论文标题
散点半空间深度的精确计算
Exact and approximate computation of the scatter halfspace depth
论文作者
论文摘要
散点半空间深度（SHD）是位置半空间（也称为Tukey）深度的延伸，适用于散点的非参数分析。使用SHD，可以为多变量数据定义最小的最佳稳健散射估计器。但是，文献中尚未解决尺寸数据$ d \ geq 2 $数据的精确计算的问题。我们开发了一种精确的算法，用于在任何维度$ d $中计算SHD，并为$ d \ leq 5 $有效地实施IT，并在R中使用任何尺寸$ d \ geq 1 $。由于SHD的确切计算特别是对于更高的维度，我们还提出了两种快速近似算法。我们所有的程序都可以在R套件中免费提供。
The scatter halfspace depth (sHD) is an extension of the location halfspace (also called Tukey) depth that is applicable in the nonparametric analysis of scatter. Using sHD, it is possible to define minimax optimal robust scatter estimators for multivariate data. The problem of exact computation of sHD for data of dimension $d \geq 2$ has, however, not been addressed in the literature. We develop an exact algorithm for the computation of sHD in any dimension $d$ and implement it efficiently using C++ for $d \leq 5$, and in R for any dimension $d \geq 1$. Since the exact computation of sHD is slow especially for higher dimensions, we also propose two fast approximate algorithms. All our programs are freely available in the R package scatterdepth.
