ABSTRACT Technological advances in science and engineering have led to the routine collection of large and complex data objects, where the dependence structure among those objects is often of great interest. Those complex objects (e.g., different brain subcortical structures) often reside in some Banach spaces, and hence their relationship cannot be well characterized by most of the existing measures of dependence such as correlation coefficients developed in Hilbert spaces. To overcome the limitations of the existing measures, we propose Ball Covariance as a generic measure of dependence between two random objects in two possibly different Banach spaces. Our Ball Covariance possesses the following attractive properties: (i) It is nonparametric and model-free, which make the proposed measure robust to model mis-specification; (ii) It is nonnegative and equal to zero if and only if two random objects in two separable Banach spaces are independent; (iii) Empirical Ball Covariance is easy to compute and can be used as a test statistic of independence. We present both theoretical and numerical results to reveal the potential power of the Ball Covariance in detecting dependence. Also importantly, we analyze two real datasets to demonstrate the usefulness of Ball Covariance in the complex dependence detection. Supplementary materials for this article are avaiable online.

中文翻译：

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Ball 协方差：Banach 空间中依赖性的通用度量


摘要 科学和工程领域的技术进步导致了大型复杂数据对象的常规收集，其中这些对象之间的依赖结构通常引起人们极大的兴趣。这些复杂的对象（例如，不同的大脑皮层下结构）通常驻留在一些巴纳赫空间中，因此它们的关系不能通过大多数现有的依赖性度量（例如在希尔伯特空间中开发的相关系数）来很好地表征。为了克服现有度量的局限性，我们提出 Ball 协方差作为两个可能不同的 Banach 空间中两个随机对象之间依赖关系的通用度量。我们的球协方差具有以下吸引人的特性：（i）它是非参数且无模型的，这使得所提出的度量对模型错误指定具有鲁棒性； (ii) 当且仅当两个可分离巴拿赫空间中的两个随机对象是独立的时，它是非负的并且等于零； (iii) 经验球协方差很容易计算，可以用作独立性检验统计量。我们提出了理论和数值结果来揭示球协方差在检测依赖性方面的潜在力量。同样重要的是，我们分析了两个真实数据集，以证明球协方差在复杂依赖性检测中的有用性。本文的补充材料可在线获取。