Granger causality based on vector time series and quaternion algebra with possible applications to molecular dynamics...

Area of Science:

• Physics
• Chemistry
• Biology
• Nanoscience
• Engineering

Background:

• Complex dynamical systems generate time-dependent vector signals (e.g., momenta, torques).
• Analyzing causal relationships in these multidimensional time-series is crucial for understanding system functionality.
• Existing Granger causality methods are limited to scalar signals.

Purpose of the Study:

• Extend Granger causality to analyze vector signals in complex dynamical systems.
• Develop a robust analytical model and numerical implementation for vector signal causality.
• Validate the extended formalism using established dynamic models.

Main Methods:

• Utilized quaternion algebra to represent vector signals as time-dependent quaternions.
• Developed an analytical model based on the autoregressive formalism (Q-MVAR).
• Validated the Q-MVAR formalism and implementation with two distinct dynamic models.

Main Results:

• Successfully extended Granger causality to handle multidimensional vector signals.
• The Q-MVAR formalism demonstrated accurate causal relationship identification in validated models.
• The method is applicable to classical motions in biomolecular systems and beyond.

Conclusions:

• The developed Q-MVAR formalism provides a powerful tool for causal analysis of vector signals.
• This extended causality framework has broad applicability across physical, natural, and engineering sciences.
• Further development and application of this formalism will enhance understanding of complex system dynamics.