Efficient Computation of Curve Stabbing Depth: A Novel Data Depth Measure for Plane Curves

Kernkonzepte

Curve stabbing depth is a new data depth measure that quantifies how deeply a given planar curve is located relative to a set of curves. It evaluates the average number of curves stabbed by rays rooted along the length of the query curve.

Zusammenfassung

The paper introduces a new data depth measure called "curve stabbing depth" for quantifying the centrality of a given planar curve relative to a set of curves.
The key highlights are:

Curve stabbing depth is defined as the average number of curves in the set that are intersected by rays rooted along the length of the query curve. This generalizes the notion of data depth from point data to curve data.

An O(n^3 + n^2m log^2 m + nm^2 log^2 m)-time algorithm is presented for computing the curve stabbing depth of an m-vertex polyline query curve relative to a set of n polylines, each with O(m) vertices.

Curve stabbing depth is analyzed for properties such as equivariance under transformations, stability of the median element, and robustness to perturbations, showing it behaves similarly to common depth measures for point data.

A randomized approximation algorithm is provided and contrasted with the deterministic algorithm for computing curve stabbing depth.

The paper develops efficient techniques for partitioning the query curve into cyclically invariant segments, maintaining dynamic updates to wedge tangent points and stabbing numbers, and integrating the angular area swept by wedges to compute the final curve stabbing depth.

Statistiken

The number of vertices in the query polyline Q is m.
The number of polylines in the set P is n, and each polyline in P has O(m) vertices.

Zitate

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

Wichtige Erkenntnisse aus

Curve Stabbing Depth: Data Depth for Plane Curves

by Stephane Dur... um arxiv.org 04-30-2024

https://arxiv.org/pdf/2311.07907.pdf

Tiefere Fragen

How can curve stabbing depth be extended to handle more general curve representations beyond polylines

To extend curve stabbing depth to handle more general curve representations beyond polylines, we can consider representing curves as a sequence of points in a higher-dimensional space. By treating each point as a vertex and connecting them with edges, we can create a graph structure that represents the curve. This graph can then be used to calculate the stabbing depth by considering the intersections of rays with the edges of the graph. Additionally, we can explore using spline curves or other curve-fitting techniques to approximate the curves and calculate stabbing depth based on these approximations.

What are the limitations of curve stabbing depth compared to other functional data depth measures, and how can it be further generalized

One limitation of curve stabbing depth compared to other functional data depth measures is its reliance on the concept of wedges and tangent points, which may not be applicable to all types of curves. To further generalize curve stabbing depth, we can explore incorporating distance-based measures that consider the proximity of points on the curve to the set of curves being analyzed. Additionally, we can investigate incorporating curvature information or shape descriptors to enhance the depth measure's ability to capture the complexity and variability of different curve shapes. By integrating these additional features, curve stabbing depth can be extended to provide a more comprehensive and robust measure of centrality and outlyingness for curve data.

What are the potential applications of curve stabbing depth in areas like trajectory analysis, shape analysis, and functional data mining

Curve stabbing depth has various potential applications in trajectory analysis, shape analysis, and functional data mining. In trajectory analysis, curve stabbing depth can be used to quantify the similarity or dissimilarity between different trajectories, helping in trajectory clustering, anomaly detection, and pattern recognition. In shape analysis, curve stabbing depth can assist in comparing and classifying shapes based on their internal structure and relative positions. In functional data mining, curve stabbing depth can aid in identifying representative elements in sets of curves, clustering similar curves, and detecting outliers or anomalies in curve data. Overall, curve stabbing depth provides a versatile tool for analyzing and understanding complex curve data in various domains.