Open or closed? Information flow decided by transfer operators and forecastability quality metric

Area of Science:

• Systems Theory
• Information Theory
• Causality Analysis

Background:

• System closure, defined as autonomy or openness to external influence, is fundamental to understanding information flow and causality.
• Weiner-Granger causality provides a forecasting-based framework for inferring causal relationships between subsystems.
• Existing methods for causality analysis, such as transfer entropy, can be ill-defined for certain system dynamics.

Purpose of the Study:

• To develop a direct, analytic framework for assessing system closure and its implications for information flow and causality.
• To introduce a novel Forecastability Quality Metric (FQM) robust to the challenges of analyzing open systems.
• To contrast the theoretical underpinnings of closed (deterministic) and open (stochastic) system dynamics.

Main Methods:

• Utilized the Frobenius-Perron (FP) transfer operator and its restricted variants to model the evolution of system densities.
• Interpreted closed systems using the deterministic FP operator and open systems using the stochastic FP operator.
• Developed the Forecastability Quality Metric (FQM) based on Jensen-Shannon divergence, a symmetrized form of Kullback-Leibler divergence.

Main Results:

• Demonstrated that system closure directly influences the conceptualization of information flow and causal inference.
• The Forecastability Quality Metric (FQM) provides a well-defined measure for assessing forecastability in coupled chaotic systems.
• Highlighted the theoretical limitations of differential entropy (Kullback-Leibler divergence) in certain system analyses.

Conclusions:

• System closure is a critical determinant of how information flow and causality are modeled and understood.
• The developed Forecastability Quality Metric (FQM) offers a promising theoretical direction for future causality research.
• The study provides a foundation for both theoretical advancements and future data-oriented investigations into system dynamics.