A New Dimension-Reduction Method for High-Dimensional Nonlinear Time Series

What is it about?

Time series models often become difficult to estimate and interpret when many past observations are used as predictors, particularly when the relationships are nonlinear. This study develops a new convolution-transformation approach to sufficient dimension reduction for time series. The method identifies a small number of informative combinations of lagged observations while preserving the information needed to describe the future behavior of the series. We develop estimators for both the time series central mean subspace, which focuses on conditional mean dynamics, and the broader time series central subspace, which captures information about the full conditional distribution. We also extend a Fourier-transform-based approach to this broader setting and establish theoretical properties of the proposed estimators. Simulation studies and real-data applications show that the convolution and Fourier approaches offer complementary strengths for dimension reduction in complex time series. Related keywords: sufficient dimension reduction SDR time series dimension reduction nonlinear time series high-dimensional time series high-dimensional lag structure time series central mean subspace TS-CMS time series central subspace TS-CS convolution transformation Fourier transformation dimension reduction time series forecasting statistical learning curse of dimensionality Convolution-based estimation Fourier-based estimation Nonlinear time series Sufficient dimension reduction Time series central mean subspace.

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Photo by Bozhin Karaivanov on Unsplash

Photo by Bozhin Karaivanov on Unsplash

Why is it important?

High-dimensional lag structures create a major challenge in time series analysis because model complexity can increase rapidly as more past observations are considered. Sufficient dimension reduction offers a way to compress these predictors without discarding the information that is relevant to the response. The proposed framework extends transformation-based dimension reduction to important time-series settings, including both conditional-mean and full-distribution dynamics. This can support more parsimonious, interpretable, and computationally manageable models in areas such as forecasting, econometrics, finance, environmental analysis, health, and other applications involving complex temporal data.

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