On the Modeling and Prediction of High-Dimensional Functional Time Series

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.本文提出一种两步建模与预测方法，用于高维函数型时间序列（high-dimensional functional time series）的分析与预测——此类序列中函数型时间序列的维度p远大于时间序列长度n。第一步对正定矩阵（positive definite matrix）开展特征分析，由此得到针对原始高维函数型时间序列的一一对应线性变换；经变换后的曲线序列可被划分为若干组，使得任意两组间的任意两个子序列在同期与序列相关性层面均互不相关。因此在第二步中，我们可独立处理各个分组，且不会损失整体线性动态结构的相关信息。第二步的核心是为每个分组内的所有变换后函数型时间序列构建有限维动态结构，且该有限维结构以向量时间序列（vector time series）的形式进行表征。通过对所有分组内的向量时间序列进行建模与预测，即可实现对原始高维函数型时间序列的建模与预测。本文对所提方法的理论性质展开了推导分析，并通过大规模模拟实验与两个真实数据集，验证了该方法在有限样本下的表现。本文的补充材料可在线获取，其中包含了用于复现本研究工作的相关材料的标准化说明。