Overcoming feature scarcity in complex system prediction: An alternative delay embedding

Predicting the future dynamics of complex systems remains challenging in scientific research. Traditional methods generally approach this issue by approximating the correlations between predictors (features) and target variables, often employing predictors from a pool of target-related variables (system components). However, these approaches are inherently constrained when reliable target-related features are scarce. Here, we introduce a framework, alternative delay embedding (ADE), to effectively integrate the delay embedding technique with a machine learning algorithm based on Gaussian process regression. Rather than relying on the identification of target-related variables, ADE exploits the target's sequential information to generate reconstructions that function as robust predictors. The reliability of the ADE framework is validated across multiple benchmark model systems, e.g., the logistic map, the Mackey-Glass equation, and the Lorenz system, as well as real-world datasets spanning diverse domains, e.g., sea surface temperature, physiological signals, electroencephalography signals, and financial exchange rates. Demonstrations on short input data highlight enhanced robustness of ADE compared to several classic methods. The ADE framework serves as a useful complement to existing predictive approaches, particularly in cases where reliable target-related features are either scarce or elusive.