Curve Stabbing Depth: Data Depth for Plane Curves

TL;DR

This work introduces curve stabbing depth, a principled depth measure for plane curves defined by D(Q,𝒞) = \frac{1}{\pi L(Q)} ∫_{q∈Q} ∫_{0}^{π} \min\{stab_{𝒞}(\overrightarrow{q_θ}), stab_{𝒞}(\overrightarrow{q_{θ+π}})\} dθ dq, capturing how deeply Q lies inside a family 𝒞. It grounds the measure in wedges and tangent points, and develops an exact algorithm for computing the depth when Q and 𝒞 are polylines, with a worst-case time of O(n^3 + n^2 m log^2 m + n m^2 log^2 m). The paper also studies fundamental properties, showing similarity invariance and a bounded, nondegenerate, and vanishing-at-infinity behavior, alongside a Monte Carlo approximation framework. Together, these results enable robust analysis of curve data with potential applications to trajectory classification, clustering, and central-trajectory discovery, while outlining extensions to higher dimensions and alternative depth measures.

Abstract

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 $\cal C$ of curves in $\mathbb{R}^2$. Curve stabbing depth evaluates the average number of elements of $\cal C$ stabbed by rays rooted along the length of $Q$. We describe an $O(n^3 + n^2 m\log^2m+nm^2\log^2 m)$-time algorithm for computing curve stabbing depth when $Q$ is an $m$-vertex polyline and $\cal C$ is a set of $n$ polylines, each with $O(m)$ vertices.

Figures (10)

• Figure 1: The set of wedges $w_1,w_2,w_3,$ and $w_4$ induced by curves $C_1,C_2,C_3$, and $C_4$ rooted at the point $q$ on the curve $Q$. Moving counterclockwise around $q$, the positive angle between $\tau_1(w_2)$ with the horizontal is indicated by $\phi(\tau_1(w_2))$, the tangent points of $w_2$ are labelled $\tau_1(w_2)$ and $\tau_2(w_2)$, and the internal angle of $w_3$ is highlighted by $\theta(w_3)$ with $\phi(w_3)=\phi(\tau_2(w_3))-\phi(\tau(w_3))$.
• Figure 2: Two ways a query line segment $Q$ and a wedge rooted at a point on $Q$ can be arranged under general position. Case 1 is drawn in black while Case 2 is outlined in blue.
• Figure 3: A configuration similar to that shown in Figure \ref{['fig:wedge']} for three curves $C_1,C_2$, and $C_3$ is depicted, with their respective wedge boundaries extended through the origin. The circular partition induced is shown by the sequence of angles towards the right-hand side of the figure.
• Figure 4: A configuration similar to that shown in Figure \ref{['fig:wedge']} for two curves $C_1$ and $C_2$ is depicted, the highlighted segment being cyclically invariant with respect to the given population, as can be seen by inspecting the wedge boundaries
• Figure 5: The hierarchy of convex hulls associated with a polyline $P$ as indexed by segments $I_1$, $I_2$, and $I_3$ of $Q$. Indicated in red are the convex hulls that encapsulate subpaths of $P$ generated by the intersecting segment of $Q$. Note the singular dangling segment formed by cutting $P$ by $I_3$ is considered a degenerate convex hull.

Theorems & Definitions (50)

• Definition 1: Polyline
• Definition 2: Stabbing Number
• Definition 3: Subpath Uniqueness
• Definition 4: Curve Stabbing Depth
• Definition 5: Wedge
• Definition 6: Tangent Points
• Definition 7: Circular Partition
• Definition 9: Cyclically Invariant Segments
• Lemma 10: Invariant Segments along Polylines
• proof