Curve Stabbing Depth: Data Depth for Plane Curves

Measures of data depth have been studied extensively for point data. Motivated by recent work on analysis, clustering, and identifying representative elements in sets of trajectories, we introduce {\em curve stabbing depth} to quantify how deeply a given curve $Q$ is located relative to a given set $\mathcal{C}$ of curves in ${\mathbb{R}}^2$. Curve stabbing depth evaluates the average number of elements of $\mathcal{C}$ stabbed by rays rooted along the length of $Q$. We describe an $O(n^3+n^{2}m{\log }^{2}m+n{m}^2{\log }^{2}m)$-time algorithm for computing curve stabbing depth when $Q$ is an $m$-vertex polyline and $\mathcal{C}$ is a set of $n$ polylines, each with $O(m)$ vertices.