Bibliographic Details
| Title: |
Hyperplane Distance Depth |
| Authors: |
Mashghdoust, Amirhossein, Durocher, Stephane |
| Source: |
36th Canadian Conference on Computational Geometry (CCCG 2024) 36th Canadian Conference on Computational Geometry (CCCG 2024) 36th Canadian Conference on Computational Geometry (CCCG 2024) |
| Publication Year: |
2024 |
| Collection: |
Computer Science |
| Subject Terms: |
Computer Science - Computational Geometry |
| More Details: |
Depth measures quantify central tendency in the analysis of statistical and geometric data. Selecting a depth measure that is simple and efficiently computable is often important, e.g., when calculating depth for multiple query points or when applied to large sets of data. In this work, we introduce \emph{Hyperplane Distance Depth (HDD)}, which measures the centrality of a query point $q$ relative to a given set $P$ of $n$ points in $\mathbb{R}^d$, defined as the sum of the distances from $q$ to all $\binom{n}{d}$ hyperplanes determined by points in $P$. We present algorithms for calculating the HDD of an arbitrary query point $q$ relative to $P$ in $O(d \log n)$ time after preprocessing $P$, and for finding a median point of $P$ in $O(d n^{d^2} \log n)$ time. We study various properties of hyperplane distance depth and show that it is convex, symmetric, and vanishing at infinity. |
| Document Type: |
Working Paper |
| Access URL: |
http://arxiv.org/abs/2411.06114 |
| Accession Number: |
edsarx.2411.06114 |
| Database: |
arXiv |
