Inquiry Regarding the Influence of Facilitative/Inhibitory Effects on Causal Leakage Estimation in SURD

[Juzhongwang]

Hello Alvaro,

I have recently undertaken an in-depth study of your proposed SURD (Synergistic-Unique-Redundant Decomposition of causality) framework. In particular, I have carefully worked through the illustrative examples provided in your paper (Examples 1–10), including logic gates, basic coupling modules, energy cascades, Moran effects, and turbulent boundary layers. These examples have greatly helped me understand the methodological strengths of SURD in resolving complex causal structures in dynamical systems.

As I explore potential applications of SURD to vegetation-climate coupling systems, I would like to seek your advice on two specific issues—especially regarding the possible impact of facilitative vs. inhibitory effect directions on the estimation of causal leakage (ΔI_leak).

1. Can SURD distinguish between facilitative and inhibitory causal directions?
From my understanding, SURD is built upon an information-theoretic foundation that focuses on the structural types of informational contributions—redundant (R), unique (U), and synergistic (S)—rather than the sign or directionality (i.e., positive/facilitative vs. negative/inhibitory effects) of influence. That is, SURD aims to determine whether a variable contributes information about the target's future, not how it affects the target (e.g., enhancing or suppressing its dynamics).

In your examples, SURD performs remarkably well in uncovering hidden synergistic mechanisms and masking effects. However, it seems that the framework does not explicitly distinguish between “synergistic facilitation” (e.g., two variables jointly enhancing a target) and “synergistic inhibition” (e.g., two variables jointly suppressing a target). If this is correct, then my question is:

Is there any mechanism within SURD that allows differentiation between facilitative and inhibitory interactions within synergistic or unique contributions?

I am particularly concerned that in nonlinear systems where opposing directional effects (e.g., one variable facilitates and another inhibits) occur together, the information content may still appear “synergistic” in an abstract sense, while in practice the directionality conflict could degrade the interpretability of the results.

In my own analysis, I attempted to address this by complementing the SURD results with sign-based correlation analyses (e.g., partial correlations or symbolic sign graphs) to infer potential directionality behind the informational structure. I would appreciate your opinion on whether such post hoc diagnostics are advisable or theoretically consistent with the SURD paradigm.

2. Does directional conflict contribute to high ΔI_leak values?
In my current research, I apply SURD to analyze the causal effects of 11 environmental and anthropogenic drivers (e.g., precipitation, temperature, human activity) on vegetation growth indicators (e.g., GPP, NDVI,). To account for time-lagged effects, I have already aggregated explanatory variables to reflect their impact in the 2–3 months preceding the vegetation response. Therefore, I set lag = 0 in the SURD analysis, focusing on contemporaneous relationships after variable preprocessing.

In some regions, however, I observed very high ΔI_leak values (up to 83%), suggesting a large portion of information not captured by the current model. Based on your paper, this causal leakage may reflect:

• missing latent variables (e.g., soil properties, CO₂, extreme events),
• structural mismatches in the model,
• or unresolved synergistic interactions possibly involving directional conflicts, such as:

• temperature promoting vegetation growth up to a threshold but becoming inhibitory afterward,
• or non-monotonic human impacts with mixed positive/negative effects.

This led me to ask:

①Could conflicts in directional influence among synergistic variables reduce the overall detectability of synergistic contributions and thereby inflate ΔI_leak?

②If a variable set is rich in mutual information but the directional signs of influence are inconsistent, could SURD interpret this as “unexplained structure,” thus contributing to higher causal leakage estimates?

I understand that SURD focuses on informational synergy rather than causal polarity. However, in real-world systems—especially climate-vegetation interactions—nonlinear thresholds and shifting facilitative/inhibitory roles are common. In such cases, high ΔI_leak values may reflect not only unmeasured variables, but also confounding directionality that challenges the information structure itself.

These are the key questions I’m currently grappling with. I would be extremely grateful for your insights or any clarification regarding how facilitative/inhibitory interactions might affect causal interpretation under the SURD framework. Your work has provided me with invaluable conceptual tools, and I would greatly appreciate your guidance as I seek to apply these tools more robustly in empirical studies.

