Exact computation of the halfspace depth

Dyckerhoff, Rainer; Mozharovskyi, Pavlo

doi:10.1016/j.csda.2015.12.011

Cited by 64 publications

(55 citation statements)

References 41 publications

(44 reference statements)

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“…Computation of the depth of a curve demands algorithmic elaboration for the point Tukey curve depth D(x| Q m,n , µ m , H n,m ∆ ) (3.4). Calculation of D(x| Q m,n , µ m , H n,m ∆ ) relies on the works by Rousseeuw & Ruts (1996) in dimension 2 and by Dyckerhoff & Mozharovskyi (2016) for higher dimensions, which develop algorithms for computation of the multivariate Tukey depth.…”

Section: Methodsmentioning

confidence: 99%

Depth for Curve Data and Applications

Micheaux

Mozharovskyi

Vimond

2020

Journal of the American Statistical Association

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John W. Tukey (1975) defined statistical data depth as a function that determines centrality of an arbitrary point with respect to a data cloud or to a probability measure. During the last decades, this seminal idea of data depth evolved into a powerful tool proving to be useful in various fields of science. Recently, extending the notion of data depth to the functional setting attracted a lot of attention among theoretical and applied statisticians. We go further and suggest a notion of data depth suitable for data represented as curves, or trajectories, which is independent of the parametrization. We show that our curve depth satisfies theoretical requirements of general depth functions that are meaningful for trajectories. We apply our methodology to diffusion tensor brain images and also to pattern recognition of hand written digits and letters. Supplementary materials are available online.

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Section: Methodsmentioning

confidence: 99%

Depth for Curve Data and Applications

Micheaux

Mozharovskyi

Vimond

2020

Journal of the American Statistical Association

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“…The Tukey depth can be computed exactly (Dyckerhoff and Mozharovskyi, 2016) with complexity O(n d−1 log n), although to avoid computational burden we also implement its approximation with random directions (Dyckerhoff, 2004) having complexity O(kn), with k denoting the number of random directions. All of the experiments are performed with exactly computed Tukey depth, unless stated otherwise.…”

Section: Tukey Depthmentioning

confidence: 99%

Nonparametric Imputation by Data Depth

Mozharovskyi

Josse

Husson

2019

Journal of the American Statistical Association

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We present single imputation method for missing values which borrows the idea of data depth-a measure of centrality defined for an arbitrary point of a space with respect to a probability distribution or data cloud. This consists in iterative maximization of the depth of each observation with missing values, and can be employed with any properly defined statistical depth function. For each single iteration, imputation reverts to optimization of quadratic, linear, or quasiconcave functions that are solved analytically by linear programming or the Nelder-Mead method. As it accounts for the underlying data topology, the procedure is distribution free, allows imputation close to the data geometry, can make prediction in situations where local imputation (k-nearest neighbors, random forest) cannot, and has attractive robustness and asymptotic properties under elliptical symmetry. It is shown that a special case-when using the Mahalanobis depth-has direct connection to well-known methods for the multivariate normal model, such as iterated regression and regularized PCA. The methodology is extended to multiple imputation for data stemming from an elliptically symmetric distribution. Simulation and real data studies show good results compared with existing popular alternatives. The method has been implemented as an R-package. Supplementary materials for the article are available online. The major part of this project has been conducted during the postdoc of Pavlo Mozharovskyi at Agrocampus Ouest (Rennes) granted by Centre Henri Lebesgue due to program PIA-ANR-11-LABX-0020-01. arXiv:1701.03513v2 [stat.ME] 6 Aug 2018 = µ X miss(i) + Σ X miss(i),obs(i) Σ −1 X obs(i),obs(i) x i,obs(i) − µ X obs(i) .The last expression is the closed-form solution to min z miss(i) ∈R |miss(i)| , z obs(i) =x obs(i) d M (z, µ X |Σ X )

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“…In the last phase (lines [11][12][13][14][15][16][17][18], algorithm calculates level sets by intersecting balls constructed in the previous steps. In every iteration (line 12), all n balls that contain n(1−α k )+1 are intersected (line 13).…”

Section: Implementation: Algorithm 1 For Finding Deepest Points (Tukementioning

confidence: 99%

“…Another exact algorithm for finding Tukey depth in R d is proposed by Liu and Zuo in [17], which proves to be extremely time-consuming (see Table 5.1 of Section 5.3 in [20]) and the algorithm involves heavy computations, but can serve as a benchmark. Recently, Dyckerhoff and Mozharovskyi in [13] proposed two exact algorithms for finding halfspace depth that run in O(n d ) and O(n d−1 log n) time. Table 3 shows execution times of ABCDepth algorithm for finding a depth of a sample point.…”

Section: Performance and Comparisonsmentioning

confidence: 99%

Approximate calculation of Tukey's depth and median with high-dimensional data

Merkle¹

2018

YUGOSLAV J OPERATION

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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 ≥ 1, the algorithm is based on a representation of level sets as intersections of balls in R d . Our approach does not need calculations of projections of sample points to directions. This novel idea enables calculations of approximate level sets in very high dimensions with complexity which is linear in d, which provides a great advantage over all other approximate algorithms. Using different versions of this algorithm we demonstrate approximate calculations of the deepest set of points ("Tukey median") and Tukey's depth of a sample point or out-of-sample point, all with a linear in d complexity. An additional theoretical advantage of this approach is that the data points are not assumed to be in "general position". Examples with real and synthetic data show that the executing time of the algorithm in all mentioned versions in high dimensions is much smaller than the time of other implemented algorithms. Also, our algorithms can be used with thousands of multidimensional observations.

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Exact computation of the halfspace depth

Cited by 64 publications

References 41 publications

Depth for Curve Data and Applications

Depth for Curve Data and Applications

Nonparametric Imputation by Data Depth

Approximate calculation of Tukey's depth and median with high-dimensional data

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