论文标题
半空间深度的措施部分重建
Partial reconstruction of measures from halfspace depth
论文作者
论文摘要
相对于有限的（或概率）borel量$ x $的半空间深度$ x $，$ \ mathbb {r}^d $中的$μ$定义为所有包含$ x $的$μ$ $的$μ$ $ $。一个自然的问题是，半空间深度作为$ x \ in \ mathbb {r}^d $的函数是否完全确定了该度量$μ$。通常，事实证明并非如此，并且有可能在$ \ mathbb {r}^d $中具有相同的半空间深度功能。在本文中，我们表明，尽管结果负面结果，但仍然可以从其半空间深度中获得有关$μ$的支撑和位置的大量信息。我们在一个非平凡的双变量概率分布的示例中说明了我们的部分重建程序，该概率分布从其半空间深度成功地确定了原子部分。
The halfspace depth of a $d$-dimensional point $x$ with respect to a finite (or probability) Borel measure $μ$ in $\mathbb{R}^d$ is defined as the infimum of the $μ$-masses of all closed halfspaces containing $x$. A natural question is whether the halfspace depth, as a function of $x \in \mathbb{R}^d$, determines the measure $μ$ completely. In general, it turns out that this is not the case, and it is possible for two different measures to have the same halfspace depth function everywhere in $\mathbb{R}^d$. In this paper we show that despite this negative result, one can still obtain a substantial amount of information on the support and the location of the mass of $μ$ from its halfspace depth. We illustrate our partial reconstruction procedure in an example of a non-trivial bivariate probability distribution whose atomic part is determined successfully from its halfspace depth.