Thank you very much for your time and consideration.

[kolkov]

Sorry for jumping in here (I'm not one of the authors), but I recently implemented SURD in Go and spent quite a bit of time validating it, so I might have some practical observations.

From what I understand, you're right that SURD focuses on information structure rather than causal polarity. It measures how much information is gained, not in which direction the influence acts. So a temperature variable that switches from facilitative to inhibitory would still contribute the same synergistic information - SURD just can't tell you the sign.

For your vegetation-climate work with high ΔI_leak (83%):

1. Directional conflicts and leakage

I don't think directional conflicts directly inflate ΔI_leak, but they might be a symptom of the same underlying issue - missing nonlinear structure. If temperature has a threshold effect (positive below, negative above), that threshold itself might be driven by some unmeasured variable (soil moisture? accumulated heat stress?). So the high leakage could be pointing to that missing piece rather than being caused by the conflict itself.

2. Practical approach

What I'd probably try:

• Run SURD to identify synergistic groups
• Then use correlation/regression on those groups to infer directionality
• If you see strong synergy but weak correlation, that's a red flag - maybe the relationship is more complex than linear

For thresholds, you might need to bin the data (e.g., temperature <25°C vs >25°C) and run SURD separately on each regime. Not elegant, but it might reveal whether the synergy structure changes across the threshold.

3. Testing idea

If you want to validate the implementation or test different preprocessing approaches, I just released a Go version of SURD that matches the Python reference exactly (validated on turbulence data). It's faster for large datasets, so might be useful if you're iterating on 11 variables × multiple regions. No pressure, just mentioning in case it helps: https://github.com/causalgo/causalgo

Again, sorry if this is off-base - I'm definitely not an expert on vegetation dynamics. But from an implementation perspective, that's how I'd approach the directional ambiguity.

Good luck with your research!

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Update (26 November 2025): SCIC Algorithm

Following up on the above - we've now released SCIC (Signed Causal Information Components) in v0.4.0, which directly addresses the directionality question.

What SCIC Does

SCIC extends SURD's information-theoretic framework to capture directional causality:

• Positive causality: Target increases when agent increases
• Negative causality: Target decreases when agent increases (inhibitory)
• Sign stability: Bootstrap confidence for directional inference
• Conflict detection: Identifies variables with mixed directional effects

Relevance to Your Vegetation-Climate Case

For high ΔI_leak (83%) and suspected directional conflicts:

1. SCIC can detect threshold effects: If temperature is facilitative <25°C but inhibitory >25°C, SCIC will show this as a conflict
2. Signed synergy: You'll see whether synergistic groups act positively or negatively
3. Confidence estimates: Bootstrap sign stability tells you how consistent the directionality is

Example

import "github.com/causalgo/causalgo/internal/scic"

cfg := scic.Config{
    DirectionalityMethod: scic.QuartileMethod,  // or GradientMethod for smooth transitions
    NumBootstrap:        100,
}

result, _ := scic.AnalyzeFromData(data, bins, cfg)

// Check for conflicts (mixed directionality)
for i, conflict := range result.Conflicts {
    if conflict {
        fmt.Printf("Variable %d shows directional conflict\n", i)
    }
}

Would Love Your Feedback

Since you're working on exactly the use case SCIC was designed for, we'd really appreciate if you could test it on your vegetation-climate data:

• Does it capture the facilitative/inhibitory transitions you expect?
• How do sign stability values correlate with known thresholds?
• Any edge cases or unexpected behavior?

SCIC is brand new (just released today), so feedback from domain experts like you would be invaluable for validation.

Docs: https://pkg.go.dev/github.com/causalgo/causalgo/internal/scic
Installation: go get github.com/causalgo/causalgo@v0.4.0

Happy to discuss implementation details or help troubleshoot if you give it a try!

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P.S. Your question about directional effects actually helped accelerate this work - we'd been exploring the idea for a while, but seeing a real-world use case where it was needed pushed us to finalize and release SCIC. So thank you for the motivation!