论文标题
投影深度的近似计算
Approximate computation of projection depths
论文作者
论文摘要
数据深度是多元统计数据中的一个概念，可以测量给定数据云中$ \ ir^d $中的点的中心性。如果一个点的深度可以表示为数据的所有一维投影，则深度满足所谓的投影属性。这种深度构成了重要的阶级，其中包括文献中提出的许多深度。对于满足投影属性的深度，很容易构建近似算法，因为仅根据有限数量的一维投影将最小的深度占据最小的深度就可以在多元数据相对于多变量数据的深度上产生上限。如果不存在确切的算法或确切的算法具有较高的计算复杂性，那么这种算法特别有用，就像半空间深度或投影深度一样。为了在高维度中计算这些深度，肯定可以使用具有更好复杂性的近似算法。我们没有专注于单一方法，我们提供了几种方法的全面，公平的比较，这些方法都已经在文献和原始方法中进行了描述。
Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in $\IR^d$. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional projections of the data, then the depth satisfies the so-called projection property. Such depths form an important class that includes many of the depths that have been proposed in literature. For depths that satisfy the projection property an approximate algorithm can easily be constructed since taking the minimum of the depths with respect to only a finite number of one-dimensional projections yields an upper bound for the depth with respect to the multivariate data. Such an algorithm is particularly useful if no exact algorithm exists or if the exact algorithm has a high computational complexity, as is the case with the halfspace depth or the projection depth. To compute these depths in high dimensions, the use of an approximate algorithm with better complexity is surely preferable. Instead of focusing on a single method we provide a comprehensive and fair comparison of several methods, both already described in the literature and original.
