Ultralow‐Dimensionality Reduction for Identifying Critical Transitions by Spatial‐Temporal PCA

Abstract Discovering dominant patterns and exploring dynamic behaviors especially critical state transitions and tipping points in high‐dimensional time‐series data are challenging tasks in study of real‐world complex systems, which demand interpretable data representations to facilitate comprehension of both spatial and temporal information within the original data space. This study proposes a general and analytical ultralow‐dimensionality reduction method for dynamical systems named spatial‐temporal principal component analysis (stPCA) to fully represent the dynamics of a high‐dimensional time‐series by only a single latent variable without distortion, which transforms high‐dimensional spatial information into one‐dimensional temporal information based on nonlinear delay‐embedding theory. The dynamics of this single variable is analytically solved and theoretically preserves the temporal property of original high‐dimensional time‐series, thereby accurately and reliably identifying the tipping point before an upcoming critical transition. Its applications to real‐world datasets such as individual‐specific heterogeneous ICU records demonstrate the effectiveness of stPCA, which quantitatively and robustly provides the early‐warning signals of the critical/tipping state on each patient.