Slicing-free supervised dimension reduction for multivariate time series

Sufficient dimension reduction (SDR) methods have been extensively studied for regression models with independent data, but options for time series are limited, focusing mainly on scalar responses with TSIR, TSAVE, and TSSH. Although valuable, these SDR methods rely on the slice approach. Extending them to multivariate responses via marginal slicing leads to numerous slices. Furthermore, the slice approach also poses two main questions: how many slices should be chosen and how to divide all samples into different slices. To overcome these, we introduce TMDDM, a slicing-free SDR method for time series, using approximate joint diagonalization of supervised lagged martingale difference divergence matrices (MDDM) to account for the data temporal characteristics. We also discuss lag selection strategies and the dimensionality of dimension reduction space. Simulations and real data analysis demonstrate its favourable finite-sample performance.