Abstract
Statistical analysis of high-dimensional functional times series arises in various applications. Under this scenario, in addition to the intrinsic infinite-dimensionality of functional data, the number of functional variables can grow with the number of serially dependent observations. In this paper, we focus on the theoretical analysis of relevant estimated cross-(auto)covariance terms between two multivariate functional time series or a mixture of multivariate functional and scalar time series beyond the Gaussianity assumption. We introduce a new perspective on dependence by proposing functional cross-spectral stability measure to characterize the effect of dependence on these estimated cross terms, which are essential in the estimates for additive functional linear regressions. With the proposed functional cross-spectral stability measure, we develop useful concentration inequalities for estimated cross-(auto)covariance matrix functions to accommodate more general sub-Gaussian functional linear processes and, furthermore, establish finite sample theory for relevant estimated terms under a commonly adopted functional principal component analysis framework. Using our derived non-asymptotic results, we investigate the convergence properties of the regularized estimates for two additive functional linear regression applications under sparsity assumptions including functional linear lagged regression and partially functional linear regression in the context of high-dimensional functional/scalar time series.
Highlights
Before presenting relevant non-asymptotic results beyond Gaussian functional time series, we introduce the definitions of sub-Gaussian process and multivariate functional linear process
We focus on sample cross-(auto)covariance between estimated scores, σphX,jYklm pnhq1 řnh t“1 ζptj l ξppthqkm, and establish a normalized deviation bound in elementwise 8 norm on how σphX,jYklm concentrates around σhX,jYklm “ Covpζtjl, ξpthqkmq
We examine the performance of 1{ 2-LS based on area under the ROC curve (AUROC) and estimation errors, and compare it with the performance of ordinary least squares in the oracle case (OLS-O), where the sparsity structures in the estimates are determined by the true model in advance
Summary
Examples of high-dimensional functional time series include daily electricity consumption curves (Cho et al, 2013) for a large collection of households, half-hourly measured PM10 curves (Aue, Norinho and Hormann, 2015) over a large number of sites and cumulative intraday return curves (Horvath, Kokoszka and Rice, 2014) for hundreds of stocks. These applications require developing learning techniques to handle such type of data. One large class considers imposing various functional sparsity assumptions on the model parameter space, e.g. vector functional autoregressions (VFAR) (Guo and Qiao, 2021) and, under a special independent setting, functional graphical models (Qiao, Guo and James, 2019) and functional additive regressions (Fan et al, 2014; Fan, James and Radchenko, 2015; Kong et al, 2016; Luo and Qi, 2017; Xue and Yao, 2021), where the corresponding regularized estimates are proposed
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