AI 中文总结
本文提出曲线带深度（CBD），一种基于带的深度度量，用于无参数平面曲线，通过几何带区域定义，并发展出积分、下确界和快速行走变体，具备理论性质，在手写分类、签名筛选和聚类任务中有效。
AI 中文摘要
我们引入了曲线带深度（CBD），这是一种基于带的深度，用于无参数平面曲线的样本。CBD 的动机来自函数型数据的带深度和修正带深度，但针对轨迹数据。与 de2021depth 的基于半空间的曲线深度和 durocher2023csd 的曲线刺穿深度不同，CBD 通过由两条曲线生成的几何带区域来定义，并测量目标曲线位于此类带内的弧长比例。我们开发了一个 CBD 家族，包括积分版本（int-CBD）、下确界版本（inf-CBD）和快速行走变体（FW-CBD）。快速行走带是一种更窄的带构造，包含在全局凸组合带中。我们建立了这些构造的有界性、无穷远处消失性和相似性不变性，以及在温和可测性假设下诱导深度图的 Borel 可测性结果。针对曲线长度异质的样本，我们提出了一个长度惩罚变体。我们通过弧长采样和多边形近似实现这些方法，并通过重叠手写数据分类、源自 MNIST 的数字曲线、MOBISIG 上的在线签名筛选以及基于分解带贡献的探索性聚类任务来评估它们。
英文摘要
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.
