Describing statistical dependencies is foundational to empirical scientific research. For uncovering intricate and possibly non-linear dependencies between a single target variable and several source variables within a system, a principled and versatile framework can be found in the theory of Partial Information Decomposition (PID). Nevertheless, the majority of existing PID measures are restricted to categorical variables, while many systems of interest in science are continuous. In this paper, we present a novel analytic formulation for continuous redundancy--a generalization of mutual information--drawing inspiration from the concept of shared exclusions in probability space as in the discrete PID definition of $I^\mathrm{sx}_\cap$. Furthermore, we introduce a nearest-neighbor based estimator for continuous PID, and showcase its effectiveness by applying it to a simulated energy management system provided by the Honda Research Institute Europe GmbH. This work bridges the gap between the measure-theoretically postulated existence proofs for a continuous $I^\mathrm{sx}_\cap$ and its practical application to real-world scientific problems.

翻译：描述统计依赖性乃是实证科学研究的基础。为揭示系统中单个目标变量与多个源变量之间复杂且可能非线性的依赖关系，部分信息分解（PID）理论提供了一种通用且严谨的框架。然而，现有的大多数PID度量方法仅适用于分类变量，而科学领域中许多值得关注的系统却是连续的。本文受离散PID定义中$I^\mathrm{sx}_\cap$基于概率空间共享排除概念的启发，提出了一种针对连续冗余量（互信息的推广形式）的新型解析公式。此外，我们引入了一种基于最近邻的连续PID估计器，并通过将其应用于本田欧洲研究院（Honda Research Institute Europe GmbH）提供的模拟能源管理系统，展示了该方法的有效性。本研究弥合了连续$I^\mathrm{sx}_\cap$在测度论层面存在性证明与其在实际科学问题中应用之间的鸿沟。