相关论文: The random Tukey depth
Tukey's depth (or halfspace depth) is a widely used measure of centrality for multivariate data. However, exact computation of Tukey's depth is known to be a hard problem in high dimensions. As a remedy, randomized approximations of Tukey's…
Tukey depth, aka halfspace depth, has attracted much interest in data analysis, because it is a natural way of measuring the notion of depth relative to a cloud of points or, more generally, to a probability measure. Given an i.i.d. sample,…
Tukey's depth offers a powerful tool for nonparametric inference and estimation, but also encounters serious computational and methodological difficulties in modern statistical data analysis. This paper studies how to generalize and compute…
We present a new fast approximate algorithm for Tukey (halfspace) depth level sets and its implementation-ABCDepth. Given a $d$-dimensional data set for any $d\geq 1$, the algorithm is based on a representation of level sets as…
The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is…
For multivariate data, Tukey's half-space depth is one of the most popular depth functions available in the literature. It is conceptually simple and satisfies several desirable properties of depth functions. The Tukey median, the…
Tukey depth function is one of the most famous multivariate tools serving robust purposes. It is also very well known for its computability problems in dimensions $p \ge 3$. In this paper, we address this computing issue by presenting two…
We present a new algorithm for Tukey (halfspace) depth level sets and its implementation. Given $d$-dimensional data set for any $d\geq 2$, the algorithm is based on representation of level sets as intersections of balls in $R^d$, and can…
The Tukey (or halfspace) depth extends nonparametric methods toward multivariate data. The multivariate analogues of the quantiles are the central regions of the Tukey depth, defined as sets of points in the $d$-dimensional space whose…
Depth of the Tukey median is investigated for empirical distributions. A sharper upper bound is provided for this value for data sets in general position. This bound is lower than the existing one in the literature, and more importantly…
The concept of depth represents methods to measure how deep an arbitrary point is positioned in a dataset and can be seen as the opposite of outlyingness. It has proved very useful and a wide range of methods have been developed based on…
Determining the representativeness of a point within a data cloud has recently become a desirable task in multivariate analysis. The concept of statistical depth function, which reflects centrality of an arbitrary point, appears to be…
Tukey depth regions are important notions in nonparametric multivariate data analysis. A $\tau$-th Tukey depth region $\mathcal{D}_{\tau}$ is the set of all points that have at least depth $\tau$. While the Tukey depth regions are easily…
Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in $\IR^d$. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional…
Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…
Mean estimation is a fundamental task in statistics and a focus within differentially private statistical estimation. While univariate methods based on the Gaussian mechanism are widely used in practice, more advanced techniques such as the…
Halfspace (or Tukey) depth is a fundamental and robust measure of centrality of data points in multivariate datasets. Computing the depth of a point with respect to the uniform distribution on an open convex body in $\mathbb{R}^d$ is a…
The Tukey depth of a flat with respect to a point set is a concept that appears in many areas of discrete and computational geometry. In particular, the study of centerpoints, center transversals, Ham Sandwich cuts, or $k$-edges can all be…
The scatter halfspace depth (sHD) is an extension of the location halfspace (also called Tukey) depth that is applicable in the nonparametric analysis of scatter. Using sHD, it is possible to define minimax optimal robust scatter estimators…
Data depth is a powerful nonparametric tool originally proposed to rank multivariate data from center outward. In this context, one of the most archetypical depth notions is Tukey's halfspace depth. In the last few decades notions of depth…