Shapley Value

The sum of all feature attributions exactly equals the difference between the model's prediction for the instance and the average baseline prediction. This ensures the explanation is a complete accounting of the output.

• No importance is lost or double-counted
• Guarantees additive decomposition: f(x) = φ₀ + Σ φᵢ
• Contrast with LIME, which only approximates local behavior

Two features that contribute identically to every possible coalition of features receive the same attribution value. This prevents arbitrary bias in importance assignment.

• If v(S ∪ {i}) = v(S ∪ {j}) for all subsets S, then φᵢ = φⱼ
• Critical for regulatory compliance in financial and medical models
• Ensures consistent treatment of correlated features with equal marginal contributions

A feature that contributes nothing to any coalition—its marginal contribution is always zero—receives an attribution of exactly zero. This prevents spurious importance from being assigned to irrelevant inputs.

• If v(S ∪ {i}) = v(S) for all S, then φᵢ = 0
• Provides a natural mechanism for feature selection
• Useful for identifying and pruning redundant RF signal features

When a prediction is the sum of two separate games (or models), the Shapley value for the combined game equals the sum of the individual Shapley values. This enables consistent explanations across ensemble models.

• φᵢ(v + w) = φᵢ(v) + φᵢ(w) for any two games v and w
• Essential property for explaining random forests and gradient-boosted trees
• Allows decomposition of multi-objective RF optimization problems

Each feature's importance is computed as the weighted average of its marginal contribution when added to every possible subset of other features. The weighting follows a combinatorial formula that gives equal weight to all coalition sizes.

• Weight for a coalition of size |S|: |S|!(n - |S| - 1)! / n!
• Considers all 2ⁿ⁻¹ possible feature orderings
• Computationally expensive—exact calculation is NP-hard, motivating SHAP approximations

Shapley values are the only attribution method that simultaneously satisfies Efficiency, Symmetry, Dummy, and Additivity. This uniqueness theorem provides a rigorous mathematical foundation that alternative methods like LIME or Integrated Gradients cannot claim.

• Proven by Lloyd Shapley in 1953
• No other solution concept satisfies all four axioms
• Provides defensible explanations for mission-critical RF AI systems in regulatory environments

A unified framework that implements Shapley values for machine learning interpretability. SHAP (SHapley Additive exPlanations) connects optimal credit allocation with local explanations by approximating each feature's contribution as an additive importance score. For RF applications, SHAP can decompose a modulation classifier's prediction to reveal which time-frequency bins in a spectrogram most influenced the decision. The framework includes efficient estimation methods like KernelSHAP (model-agnostic) and TreeSHAP (for gradient-boosted models), making it practical for high-dimensional IQ data where exact Shapley computation is combinatorially prohibitive.

Local Interpretable Model-agnostic Explanations creates surrogate models to explain individual predictions. LIME works by:

• Perturbing the input signal (e.g., masking time segments of IQ data)
• Observing how the model's output changes
• Fitting a simple, interpretable model (like a sparse linear model) locally around the prediction

Unlike Shapley values, LIME does not guarantee a fair distribution of the prediction among features, but its computational efficiency makes it attractive for real-time RF anomaly detection systems where latency constraints preclude Shapley-based methods.

A gradient-based attribution method that satisfies the completeness axiom—the sum of feature attributions equals the difference between the model's output at the input and a baseline. It computes importance by accumulating gradients along a straight-line path from a neutral baseline (e.g., zero signal or Gaussian noise) to the actual RF input. This path integral approach avoids the saturation problem where gradients of saturated neurons approach zero, masking their true contribution. For RF fingerprinting models, Integrated Gradients can highlight which transient signal regions most strongly identify a specific emitter.

Explains a prediction by identifying the minimal perturbation required to flip the model's decision to a desired alternative. For a cognitive radio system that misclassifies a friendly signal as hostile, a counterfactual explanation answers: 'What is the smallest change to the signal that would cause it to be correctly identified?' This approach is inherently causal and actionable—unlike feature importance scores, counterfactuals provide a direct prescription for intervention. In RFML, counterfactuals can guide adversarial hardening by revealing the precise signal modifications that confuse classifiers.

LRP decomposes a neural network's prediction by redistributing relevance scores backwards through the network using a conservation principle—the total relevance is preserved at each layer. Key LRP rules include:

• LRP-0: Basic redistribution proportional to neuron activations
• LRP-ε: Adds a small stabilizing term to handle near-zero activations
• LRP-γ: Favors positive contributions for sharper explanations

For deep convolutional RF models processing raw IQ samples, LRP produces fine-grained heatmaps showing which samples in the complex baseband stream drove the classification, offering higher spatial precision than perturbation-based methods like SHAP.

A global model inspection technique that measures the increase in prediction error after randomly shuffling a feature's values, breaking its relationship with the target. Unlike Shapley values, which provide local explanations per prediction, permutation importance reveals which features the model relies on across the entire dataset. For RF spectrum sensing models, this can identify whether the model depends on physically meaningful signal characteristics or spurious correlations. The method is model-agnostic and straightforward to implement, but can be misleading when features are correlated—a limitation Shapley values address through their game-theoretic foundation.