Lower bounds for computing statistical depth

Aloupis, Greg; Cortés, Carmen; Gómez, Francisco; Soss, Michael; Toussaint, Godfried T.

doi:10.1016/s0167-9473(02)00032-4

Cited by 27 publications

(20 citation statements)

References 12 publications

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“…We refer the reader to [1,2] for recent results with a similar flavor to what is presented here, and for further introductory references to the topic of statistical depth. The main result in [1] involves lower bounds on the computation of halfspace depth [8] and simplicial depth [5]. For both depths, the given bound of Ω(n log n) was in the algebraic decision tree model and matched known upper bounds in the RAM model.…”

Section: Introductionmentioning

confidence: 80%

A lower bound for computing Oja depth

Aloupis

McLeish

2005

Information Processing Letters

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Let S = {s 1 , . . . , s n } be a set of points in the plane. The Oja depth of a query point θ with respect to S is the sum of the areas of all triangles (θ, s i , s j ). This depth may be computed in O(n log n) time in the RAM model of computation. We show that a matching lower bound holds in the algebraic decision tree model. This bound also applies to the computation of the Oja gradient, the Oja sign test, and to the problem of computing the sum of pairwise distances among points on a line.

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Section: Introductionmentioning

confidence: 80%

A lower bound for computing Oja depth

Aloupis

McLeish

2005

Information Processing Letters

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“…For instance, Rousseeuw and Ruts [1996] and Aloupis et al [2002] developed algorithms for the computation of the half-space and the simplicial depth functions. The Mahalanobis depth is among the simplest ones to evaluate.…”

Section: Discussionmentioning

confidence: 99%

Depth-based multivariate descriptive statistics with hydrological applications

Chebana

Ouarda

2011

J. Geophys. Res.

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Hydrological events are often described through various characteristics which are generally correlated. To be realistic, these characteristics are required to be considered jointly. In multivariate hydrological frequency analysis, the focus has been made on modeling multivariate samples using copulas. However, prior to this step, data should be visualized and analyzed in a descriptive manner. This preliminary step is essential for all of the remaining analysis. It allows us to obtain information concerning the location, scale, skewness, and kurtosis of the sample as well as outlier detection. These features are useful to exclude some unusual data, to make different comparisons, and to guide the selection of the appropriate model. In the present paper we introduce methods measuring these features, which are mainly based on the notion of depth function. The application of these techniques is illustrated on two real‐world streamflow data sets from Canada. In the Ashuapmushuan case study, there are no outliers and the bivariate data are likely to be elliptically symmetric and heavy‐tailed. The Magpie case study contains a number of outliers, which are identified to be real observed data. These observations cannot be removed and should be accommodated by considering robust methods for further analysis. The presented depth‐based techniques can be adapted to a variety of hydrological variables.

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“…, h n (q) . The optimality of their algorithm in the real RAM model was proved by Aloupis et al [2].…”

Section: Half-space Countsmentioning

confidence: 96%

On Combinatorial Depth Measures

Durocher

Fraser

Leblanc

et al. 2018

Int. J. Comput. Geom. Appl.

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Given a set [Formula: see text] of points and a point [Formula: see text] in the plane, we define a function [Formula: see text] that provides a combinatorial characterization of the multiset of values [Formula: see text], where for each [Formula: see text], [Formula: see text] is the open half-plane determined by [Formula: see text] and [Formula: see text]. We introduce two new natural measures of depth, perihedral depth and eutomic depth, and we show how to express these and the well-known simplicial and Tukey depths concisely in terms of [Formula: see text]. The perihedral and eutomic depths of [Formula: see text] with respect to [Formula: see text] correspond respectively to the number of subsets of [Formula: see text] whose convex hull contains [Formula: see text], and the number of combinatorially distinct bisections of [Formula: see text] determined by a line through [Formula: see text]. We present algorithms to compute the depth of an arbitrary query point in [Formula: see text] time and medians (deepest points) with respect to these depth measures in [Formula: see text] and [Formula: see text] time respectively. For comparison, these results match or slightly improve on the corresponding best-known running times for simplicial depth, whose definition involves similar combinatorial complexity.

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Lower bounds for computing statistical depth

Cited by 27 publications

References 12 publications

A lower bound for computing Oja depth

A lower bound for computing Oja depth

Depth-based multivariate descriptive statistics with hydrological applications

On Combinatorial Depth Measures

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