Computing the halfspace depth with multiple try algorithm and simulated annealing algorithm

Shao, Wei; Zuo, Yijun

doi:10.1007/s00180-019-00906-x

Cited by 10 publications

(10 citation statements)

References 25 publications

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“…In extreme situations, if we select g(x) ∝ h(x), that is, g(x) � c • h(x) (where c � 1/􏽒 A h(x)dx ), the variance of 􏽥 μ will drop to zero, and 􏽥 μ is equal to the exact value 􏽒 A h(x)dx. However, we cannot directly use the IS method defined in (6) during such an extreme situation because we do not know the exact value of 􏽒 A h(x)dx in advance.…”

Section: New Algorithm For Sd In R Dmentioning

confidence: 99%

“…Using the definition of SD in (1) is not appropriate in computing the SD value of a data point with respect to a dataset. e MC method in (6) becomes extremely inefficient when dimension p or sample size n is excessively large because the number of simplices containing the original data point decreases with the increase in p or n.…”

Section: Theoremmentioning

confidence: 99%

“…As a powerful multivariate nonparametric and robust statistical tool, the statistical depth function extends the concept of one-dimensional data order statistics and provides the central-outward sorting of multivariate data [1][2][3][4]. In recent years, the interest of researchers in statistical depth has increased due to the extensive application of the statistical depth function in multivariate statistical analysis, robust estimation, discriminant analysis, hypothesis testing, machine learning, economics, and hydrological data analysis [5,6].…”

Section: Introductionmentioning

confidence: 99%

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Computing Simplicial Depth by Using Importance Sampling Algorithm and Its Application

Meng

Shao

2021

Mathematical Problems in Engineering

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Simplicial depth (SD) plays an important role in discriminant analysis, hypothesis testing, machine learning, and engineering computations. However, the computation of simplicial depth is hugely challenging because the exact algorithm is an NP problem with dimension d and sample size n as input arguments. The approximate algorithm for simplicial depth computation has extremely low efficiency, especially in high-dimensional cases. In this study, we design an importance sampling algorithm for the computation of simplicial depth. As an advanced Monte Carlo method, the proposed algorithm outperforms other approximate and exact algorithms in accuracy and efficiency, as shown by simulated and real data experiments. Furthermore, we illustrate the robustness of simplicial depth in regression analysis through a concrete physical data experiment.

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Section: New Algorithm For Sd In R Dmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing Simplicial Depth by Using Importance Sampling Algorithm and Its Application

Meng

Shao

2021

Mathematical Problems in Engineering

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“…Due to the curse of dimensionality, µ • should behave more and more like a boundary point as m increases. Table 1 shows a performance comparison between SAP and some methods implemented in R packages ddalpha (Pokotylo et al, 2016), depth (Genest et al, 2017), and DepthProc (Kosiorowski and Zawadzki, 2017) and MTMSA (Shao and Zuo, 2020). In calling the first three packages, we used the "approximate" option (since no algorithm can compute the exact depth when m > 6) and increased the number of initial random directions from the default 1000 to 20,000 to boost their accuracy; the other parameters are taken their default values.…”

Section: Location Depthmentioning

confidence: 99%

On Generalization and Computation of Tukey's Depth: Part I

She¹,

Tang²,

Liu³

2021

Preprint

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Tukey's depth offers a powerful tool for nonparametric inference and estimation, but also encounters serious computational and methodological difficulties in modern statistical data analysis. This paper studies how to generalize and compute Tukey-type depths in multidimensions. A general framework of influence-driven polished subspace depth, which emphasizes the importance of the underlying influence space and discrepancy measure, is introduced. The new matrix formulation enables us to utilize state-of-the-art optimization techniques to develop scalable algorithms with implementation ease and guaranteed fast convergence. In particular, half-space depth as well as regression depth can now be computed much faster than previously possible, with the support from extensive experiments. A companion paper is also offered to the reader in the same issue of this journal.

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“…For instance, the computation of the halfspace depth of a single point in arbitrary dimension is known to be NP-hard [19]. Therefore, a great deal of research has focused on procedures that approximate the true depth [13,6,4,32,2,33]. A particularly simple upper bound on the halfspace and projection depth can be devised if one uses their so-called projection property [13], which means that the overall (multivariate) depth of a point x is expressed as the infimum of (univariate) depths of projections of x with respect to the projected dataset.…”

Section: Introductionmentioning

confidence: 99%

Uniform convergence rates for the approximated halfspace and projection depth

Nagy,

Dyckerhoff,

Mozharovskyi

2019

Preprint

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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 dimensionality. In such scenarios, the exact depth is frequently approximated using a randomized approach: The data are projected into a finite number of directions uniformly distributed on the unit sphere, and the minimal depth of these univariate projections is used to approximate the true depth. We provide a theoretical background for this approximation procedure. Several uniform consistency results are established, and the corresponding uniform convergence rates are provided. For elliptically symmetric distributions and the halfspace depth it is shown that the obtained uniform convergence rates are sharp. In particular, guidelines for the choice of the number of random projections in order to achieve a given precision of the depths are stated.

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Computing the halfspace depth with multiple try algorithm and simulated annealing algorithm

Cited by 10 publications

References 25 publications

Computing Simplicial Depth by Using Importance Sampling Algorithm and Its Application

Computing Simplicial Depth by Using Importance Sampling Algorithm and Its Application

On Generalization and Computation of Tukey's Depth: Part I

Uniform convergence rates for the approximated halfspace and projection depth

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