Uniform convergence rates for the approximated halfspace and projection depth

Nagy, Stanislav; Dyckerhoff, Rainer; Mozharovskyi, Pavlo

doi:10.1214/20-ejs1759

Cited by 11 publications

(7 citation statements)

References 44 publications

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“…For d = 10, 50, we measure (16) over 50 repetitions and visualize it in boxplots of Figure 11, approximating projection depth (12) using random search (RS) and refined random search (RRS) algorithms, with the last one containing elements of optimization. Thus, this experiment shall also illustrate advantage of optimizationinvolving approximation over purely random one, with the last one suffering from curse of dimension as it has been shown by Nagy et al (2020). As we can see from Figure 11 (left), depth-based anomaly detection rule copes with the task for d = 10 perfectly even with small number of directions when using RRS approximation (e.g., with 200 directions depth calculation for all 1000 points of one data set took less than 20 seconds on a single core of Apple M1 Max chip).…”

Section: Computational Tractabilitymentioning

confidence: 70%

Anomaly detection using data depth: multivariate case

Mozharovskyi¹

2022

Preprint

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Anomaly detection is a branch of machine learning and data analysis which aims at identifying observations that exhibit abnormal behaviour. Be it measurement errors, disease development, severe weather, production quality default(s) (items) or failed equipment, financial frauds or crisis events, their on-time identification, isolation and explanation constitute an important task in almost any branch of industry and science. By providing a robust ordering, data depth-statistical function that measures belongingness of any point of the space to a data set-becomes a particularly useful tool for detection of anomalies. Already known for its theoretical properties, data depth has undergone substantial computational developments in the last decade and particularly recent years, which has made it applicable for contemporary-sized problems of data analysis and machine learning.In this article, data depth is studied as an efficient anomaly detection tool, assigning abnormality labels to observations with lower depth values, in a multivariate setting. Practical questions of necessity and reasonability of invariances and shape of the depth function, its robustness and computational complexity, choice of the threshold are discussed. Illustrations include use-cases that underline advantageous behaviour of data depth in various settings.

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Section: Computational Tractabilitymentioning

confidence: 70%

Anomaly detection using data depth: multivariate case

Mozharovskyi¹

2022

Preprint

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“…For γ = 0, the model in ( 9) is very useful in real multivariate reliability analysis; see [12,13]. The model in (10) for γ = 0 corresponds to a multivariate version of Student-t, Cauchy, multivariate F, and related distributions; see [14].…”

Section: Evaluation Of the Normalizing Constantsmentioning

confidence: 99%

Entropy Optimization, Maxwell–Boltzmann, and Rayleigh Distributions

Sebastian

Mathai

Haubold

2021

Entropy

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In physics, communication theory, engineering, statistics, and other areas, one of the methods of deriving distributions is the optimization of an appropriate measure of entropy under relevant constraints. In this paper, it is shown that by optimizing a measure of entropy introduced by the second author, one can derive densities of univariate, multivariate, and matrix-variate distributions in the real, as well as complex, domain. Several such scalar, multivariate, and matrix-variate distributions are derived. These include multivariate and matrix-variate Maxwell–Boltzmann and Rayleigh densities in the real and complex domains, multivariate Student-t, Cauchy, matrix-variate type-1 beta, type-2 beta, and gamma densities and their generalizations.

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“…Notice that the same assumptions are involved in the non-asymptotic rate bound analysis carried out for the halfspace depth estimator in [40] and are used to establish limit results related to its approximation in [41]. The Lipschitz conditions are satisfied by a large class of probability distributions, for which Lipschitz constants L R and L p can be both explicitely derived.…”

Section: Finite-sample Analysis -Concentration Boundsmentioning

confidence: 99%

“…Recently, this result has been refined under the Assumptions 2 and 3 in [40]. Asymptotic rates of convergence for the Monte Carlo approximation of the halfspace depth, i.e., when the minimum over the unit hypersphere is approximated from a finite number of directions, have been recently established in [41]. In contrast to the finite-sample framework, uniform asymptotic rates have been proved in several settings.…”

Section: Remark 2 (Related Work)mentioning

confidence: 99%

“…Indeed when the distribution is assumed to belong to a bounded subset of R d with bounded density, the authors obtain slow rates of order O((log(m)/m) 1/(d−1) ) suffering from the curse of dimensionality. Futhermore, they show that obtaining uniform rates of the halfspace depth approximation is not possible in absence of the bounded density assumption (see second example in Section 4.2 in [41]).…”

Section: Remark 2 (Related Work)mentioning

confidence: 99%

See 1 more Smart Citation

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

Staerman¹,

Mozharovskyi²,

Clémençon³

2021

Preprint

Self Cite

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Because it determines a center-outward ordering of observations in R d with d ≥ 2, the concept of statistical depth permits to define quantiles and ranks for multivariate data and use them for various statistical tasks (e.g. inference, hypothesis testing). Whereas many depth functions have been proposed ad-hoc in the literature since the seminal contribution of [1], not all of them possess the properties desirable to emulate the notion of quantile function for univariate probability distributions. In this paper, we propose an extension of the integrated rank-weighted statistical depth (IRW depth in abbreviated form) originally introduced in [2], modified in order to satisfy the property of affine-invariance, fulfilling thus all the four key axioms listed in the nomenclature elaborated by [3]. The variant we propose, referred to as the Affine-Invariant IRW depth (AI-IRW in short), involves the covariance/precision matrices of the (supposedly square integrable) d-dimensional random vector X under study, in order to take into account the directions along which X is most variable to assign a depth value to any point x ∈ R d . The accuracy of the sampling version of the AI-IRW depth is investigated from a nonasymptotic perspective. Namely, a concentration result for the statistical counterpart of the AI-IRW depth is proved. Beyond the theoretical analysis carried out, applications to anomaly detection are considered and numerical results are displayed, providing strong empirical evidence of the relevance of the depth function we propose here.

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Uniform convergence rates for the approximated halfspace and projection depth

Cited by 11 publications

References 44 publications

Anomaly detection using data depth: multivariate case

Anomaly detection using data depth: multivariate case

Entropy Optimization, Maxwell–Boltzmann, and Rayleigh Distributions

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

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