Temporal Integrated Gradients

TIG is built on two critical axioms that guarantee explanation validity:

• Sensitivity: If a single time step changes the prediction, it receives a non-zero attribution. This is also known as Completeness.
• Implementation Invariance: Two functionally equivalent networks always produce identical attributions, regardless of their internal architecture. This mathematical rigor avoids the pitfalls of simpler gradient-based methods that can saturate or miss critical features.

The core computation accumulates gradients along a straight-line path from a neutral baseline to the actual input:

1. A baseline sequence representing 'no signal' is defined (e.g., all zeros or mean padding).
2. The input is interpolated in small steps from the baseline to the original sequence.
3. The gradient of the model's output with respect to each time step is computed at every interpolation step.
4. These gradients are integrated (averaged), ensuring the total attribution sums exactly to the difference between the model's output for the input and the baseline.

The choice of baseline is the most critical design decision in TIG. It represents the 'absence of signal' in the temporal domain:

• Zero Baseline: A sequence of zeros. Simple but can be out-of-distribution for the model.
• Global Mean: A constant sequence of the feature's average value. Represents a neutral, non-informative signal.
• Local Masking: Replacing segments with local historical averages or Gaussian noise to create a contextually plausible neutral reference. The baseline directly encodes the counterfactual question: 'Important compared to what?'

The primary output is a saliency map over the temporal axis. For each time step t, TIG assigns a single scalar value representing its marginal contribution to the prediction.

• Positive attribution: The time step increased the output value (e.g., pushed a forecast higher).
• Negative attribution: The time step decreased the output value. This allows engineers to visualize exactly when a model 'paid attention' and which historical events drove a forecast, such as a specific sales spike from a promotion 7 days prior.

Unlike attention-weight analysis, TIG is not tied to a specific model architecture. It can be applied to any differentiable sequence model:

• LSTMs and GRUs: Explains gating dynamics without probing internal gate states.
• Temporal Convolutional Networks (TCNs): Attributes importance across the receptive field.
• Transformers: Provides a unified view of cross-time interactions, often serving as a ground-truth check for raw attention weights. This makes TIG a universal tool for auditing black-box temporal models in finance and IoT.

TIG satisfies the Summation-to-Delta property. The sum of all attributions across every time step and feature exactly equals the difference between the model's prediction for the actual input and its prediction for the baseline.

• This guarantees no attribution is 'lost' or 'created' during the explanation process.
• It provides a closed accounting system for the model's output, which is essential for high-stakes quantitative finance where basis-point-level precision in explanation is required for audit trails.