“Local positive” vs “sum weak/negative” — causal model with separate noise

Scenario: Sea Ice Paradox

This simulation explores how local correlations can be reversed when aggregated (a form of Simpson's Paradox or Ecological Fallacy). We model two sea ice regions driven by a common climate mode.

Key Variables

• Meridional Wind (\(v\)): Modeled as $v = a \cdot z + \text{noise}$. The wind in each sector depends on the climate mode $z$.
• Sea Ice Tendency (\(st\)): Modeled as $st = b \cdot v + \text{noise}$. Physically, wind pushes ice, driving changes in concentration.
• Climate Mode (\(z\)): The hidden large-scale atmospheric pattern (e.g., NAO, SAM) driving winds in both sectors.

Parameters

• \(a_1, a_2\) (Climate Coupling): Determine the sign of the wind in each sector for a given climate mode phase. If they have opposite signs, the climate mode creates a "dipole" pattern.
• \(b_1, b_2\) (Wind Forcing): Determine how wind affects ice locally. Usually positive (wind pushes ice).

The Paradox: Even if wind increases ice locally in both sectors ($b_1, b_2 > 0$), the summed correlation can be negative. This happens if the climate mode drives winds in opposite directions ($a_1, a_2$ opposite), causing the "signal" term to be negative and overwhelm the positive noise coupling.