Conditional mean dimension reduction for tensor time series

The dimension reduction problem for a stationary tensor time series is addressed. The goal is to remove linear combinations of the tensor time series that are mean independent of the past, without imposing any parametric models or distributional assumptions. To achieve this goal, a new metric called cumulative tensor martingale difference divergence is introduced and its theoretical properties are studied. Unlike existing methods, the proposed approach achieves dimension reduction by estimating a distinctive subspace that can fully retain the conditional mean information. By focusing on the conditional mean, the proposed dimension reduction method is potentially more accurate in prediction. The method can be viewed as a factor model-based approach that extends the existing techniques for estimating central subspace or central mean subspace in vector time series. The effectiveness of the proposed method is illustrated by extensive simulations and two real-world data applications.