SHAP

Overview

SHAP computes parameter importance using Shapley values from cooperative game theory (Lundberg & Lee, 2017). Each parameter receives a contribution value that represents its average marginal contribution across all possible feature coalitions.

The Importance Chart uses TreeSHAP for efficient exact Shapley value computation on Random Forest trees. Global importance is the mean $|\varphi_j(x)|$ across the training split, normalized to sum to 1.

Formula

Shapley value for parameter $j$ at sample $x$:

$$\varphi_j(x) = \sum_{S \subseteq F \setminus {j}} \frac{|S|!,(|F| - |S| - 1)!}{|F|!} \left[ f(S \cup {j}) - f(S) \right]$$

Global SHAP importance:

$$\operatorname{score}_j = \operatorname{mean} |\varphi_j(x)| \text{ over the training split}$$

Shapley values satisfy four axioms: efficiency, symmetry, linearity, dummy (a feature with no effect gets zero).

Comparison with Other Methods

Aspect	SHAP	MDI	RF-ANOVA
Theoretical basis	Shapley axioms	Impurity reduction	Variance decomposition over leaf boxes (fANOVA)
High-cardinality bias	None	High	Low
Local interpretability	Yes	No	No
Cost	Medium–High	Low	Medium

Hyperparameters

Parameter	Value
Trees	64
Max depth	10
Max rows	1,000 (downsampled)
Seed	42

R² Interpretation

R²	Meaning
≥ 0.8	Good fit. Scores are reliable.
0.5–0.8	Moderate. Use with caution.
< 0.5	Poor fit.

Notes

• SHAP shows global importance (mean $|\varphi|$). Per-sample local $\varphi$ values are not displayed.
• When features are strongly correlated, path-dependent TreeSHAP can be unstable.

When to Use

• When explainability and theoretical consistency are priorities.
• For reports requiring rigorous attribution.
• After RF-ANOVA / Permutation screening has identified top parameters.

References

• Lundberg, S. M., & Lee, S.-I. (2017). A Unified Approach to Interpreting Model Predictions. NeurIPS 30. https://proceedings.neurips.cc/paper_files/paper/2017/hash/8a20a8621978632d76c43dfd28b67767-Abstract.html
• Lundberg, S. M. et al. (2020). From local explanations to global understanding with explainable AI for trees. Nature Machine Intelligence, 2(1), 56–67. https://doi.org/10.1038/s42256-019-0138-9