{
  "abstract": "We introduce \\emph{curve band depth} (CBD), a band-based data depth for samples of \\emph{unparameterized} planar curves. CBD is motivated by band depth and modified band depth for functional data, but targets trajectory data. Unlike the halfspace-based curve depth of \\citet{de2021depth} and the curve stabbing depth of \\citet{durocher2023csd}, CBD is defined through a geometric band region generated by two curves, and measures the arc-length proportion of a target curve lying inside such bands. We develop a CBD family consisting of an integral version (int-CBD), an infimal version (inf-CBD), and a fast-walk variant (FW-CBD). The fast-walk band is a narrower band construction contained in the global convex-combination band. We establish boundedness, vanishing at infinity, and similarity invariance for these constructions, together with a Borel-measurability result for the induced depth maps under a mild measurability assumption. A length-penalized variant is proposed for samples with heterogeneous curve lengths. We implement the methods via arc-length sampling and polygonal approximations, and evaluate them through classification of overlapping handwriting data and MNIST-derived digit curves, online-signature screening on \\texttt{MOBISIG}, and an exploratory clustering task based on decomposed band contributions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32648",
  "author_total": 3,
  "authors": [
    {
      "name": "Siyi Wang"
    },
    {
      "name": "Alexandre Leblanc"
    },
    {
      "name": "Paul D. McNicholas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ckb0fsfjpbjvt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32648",
  "fulltext_url": null,
  "id": "ckb0fsfjpbjvt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32648",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Advanced Statistical Methods and Models"
    ]
  ],
  "openalex": "W7214778357",
  "pdf_url": "https://arxiv.org/pdf/2609.32648",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ckb0fsfjpbjvt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32648",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32648"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Curve Band Depth: A Band-Based Data Depth for Unparameterized Planar Curves",
  "title_text": "Curve Band Depth: A Band-Based Data Depth for Unparameterized Planar Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/ckb0fsfjpbjvt",
  "venue": "arXiv",
  "versions": []
}