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Causality can be quantified between time series, even when correlation is zero. Introducing noise to a harmonic system reveals normalized information flow, demonstrating causality despite initial zero correlation.

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Area of Science:

  • Information theory
  • Time series analysis
  • Causality inference

Background:

  • Normalized information flow quantifies causality between time series.
  • In linear systems, causation implies correlation, but not vice versa.
  • Harmonic systems present a challenge where causality exists but correlation is zero.

Purpose of the Study:

  • To investigate causality inference in systems with zero correlation.
  • To resolve the apparent contradiction between causality and zero correlation in harmonic systems.
  • To demonstrate the effectiveness of normalized information flow under noise perturbation.

Main Methods:

  • Analysis of information flow and causality between two time series.
  • Modeling a harmonic system (event A) generating a phase-shifted event (B).
  • Perturbation with noise and solving stochastic differential equations.

Main Results:

  • A harmonic system with a phase lag of π/2 shows zero correlation despite clear causality.
  • Absolute information flow (TA→B) is zero, leading to an indeterminate form (0/0) for normalized information flow (τA→B).
  • Noise perturbation and subsequent limit removal show normalized information flow approaching 100%.

Conclusions:

  • Normalized information flow accurately captures causality even when correlation is zero.
  • The study resolves the paradox of zero correlation in causally linked harmonic systems.
  • This method provides a robust measure of causality in the presence of noise.