Problem
Research questions and friction points this paper is trying to address.
data depth
unparameterized planar curves
trajectory data
band depth
curve analysis
Curve Band Depth: A Band-Based Data Depth for Unparameterized Planar Curves
🤖 AI Summary
This study addresses the absence of effective depth measures for nonparametric planar curves. To this end, it proposes "curve band depth," which defines data depth by computing the proportion of arc length of a target curve lying within a geometric band generated by two reference curves. Departing from traditional halfspace approaches, this construction establishes three variants—integral, infimum, and fast-walking—and proves their similarity invariance. The implementation integrates arc-length sampling, polygonal approximation, and length penalization techniques. The effectiveness and practical utility of the proposed method are validated through experiments on handwriting classification, MNIST digit recognition, and online signature screening tasks.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Curve Band Depth
Unparameterized Planar Curves
Data Depth
Band Region
Trajectory Data
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