Many economic and scientific problems involve the analysis of high-dimensional functional time series, where the number of functional variables $p$ diverges as the number of serially dependent observations $n$ increases. In this paper, we present a novel functional factor model for high-dimensional functional time series that maintains and makes use of the functional and dynamic structure to achieve great dimension reduction and find the latent factor structure. To estimate the number of functional factors and the factor loadings, we propose a fully functional estimation procedure based on an eigenanalysis for a nonnegative definite and symmetric matrix. Our proposal involves a weight matrix to improve the estimation efficiency and tackle the issue of heterogeneity, the rationale of which is illustrated by formulating the estimation from a novel regression perspective. Asymptotic properties of the proposed method are studied when $p$ diverges at some polynomial rate as $n$ increases. To provide a parsimonious model and enhance interpretability for near-zero factor loadings, we impose sparsity assumptions on the factor loading space and then develop a regularized estimation procedure with theoretical guarantees when $p$ grows exponentially fast relative to $n.$ Finally, we demonstrate that our proposed estimators significantly outperform the competing methods through both simulations and applications to a U.K. temperature data set and a Japanese mortality data set.

翻译：许多经济和科学问题涉及高维函数时间序列的分析，其中函数变量数量$p$随着序列相关观测值数量$n$的增加而发散。本文提出了一种针对高维函数时间序列的新型函数因子模型，该模型保持并利用函数结构和动态结构，以实现显著的降维并发现潜在因子结构。为估计函数因子数量及因子载荷，我们提出了一种完全基于函数估计的程序，该程序通过对非负定对称矩阵进行特征分析来实现。我们的方案引入了权重矩阵以提高估计效率并处理异质性问题，其原理通过从一种新颖的回归视角构建估计过程得以阐明。当$p$随$n$增加以多项式速率发散时，我们研究了所提方法的渐近性质。为建立简约模型并增强对近零因子载荷的可解释性，我们对因子载荷空间施加稀疏性假设，随后开发了具有理论保证的正则化估计程序，该程序适用于$p$相对于$n$呈指数级快速增长的情形。最后，通过模拟实验以及对英国温度数据集和日本死亡率数据集的应用，我们证明所提出的估计量显著优于现有竞争方法。