## 8 Citations

### Depth-based pseudo-metrics between probability distributions

- Computer ScienceArXiv
- 2021

This work proposes two new pseudo-metrics between continuous probability measures based on data depth and its associated central regions that highlight similarities with the Wasserstein distance and proposes an efficient approximation possessing linear time complexity w.r.t. the size of the data set and its dimension.

### Exact and approximate computation of the scatter halfspace depth

- Computer Science, Mathematics
- 2022

An exact algorithm for the computation of sHD in any dimension d is developed and implemented using C++ for d ≤ 5, and in R for any Dimension d ≥ 1 is proposed.

### A Pseudo-Metric between Probability Distributions based on Depth-Trimmed Regions

- Computer Science, Mathematics
- 2021

This work introduces a novel pseudo-metric between probability distributions by leveraging the extension of univariate quantiles to multivariate spaces and proposes an efﬁcient approximation method with linear time complexity w.r.t. the size of the dataset and its dimension.

### Affine-Invariant Integrated Rank-Weighted Depth: Definition, Properties and Finite Sample Analysis

- MathematicsArXiv
- 2021

An extension of the integrated rank-weighted statistical depth (IRW depth in abbreviated form) originally introduced in Ramsay et al. (2019), modified in order to satisfy the property of affine-invariance is proposed, fulfilling thus all the four key axioms listed in the nomenclature elaborated by Zuo and Serfling (2000).

### Tukey Depths and Hamilton-Jacobi Differential Equations

- MathematicsSIAM J. Math. Data Sci.
- 2022

It is proved that this equation possesses a unique viscosity solution, and that this solution always bounds the Tukey depth from below.

### Anomaly detection using data depth: multivariate case

- Computer ScienceArXiv
- 2022

In this article, data depth is studied as an anomaly detection tool, assigning abnormality labels to observations with lower depth values, in a multivariate setting, and practical questions of necessity and reasonability, its robustness and computational complexity, choice of the threshold are discussed.

### A Halfspace-Mass Depth-Based Method for Adversarial Attack Detection

- Computer Science
- 2022

HAMPER is introduced, a new method to detect adversarial examples by leveraging the concept of data depths, a statistical notion that provides center-outward ordering of points with respect to (w.r.t.) a probability distribution, which makes it a natural candidate for adversarial attack detection in high-dimensional spaces.

### Statistical monitoring of models based on artificial intelligence

- Computer ScienceArXiv
- 2022

This work proposes to consider the latent feature representation of the data (called “embedding”) generated by the NN for determining the time point when the data stream starts being nonstationary, and monitors embeddings by applying multivariate control charts based on the calculation of theData depth and normalized ranks.

## References

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- Mathematics
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Summary:Data depth is a concept that measures the centrality of a point in a given data cloud x1, x2,...,xn ∈ ℝ or in a multivariate distribution PX on ℝdd. Every depth defines a family of so–called…

### Computing location depth and regression depth in higher dimensions

- Computer ScienceStat. Comput.
- 1998

An exact algorithm to compute the location depth in three dimensions in O(n2logn) time and an approximate algorithm that computes the depth of a regression fit in O3+mpn+mnlogn time are constructed.

### Uniform convergence rates for the approximated halfspace and projection depth

- MathematicsElectronic Journal of Statistics
- 2020

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…

### Simulated annealing for higher dimensional projection depth

- Computer ScienceComput. Stat. Data Anal.
- 2012

### Depth statistics

- Mathematics
- 2012

In 1975 John Tukey proposed a multivariate median which is th e ‘deepest’ point in a given data cloud inRd (Tukey, 1975). In measuring the depth of an arbitrary point z with respect to the data,…

### Computing projection depth and its associated estimators

- MathematicsStat. Comput.
- 2014

An exact algorithm is proposed from the view of cutting a convex polytope with hyperplanes to compute the projection depth and most of its associated estimators of dimension p≥2, including Stahel-Donoho location and scatter estimators, projection trimmed mean, projection depth contours and median, etc.

### Classifying real-world data with the $${ DD}\alpha $$DDα-procedure

- MathematicsAdv. Data Anal. Classif.
- 2015

The $${ DD}\alpha $$DDα-classifier, a nonparametric fast and very robust procedure, is described and applied to fifty classification problems regarding a broad spectrum of real-world data. The…

### On a Notion of Data Depth Based on Random Simplices

- Mathematics
- 1990

For a distribution F on R p and a point x in R p , the simplicial depth D(x) is introduced, which is the probability that the point x is contained inside a random simplex whose vertices are p+1…