Tukey’s Depth for Object Data
Tukey’s Depth for Object Data
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DOI：
10.1080/01621459.2021.2011298
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发表时间：
2021-09
影响因子：
3.7
通讯作者：
Xiongtao Dai;S. López-Pintado
中科院分区：
文献类型：
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作者：
Xiongtao Dai;S. López-Pintado
摘要本文提出了一种新的基于数据深度的非欧对象数据探索工具，扩展了著名的Tukey的欧几里德数据深度。建议的度量半空间深度，适用于一般度量空间中的数据对象，分配给数据点的深度值，这些深度值表征这些点相对于分布的中心性，并提供了一个可解释的中心向外排名。理想的理论性质，推广标准的深度性质假设为欧几里德数据建立度量半空间深度。深度中位数，定义为最深点，被证明具有高的鲁棒性作为一个位置描述符在理论和仿真。我们提出了一个有效的算法来近似度量半空间深度，并说明它的能力，以适应内在的数据几何。度量半空间深度应用于阿尔茨海默氏症的研究，揭示组的差异，在大脑的连接，建模为协方差矩阵，在不同阶段的痴呆症的受试者。基于7种致病性寄生虫的系统发育树，我们提出的度量半空间深度也被用来构建一个有意义的共识估计的进化历史，并确定潜在的离群树。本文的补充材料可在网上查阅。
Abstract We develop a novel exploratory tool for non-Euclidean object data based on data depth, extending celebrated Tukey’s depth for Euclidean data. The proposed metric halfspace depth, applicable to data objects in a general metric space, assigns to data points depth values that characterize the centrality of these points with respect to the distribution and provides an interpretable center-outward ranking. Desirable theoretical properties that generalize standard depth properties postulated for Euclidean data are established for the metric halfspace depth. The depth median, defined as the deepest point, is shown to have high robustness as a location descriptor both in theory and in simulation. We propose an efficient algorithm to approximate the metric halfspace depth and illustrate its ability to adapt to the intrinsic data geometry. The metric halfspace depth was applied to an Alzheimer’s disease study, revealing group differences in the brain connectivity, modeled as covariance matrices, for subjects in different stages of dementia. Based on phylogenetic trees of seven pathogenic parasites, our proposed metric halfspace depth was also used to construct a meaningful consensus estimate of the evolutionary history and to identify potential outlier trees. Supplementary materials for this article are available online.