A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs

The functional ANOVA, or Hoeffding decomposition, provides a principled framework for interpretability by decomposing a model prediction into main effects and higher-order interactions. For independent inputs, this classical decomposition is explicit. It is closely connected to SHAP values, generalized additive models, and orthogonal polynomial expansions, and therefore constitutes a fundamental tool for additive explainability. In the more general and realistic dependent setting, however, obtaining a tractable representation and estimating the decomposition from data remain challenging. In this work, we address this problem for continuous inputs. By combining Hilbert space methods with the generalized functional ANOVA, we build an explicit decomposition Riesz Basis allowing to easily compute the decomposition. Our formulation recovers the classical independent case and its associated orthogonal decomposition. Building on this representation, we propose a simple but mighty algorithm to estimate the decomposition from a data sample in a model-agnostic setting and we compare it empirically with several state-of-the-art explanation methods, demonstrating the power of the approach.

While many explainable AI (XAI) methods have been proposed, most are not designed for time-series forecasting models and often rely on the implicit assumption that timestamp features are independent. This assumption ignores the fundamental property of temporal dependence and can lead to explanations that violate the sequential and causal structure of the data. We introduce \textsc{KARMA}, a method for explaining time-series predictors by constructing a Markov surrogate model that captures the temporal dependencies learned by the predictor. Our approach revolves around three main aspects: identifying the minimal history length $K$ that is predictively sufficient for the model, estimating the best-fitting $K$-order Markov transition kernel from the discretized history space, and a five-level global explanation hierarchy that can be derived from the Markov transition kernel, which we illustrate using real-world weather data (Beijing PM 2.5). We also certify using complex synthetic data with known true causal edges that KARMA (i) recovers the data causal structure as learned by the model via a controlled experiment and (ii) identifies temporal dependencies better than established attribution methods such as TimeSHAP.

Recent advances in explaining time series (TS) models have produced methods that identify which past values a prediction depends on. However, most existing methods return a single importance vector, assuming that every predicted step depends on the same past values. In this paper, we show that this assumption does not hold, as different forecast steps depend on different past values. Motivated by this observation, we propose Horizon-Resolved eXplanation (HRX), which adds a horizon axis to the explanation, so that every forecast step receives its own importance map. HRX is a simple yet effective plug-in framework with three components: 1) an estimator that reads these maps out of any differentiable forecaster without modifying the TS backbone, 2) an evaluation protocol that validates the horizon axis by measuring how much a single forecast step changes when the inputs an importance map ranks highest are removed, and 3) a rank criterion that predicts in advance whether the axis is worth resolving on a given TS. We further show that this step-wise dependence is low-dimensional, as the explanations of all steps are built from a few shared maps whose number does not grow with the forecast length. Extensive experiments across various backbones and datasets show that the improvement comes from the horizon axis and holds for estimators of previous explanation methods. Code is available at https://github.com/seunghan96/HRX.