Extracting a low-dimensional predictable time series

Large scale multi-dimensional time series can be found in many disciplines, including finance, econometrics, biomedical engineering, and industrial engineering systems. It has long been recognized that the time dependent components of the vector time series often reside in a subspace, leaving its complement independent over time. In this paper we develop a method for projecting the time series onto a low-dimensional time-series that is predictable, in the sense that an auto-regressive model achieves low prediction error. Our formulation and method follow ideas from principal component analysis, so we refer to the extracted low-dimensional time series as principal time series. In one special case we can compute the optimal projection exactly; in others, we give a heuristic method that seems to work well in practice. The effectiveness of the method is demonstrated on synthesized and real time series.