A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs
Abstract
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.
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BibTeX
@misc{langbein2026hilbertvalued,
title = {{A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs}},
author = {Sophie Hanna Langbein and Niklas Koenen and Marvin N. Wright and Julia Herbinger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11295},
eprint = {2609.11295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11295},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs
AU - Sophie Hanna Langbein
AU - Niklas Koenen
AU - Marvin N. Wright
AU - Julia Herbinger
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.11295
UR - https://arxiv.org/abs/2609.11295
AB - 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.
ER -