In this article, we consider the problem of testing the independence between two random variables. Our primary objective is to develop tests that are highly effective at detecting associations arising from explicit or implicit functional relationship between two variables. We adopt a multi-scale approach by analyzing neighborhoods of varying sizes within the dataset and aggregating the results. We introduce a general testing framework designed to enhance the power of existing independence tests to achieve our objective. Additionally, we propose a novel test method that is powerful as well as computationally efficient. The performance of these tests is compared with existing methods using various simulated datasets.

翻译：本文研究两个随机变量之间的独立性检验问题。我们的主要目标是开发能够高效检测由变量间显式或隐式函数关系所产生关联的检验方法。我们采用多尺度分析策略，通过考察数据集中不同尺度的邻域结构并整合分析结果来实现这一目标。我们提出了一个通用的检验框架，旨在提升现有独立性检验方法对函数关系的检测效能。此外，我们设计了一种兼具高统计功效与计算效率的新型检验方法。通过多种模拟数据集的实验，我们将这些检验方法与现有方法进行了性能比较。