Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in ${\mathbb{R}}^d$. 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.

数据深度是多元统计数据中的一个概念，用于测量点在给定数据云中的中心性。 ${\mathbb{[R}}^d$。如果一个点的深度可以表示为相对于数据的所有一维投影的最小深度，则该深度满足所谓的投影特性。这样的深度形成了重要的一类，其中包括文献中已经提出的许多深度。对于满足投影特性的深度，可以很容易地构造一个近似算法，因为仅对有限数量的一维投影采用最小的深度会产生相对于多元数据的深度上限。如果不存在精确算法，或者如果精确算法具有很高的计算复杂度（例如半空间深度或投影深度的情况），则这种算法特别有用。要在高维度上计算这些深度，当然，最好使用复杂度更高的近似算法。我们不再关注单一方法，而是对文献中已经描述过的和原始方法中的几种方法进行了全面，公正的比较。