Temporal DeepLIFT

The defining mathematical guarantee of DeepLIFT. The total importance assigned across all input time steps sums precisely to the difference between the model's output for the actual sequence and its output for a chosen reference sequence.

• Equation: Σ_i attributions_i = F(x) - F(r)
• Significance: Prevents attribution leakage, ensuring no importance is artificially created or destroyed during backpropagation.
• Contrast: Standard gradient methods only measure local sensitivity and can saturate, missing features that strongly activate a neuron but have a near-zero gradient.

Instead of computing gradients, Temporal DeepLIFT defines a reference input representing a neutral or missing signal (e.g., all zeros, a historical mean, or a padded mask). Each neuron's contribution is calculated by comparing its activation on the true input to its activation on this reference.

• Reference Choice: Critical for explanation quality. A zero-vector baseline works for normalized data, while a moving average may better represent 'expected' behavior in financial time series.
• Mechanism: Uses linear composition rules to propagate difference-from-reference scores backward through non-linear gates, avoiding the gradient saturation problem inherent in sigmoid and tanh activations.

Temporal DeepLIFT defines a multiplier for each neuron operation, representing the ratio of the output difference-from-reference to the input difference-from-reference. These multipliers are chained backward through the unrolled recurrent network.

• Linear Rule: For linear transformations (e.g., dense layers), the multiplier is the weight itself.
• Rescale Rule: For non-linear activations, the multiplier is Δoutput / Δinput, effectively linearizing the function at the operating point defined by the reference.
• RevealCancel Rule: Separately tracks positive and negative contributions to handle opposing influences within a single neuron, providing a more granular decomposition for LSTM gates.

When applied to LSTMs or GRUs, Temporal DeepLIFT unrolls the network through time and treats the hidden and cell states as separate contributing inputs at each time step. It attributes importance to how past states influence the current output.

• State Attribution: The method decomposes the final prediction into contributions from the initial hidden state and each sequential input step.
• Gate Decomposition: For an LSTM, attributions flow through the forget, input, and output gates, revealing whether a time step's importance stems from writing new information to memory or preserving old information.
• Use Case: Identifying the exact moment a financial model 'locks onto' a trend and stops being influenced by older, irrelevant data.

A primary advantage over gradient-based methods like Temporal Integrated Gradients is specificity. DeepLIFT can assign non-zero importance to a time step that saturates a neuron's output, whereas a gradient at that point would be flat and uninformative.

• Example: In a fraud detection sequence, a single large transaction may push a risk score to its maximum. A gradient method would show zero importance for any subsequent transactions, while DeepLIFT correctly identifies that the initial transaction is the sole driver.
• Discrete Handling: Effectively handles discrete or thresholded inputs common in event-based time series, where gradients are undefined or zero almost everywhere.

In practice, Temporal DeepLIFT is often accessed through the SHAP DeepExplainer framework, which adapts the DeepLIFT algorithm to approximate Shapley values for deep learning models. This combines DeepLIFT's speed with SHAP's game-theoretic fairness properties.

• Speed: Runs in a single backward pass, making it computationally efficient for long sequences compared to sampling-based Shapley estimators.
• Integration: The DeepExplainer uses a background dataset of reference samples to compute expected multipliers, stabilizing the attributions for noisy time-series data.
• Output: Produces a matrix of SHAP values with dimensions (time steps × features), directly usable for generating temporal saliency maps.