Predicting multiple observations in complex systems through low-dimensional embeddings.

Area of Science:

• Complex Systems Science
• Dynamical Systems Theory
• Data-Driven Modeling

Background:

• Forecasting all components in high-dimensional complex systems is a significant challenge.
• Existing methods often struggle with high dimensionality and identifying relevant predictors.
• The curse of dimensionality hinders accurate predictions in many real-world scenarios.

Purpose of the Study:

• To introduce a novel data-driven and model-free framework for comprehensive system forecasting.
• To address the limitations of existing methods in handling high-dimensional dynamical systems.
• To develop a generalized predictor capable of forecasting all system components.

Main Methods:

• Feature-and-Reconstructed Manifold Mapping (FRMM) framework combining feature embedding and delay embedding.
• Identification of topologically equivalent low-dimensional manifolds from high-dimensional data.
• Utilizing the low-dimensional feature manifold as a generalized predictor.

Main Results:

• Demonstrated effectiveness of FRMM on diverse datasets including Indian monsoon, EEG signals, financial markets, and traffic speed.
• FRMM successfully overcomes the curse of dimensionality in complex systems.
• A generalized predictor was identified, enabling the forecasting of all system components.

Conclusions:

• FRMM offers a robust and versatile approach for forecasting complex systems.
• The framework has broad applicability across various scientific and economic domains.
• FRMM represents a significant advancement in data-driven prediction methodologies for high-dimensional systems.