all the transformed functional time series within each group. Furthermore the finite-dimensional structure is represented by that of a vector time series. Modeling and forecasting for the original high-dimensional functional time series are realized via those for the vector time series in all the groups. We investigate the theoretical properties of our proposed methods, and illustrate the finite-sample performance through both extensive simulation and two real datasets. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

This paper addresses the fundamental task of estimating covariance matrix functions for high-dimensional functional data/functional time series. We consider two functional factor structures encompassing either functional factors with scalar loadings or scalar factors with functional loadings, and postulate functional sparsity on the covariance of idiosyncratic errors after taking out the common unobserved factors. To facilitate estimation, we rely on the spiked matrix model and its functional generalization, and derive some novel asymptotic identifiability results, based on which we develop DIGIT and FPOET estimators under two functional factor models, respectively. Both estimators involve performing associated eigenanalysis to estimate the covariance of common components, followed by adaptive functional thresholding applied to the residual covariance. We also develop functional information criteria for model selection with theoretical guarantees. The convergence rates of involved estimated quantities are respectively established for DIGIT and FPOET estimators. Numerical studies including extensive simulations and a real data application on functional portfolio allocation are conducted to examine the finite-sample performance of the proposed methodology.

Nonparametric estimation of the mean and covariance functions is ubiquitous in functional data analysis and local linear smoothing techniques are most frequently used. Zhang and Wang (2016) explored different types of asymptotic properties of the estimation, which reveal interesting phase transition phenomena based on the relative order of the average sampling frequency per subject TT to the number of subjects nn, partitioning the data into three categories: “sparse”, “semi-dense”, and “ultra-dense”. In an increasingly available high-dimensional scenario, where the number of functional variables pp is large in relation to nn, we revisit this open problem from a non-asymptotic perspective by deriving comprehensive concentration inequalities for the local linear smoothers. Besides being of interest by themselves, our non-asymptotic results lead to elementwise maximum rates of L2L2 convergence and uniform convergence serving as a fundamentally important tool for further convergence analysis when pp grows exponentially with nn and possibly TT. With the presence of extra logplog⁡p terms to account for the high-dimensional effect, we then investigate the scaled phase transitions and the corresponding elementwise maximum rates from sparse to semi-dense to ultra-dense functional data in high dimensions. We also discuss a couple of applications of our theoretical results. Finally, numerical studies are carried out to confirm the established theoretical properties.