Temporal Faithfulness Metric

The foundational approach measures the correlation between attributed importance and actual model sensitivity. By systematically perturbing time steps in order of their attributed importance and measuring the resulting change in model output, a faithful explanation will show a strong monotonic relationship.

• Forward selection: Remove most-important steps first; output should degrade rapidly
• Backward selection: Remove least-important steps first; output should remain stable
• Area Over the Perturbation Curve (AOPC) quantifies this relationship as a scalar score

Two complementary metrics that evaluate explanation quality from opposing directions. Comprehensiveness measures whether the attributed time steps are necessary for the prediction—removing them should cause a large output change. Sufficiency measures whether the attributed steps alone are enough to sustain the prediction.

• High comprehensiveness + low sufficiency = explanation captures all relevant steps
• Low comprehensiveness + high sufficiency = explanation misses critical context
• Both metrics are computed by masking or retaining only the top-k attributed time steps

This metric evaluates explanation continuity and stability by measuring how much the attribution scores change when small amounts of noise are added to the input sequence. A faithful explanation method should produce similar attributions for functionally identical inputs.

• Max-Sensitivity: Maximum attribution change under bounded input perturbation
• Local Lipschitz continuity: Formalizes the expected stability of the explanation function
• Critical for detecting explanation fragility in high-frequency time-series data

Infidelity quantifies the expected error between the explanation's attribution scores and the model's actual response to meaningful perturbations. It is computed as the mean squared difference between the dot product of attribution with perturbation and the actual output change.

• Captures whether attributions correctly predict the model's local behavior
• Uses randomized smoothing with Gaussian perturbations for robust estimation
• Lower infidelity indicates higher explanation faithfulness

On synthetic or semi-synthetic datasets where the true causal time steps are known by construction, faithfulness can be measured directly. The explanation's attribution ranking is compared against ground truth using ranking metrics.

• Normalized Discounted Cumulative Gain (NDCG) evaluates ranking quality
• Hit Rate @ K measures whether top-K attributed steps contain true drivers
• Common benchmarks include synthetic autoregressive processes and modified real-world series with injected anomalies

This metric tests whether the explanation method's attributions are internally consistent with the model's learned representations. It compares attributions generated through different ablation strategies—such as zero-masking, mean-imputation, and conditional resampling—to verify that the importance ordering remains stable.

• Rank correlation between attribution orders under different ablation schemes
• High consistency suggests the explanation captures genuine model reliance
• Low consistency indicates the explanation is an artifact of the perturbation method

A foundational perturbation-based method that systematically removes or masks individual time steps from an input sequence and measures the resulting change in the model's output. The magnitude of prediction shift directly quantifies importance.

• Faithfulness Link: Serves as a primary ground-truth proxy; a faithful explanation should correlate highly with ablation rankings.
• Variants: Includes block ablation for contiguous intervals and feature-time ablation for multivariate inputs.
• Limitation: Can create out-of-distribution inputs that distort model behavior.

A gradient-based attribution technique that computes the path integral of gradients from a neutral baseline (e.g., zero signal) to the actual input. Satisfies the completeness axiom, ensuring the sum of attributions equals the prediction difference.

• Faithfulness Link: Tested by measuring correlation between attributed importance and output change under perturbation.
• Baseline Sensitivity: The choice of baseline critically impacts the explanation; a constant mean or black image equivalent is standard.
• Use Case: Provides fine-grained, step-level saliency for differentiable forecasting models.

Adapts Shapley value calculations from cooperative game theory to assign importance scores to individual time steps. Each step is treated as a player, and its marginal contribution is averaged over all possible coalitions of other steps.

• Faithfulness Link: The gold standard for local accuracy; a faithful metric should validate that SHAP values correctly reflect the model's true reliance on each lag.
• Computational Cost: Exact calculation is exponential; KernelSHAP or DeepSHAP approximations are used in practice.
• Property: Uniquely satisfies efficiency, symmetry, dummy, and additivity axioms.

A robustness evaluation framework that introduces small, controlled noise or distortions to a time series and analyzes the stability of the resulting explanations. A faithful explanation should not change dramatically under trivial input variations.

• Continuity Check: Measures whether attribution scores vary smoothly or exhibit brittle, discontinuous jumps.
• Adversarial Robustness: Tests if an attacker can craft imperceptible perturbations that flip the explanation while preserving the prediction.
• Metric: Often quantified via max-sensitivity or Lipschitz continuity of the explanation function.

An interpretable proxy model (e.g., a shallow decision tree or linear model) trained to mimic the predictions of a complex temporal black-box on a local or global scale. The surrogate's transparent structure provides the explanation.

• Faithfulness Link: Measured by the fidelity of the surrogate—how accurately its predictions match the original model on the sampled neighborhood.
• Local vs. Global: Local surrogates explain a single prediction; global surrogates approximate the entire model behavior.
• Trade-off: High fidelity often requires a more complex surrogate, reducing interpretability.

Moves beyond correlation to identify causal drivers of a forecast using structural causal models or intervention analysis. Instead of asking 'which step correlates with the output?', it asks 'which step, if intervened upon, causes the output to change?'

• Faithfulness Link: The ultimate test—a causally faithful explanation identifies the true data-generating mechanisms the model has learned to exploit.
• Methods: Includes Granger Causality Saliency, PCMCI algorithms, and counterfactual interventions.
• Challenge: Requires assumptions about the causal graph structure that may not hold in observational data.