Dimensionality reduction has always been one of the most significant and challenging problems in the analysis of high-dimensional data. In the context of time series analysis, our focus is on the estimation and inference of conditional mean and variance functions. By using central mean and variance dimension reduction subspaces that preserve sufficient information about the response, one can effectively estimate the unknown mean and variance functions of the time series. While the literature presents several approaches to estimate the time series central mean and variance subspaces (TS-CMS and TS-CVS), these methods tend to be computationally intensive and infeasible for practical applications. By employing the Fourier transform, we derive explicit estimators for TS-CMS and TS-CVS. These proposed estimators are demonstrated to be consistent, asymptotically normal, and efficient. Simulation studies have been conducted to evaluate the performance of the proposed method. The results show that our method is significantly more accurate and computationally efficient than existing methods. Furthermore, the method has been applied to the Canadian Lynx dataset.

翻译：降维一直是高维数据分析中最重要且最具挑战性的问题之一。在时间序列分析的背景下，我们关注条件均值函数和条件方差函数的估计与推断。通过使用保留响应变量充分信息的中心均值降维子空间和中心方差降维子空间，可以有效估计时间序列中未知的均值和方差函数。尽管现有文献提出了多种估计时间序列中心均值子空间（TS-CMS）和中心方差子空间（TS-CVS）的方法，但这些方法通常计算量较大，难以在实际应用中推广。利用傅里叶变换，我们推导出了TS-CMS和TS-CVS的显式估计量。我们证明了所提出的估计量具有一致性、渐近正态性和有效性。通过仿真研究评估了所提方法的性能，结果表明我们的方法在精度和计算效率上显著优于现有方法。此外，该方法已应用于加拿大山猫数据集。