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. Modelling 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.

翻译：我们提出一种两步法来建模和预测高维函数时间序列，其中函数值时间序列的数量$p$相对于时间序列长度$n$较大。第一步对正定矩阵进行特征分析，从而为原始高维函数时间序列构造一一对应的线性变换，变换后的曲线序列可分割为若干组，使得来自任意两个不同组的任意两个子序列在同期和序列上均不相关。因此在第二步中，这些组可被分别处理而不会损失整体线性动态结构的信息。第二步致力于为每个组内所有变换后的函数时间序列建立有限维动态结构。此外，该有限维结构通过向量时间序列的结构来表征。原始高维函数时间序列的建模与预测通过所有组中向量时间序列的建模与预测来实现。我们研究了所提出方法的理论性质，并通过大量模拟和两个真实数据集说明了有限样本性能。