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Approximate Data Depth Revisited
Computational Geometry · Submitted 18 May 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The main results of this paper include approximating the planar halfspace depth and $\beta$-skeleton depth using two different approximation methods.
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The abstract, sentence by sentence
Halfspace depth and $\beta$-skeleton depth are two types of depth functions in nonparametric data analysis. The halfspace depth of a query point $q\in \mathbb{R}^d$ with respect to $S\subset\mathbb{R}^d$ is the minimum portion of the elements of $S$ which are contained in a halfspace which passes through $q$. For $\beta \geq 1$, the $\beta$-skeleton depth of $q$ with respect to $S$ is defined to be the total number of \emph{$\beta$-skeleton influence regions} that contain $q$, where each of these influence regions is the intersection of two hyperballs obtained from a pair of points in $S$. The $\beta$-skeleton depth introduces a family of depth functions that contain \emph{spherical depth} and \emph{lens depth} if $\beta=1$ and $\beta=2$, respectively. The main results of this paper include approximating the planar halfspace depth and $\beta$-skeleton depth using two different approximation methods. First, the halfspace depth is approximated by the $\beta$-skeleton depth values. For this method, two dissimilarity measures based on the concepts of \emph{fitting function} and \emph{Hamming distance} are defined to train the halfspace depth function by the $\beta$-skeleton depth values obtaining from a given data set. The goodness of this approximation is measured by a function of error values. Secondly, computing the planar $\beta$-skeleton depth is reduced to a combination of some range counting problems. Using existing results on range counting approximations, the planar $\beta$-skeleton depth of a query point is approximated in $O(n\;poly(1/\varepsilon,\log n))$, $\beta\geq 1$. Regarding the $\beta$-skeleton depth functions, it is also proved that this family of depth functions converge when $\beta \to \infty$. Finally, some experimental results are provided to support the proposed method of approximation and convergence of $\beta$-skeleton depth functions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Halfspace depth and $\beta$-skeleton depth are two types of depth functions in nonparametric data analysis. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s1 | The halfspace depth of a query point $q\in \mathbb{R}^d$ with respect to $S\subset\mathbb{R}^d$ is the minimum portion of the elements of $S$ which are contained in a halfspace which passes through $q$. | background 1.00 | 0.07 | 0.02 | asserted 1.00 |
| s2 | For $\beta \geq 1$, the $\beta$-skeleton depth of $q$ with respect to $S$ is defined to be the total number of \emph{$\beta$-skeleton influence regions} that contain $q$, where each of these influence regions is the intersection of two hyperballs obtained from a pair of points in $S$. | background 1.00 | 0.13 | 0.02 | asserted 1.00 |
| s3 | The $\beta$-skeleton depth introduces a family of depth functions that contain \emph{spherical depth} and \emph{lens depth} if $\beta=1$ and $\beta=2$, respectively. | background 0.96 | 0.20 | 0.04 | asserted 1.00 |
| s4 | The main results of this paper include approximating the planar halfspace depth and $\beta$-skeleton depth using two different approximation methods. | contribution 0.94 | 0.92 | 0.50 | asserted 1.00 |
| s5 | First, the halfspace depth is approximated by the $\beta$-skeleton depth values. | method 0.70 | 0.85 | 0.17 | asserted 1.00 |
| s6 | For this method, two dissimilarity measures based on the concepts of \emph{fitting function} and \emph{Hamming distance} are defined to train the halfspace depth function by the $\beta$-skeleton depth values obtaining from a given data set. | method 1.00 | 0.85 | 0.06 | asserted 1.00 |
| s7 | The goodness of this approximation is measured by a function of error values. | method 0.98 | 0.82 | 0.04 | asserted 1.00 |
| s8 | Secondly, computing the planar $\beta$-skeleton depth is reduced to a combination of some range counting problems. | method 0.84 | 0.86 | 0.11 | asserted 1.00 |
| s9 | Using existing results on range counting approximations, the planar $\beta$-skeleton depth of a query point is approximated in $O(n\;poly(1/\varepsilon,\log n))$, $\beta\geq 1$. | result 0.85 | 0.67 | 0.23 | asserted 1.00 |
| s10 | Regarding the $\beta$-skeleton depth functions, it is also proved that this family of depth functions converge when $\beta \to \infty$. | result 0.99 | 0.89 | 0.14 | asserted 1.00 |
| s11 | Finally, some experimental results are provided to support the proposed method of approximation and convergence of $\beta$-skeleton depth functions. | contribution 0.46 | 0.93 | 0.09 | asserted 0.97 |
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Original abstract
Halfspace depth and $\beta$-skeleton depth are two types of depth functions in nonparametric data analysis. The halfspace depth of a query point $q\in \mathbb{R}^d$ with respect to $S\subset\mathbb{R}^d$ is the minimum portion of the elements of $S$ which are contained in a halfspace which passes through $q$. For $\beta \geq 1$, the $\beta$-skeleton depth of $q$ with respect to $S$ is defined to be the total number of \emph{$\beta$-skeleton influence regions} that contain $q$, where each of these influence regions is the intersection of two hyperballs obtained from a pair of points in $S$. The $\beta$-skeleton depth introduces a family of depth functions that contain \emph{spherical depth} and \emph{lens depth} if $\beta=1$ and $\beta=2$, respectively. The main results of this paper include approximating the planar halfspace depth and $\beta$-skeleton depth using two different approximation methods. First, the halfspace depth is approximated by the $\beta$-skeleton depth values. For this method, two dissimilarity measures based on the concepts of \emph{fitting function} and \emph{Hamming distance} are defined to train the halfspace depth function by the $\beta$-skeleton depth values obtaining from a given data set. The goodness of this approximation is measured by a function of error values. Secondly, computing the planar $\beta$-skeleton depth is reduced to a combination of some range counting problems. Using existing results on range counting approximations, the planar $\beta$-skeleton depth of a query point is approximated in $O(n\;poly(1/\varepsilon,\log n))$, $\beta\geq 1$. Regarding the $\beta$-skeleton depth functions, it is also proved that this family of depth functions converge when $\beta \to \infty$. Finally, some experimental results are provided to support the proposed method of approximation and convergence of $\beta$-skeleton depth functions.
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