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

arXiv: https://arxiv.org/abs/2609.11295 · HTML: https://mlsanity.com/paper/2609.11295/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 53 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [A Survey on Explainable Artificial Intelligence (XAI): Towards Medical XAI](https://mlsanity.com/paper/1907.07374/) (2019)
- [All Models are Wrong, but Many are Useful: Learning a Variable's Importance by Studying an Entire Class of Prediction Models Simultaneously](https://mlsanity.com/paper/1801.01489/) (2018)
- [Explaining by Removing: A Unified Framework for Model Explanation](https://mlsanity.com/paper/2011.14878/) (2020)
- [Doctor AI: Predicting Clinical Events via Recurrent Neural Networks](https://mlsanity.com/paper/1511.05942/) (2015)
- [FastSHAP: Real-Time Shapley Value Estimation](https://mlsanity.com/paper/2107.07436/) (2021)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [SHAP-IQ: Unified Approximation of any-order Shapley Interactions](https://mlsanity.com/paper/2303.01179/) (2023)
- [SurvSHAP(t): Time-dependent explanations of machine learning survival models](https://mlsanity.com/paper/2208.11080/) (2022)
- [Unifying Feature-Based Explanations with Functional ANOVA and Cooperative Game Theory](https://mlsanity.com/paper/2412.17152/) (2024)
- [Sensitivity analysis for multidimensional and functional outputs](https://mlsanity.com/paper/1311.1797/) (2013)
- [TsSHAP: Robust model agnostic feature-based explainability for time series forecasting](https://mlsanity.com/paper/2303.12316/) (2023)
- [PAX-TS: Model-agnostic multi-granular explanations for time series forecasting via localized perturbations](https://mlsanity.com/paper/2508.18982/) (2025)
- [GRANITE: A Generalized Regional Framework for Identifying Agreement in Feature-Based Explanations](https://mlsanity.com/paper/2601.22771/) (2026)
- [Functional Decomposition and Shapley Interactions for Interpreting Survival Models](https://mlsanity.com/paper/2602.16505/) (2026)

## Abstract (arXiv)

Feature-based explanations quantify features' influence on model predictions, but are primarily designed for scalar outputs. In many applications, however, outputs are functional or multivariate, such as time-dependent trajectories in demand forecasting. Consequently, existing approaches typically explain each output location independently, ignoring dependencies across the output components. We address this limitation by developing a unified framework for feature-based explanations of time-dependent outputs. Specifically, we generalize functional decomposition to Hilbert-valued prediction functions and extend an existing feature-based explanation framework to this setting. Our framework introduces kernel-based output representations that enable time-dependency-aware explanations at multiple levels of temporal granularity, including time-specific, time-resolved, and time-aggregated, while providing a unified view in which existing methods arise as special cases. We validate our framework on synthetic and real-world data, including intraday financial market volatility prediction and energy demand forecasting.