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arXiv · 2609.32648

Curve Band Depth: A Band-Based Data Depth for Unparameterized Planar Curves

Siyi Wang·Alexandre Leblanc·Paul D. McNicholas
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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.

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Siyi Wang, Alexandre Leblanc, Paul D. McNicholas. 2026-09-26. Curve Band Depth: A Band-Based Data Depth for Unparameterized Planar Curves. https://arxiv.org/abs/2609.32648

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