# Choosing among notions of multivariate depth statistics

arXiv: https://arxiv.org/abs/2004.01927 · HTML: https://mlsanity.com/paper/2004.01927/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Apr 2020.

## Built on by (most cited first, 14 of 14)

- [A Pseudo-Metric between Probability Distributions based on Depth-Trimmed Regions](https://mlsanity.com/paper/2103.12711/) (2021)
- [Partial Rankings of Optimizers](https://mlsanity.com/paper/2402.16565/) (2024)
- [Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis](https://mlsanity.com/paper/2105.12778/) (2021)
- [Depth Functions for Partial Orders with a Descriptive Analysis of Machine Learning Algorithms](https://mlsanity.com/paper/2304.09872/) (2023)
- [Anomaly detection using data depth: multivariate case](https://mlsanity.com/paper/2210.02851/) (2022)
- [Toward Stronger Textual Attack Detectors](https://mlsanity.com/paper/2310.14001/) (2023)
- [On the quality of randomized approximations of Tukey's depth](https://mlsanity.com/paper/2309.05657/) (2023)
- [Eikonal depth: an optimal control approach to statistical depths](https://mlsanity.com/paper/2201.05274/) (2022)
- [Statistical process monitoring of artificial neural networks](https://mlsanity.com/paper/2209.07436/) (2022)
- [Fast kernel half-space depth for data with non-convex supports](https://mlsanity.com/paper/2312.14136/) (2023)
- [Depth-Based Local Center Clustering: A Framework for Handling Different Clustering Scenarios](https://mlsanity.com/paper/2505.09516/) (2025)
- [Abnormal component analysis](https://mlsanity.com/paper/2312.16139/) (2023)
- [Model-Free Kernel Conformal Depth Measures Algorithm for Uncertainty Quantification in Regression Models in Separable Hilbert Spaces](https://mlsanity.com/paper/2506.08325/) (2025)
- [Automatic depth-based local center clustering via β-integrated local depth and adaptive grouping](https://mlsanity.com/paper/2609.26748/) (2026)

## Abstract (arXiv)

Classical multivariate statistics measures the outlyingness of a point by its Mahalanobis distance from the mean, which is based on the mean and the covariance matrix of the data. A multivariate depth function is a function which, given a point and a distribution in d-space, measures centrality by a number between 0 and 1, while satisfying certain postulates regarding invariance, monotonicity, convexity and continuity. Accordingly, numerous notions of multivariate depth have been proposed in the literature, some of which are also robust against extremely outlying data. The departure from classical Mahalanobis distance does not come without cost. There is a trade-off between invariance, robustness and computational feasibility. In the last few years, efficient exact algorithms as well as approximate ones have been constructed and made available in R-packages. Consequently, in practical applications the choice of a depth statistic is no more restricted to one or two notions due to computational limits; rather often more notions are feasible, among which the researcher has to decide. The article debates theoretical and practical aspects of this choice, including invariance and uniqueness, robustness and computational feasibility. Complexity and speed of exact algorithms are compared. The accuracy of approximate approaches like the random Tukey depth is discussed as well as the application to large and high-dimensional data. Extensions to local and functional depths and connections to regression depth are shortly addressed.