{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11295",
  "author_total": 4,
  "authors": [
    {
      "name": "Sophie Hanna Langbein"
    },
    {
      "name": "Niklas Koenen"
    },
    {
      "name": "Marvin N. Wright"
    },
    {
      "name": "Julia Herbinger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dhms4mfzsj47b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11295",
  "fulltext_url": null,
  "id": "dhms4mfzsj47b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11295",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Explainable Artificial Intelligence (XAI)"
    ]
  ],
  "openalex": "W7212295906",
  "pdf_url": "https://arxiv.org/pdf/2609.11295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dhms4mfzsj47b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:36Z",
      "native_id": "2609.11295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs",
  "title_text": "A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:36Z",
  "url": "https://research.draconitic.com/paper/dhms4mfzsj47b",
  "venue": "arXiv",
  "versions": []
}