Supervised dimension reduction for functional time series

Functional time series model has been the subject of the most research in recent years, and since functional data is infinite dimensional, dimension reduction is essential for functional time series. However, the majority of the existing dimension reduction methods such as the functional principal component and fixed basis expansion are unsupervised and typically result in information loss. Then, the functional time series model has an urgent need for a supervised dimension reduction method. The functional sufficient dimension reduction method is a supervised technique that adequately exploits the regression structure information, resulting in minimal information loss. Functional sliced inverse regression (FSIR) is the most popular functional sufficient dimension reduction method, but it cannot be applied directly to functional time series model. In this paper, we examine a functional time series model in which the response is a scalar time series and the explanatory variable is functional time series. We propose a novel supervised dimension reduction technique for the regression model by combining the FSIR and blind source separation methods. Furthermore, we propose innovative strategies for selecting the dimensionality of dimension reduction space and the lags of the functional time series. Numerical studies, including simulation studies and a real data analysis are show the effectiveness of the proposed methods.