Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.
翻译:统计深度为更高层面的数据提供了量化和中位数的基本概括性。 本文基于控制理论和eikonal等式,提出了一种新的全球界定的统计深度类型,用以测量在达到分布支持范围之外的点的路径中必须经过的最小概率密度: 例如空间无限性。 这种深度很容易解释和计算, 直截了当地捕捉多模式行为, 并自然延伸到非欧洲语言的数据。 我们证明了这种深度的各种特性, 并提供了对计算考虑的讨论。 特别是, 我们证明这种深度概念在近似有几何限制的对抗模型下是稳健的, 这是Tukey深度所没有的特性。 最后, 我们用二维混合物模型和MNIST来举例说明一些例子。