Angle Based Dependence Measures in Metric Spaces

In this article, we introduce a general framework of angle based independence test using reproducing kernel Hilbert space. We consider that both random variables are in metric spaces. By making use of the reproducing kernel Hilbert space equipped with Gaussian measure, we derive the angle based dependence measures with simple and explicit forms. This framework can be adapted to different types of data, like high-dimensional vectors or symmetry positive definite matrices. And it also incorporates several well-known angle based independence tests. In addition, our framework can induce another notable dependence measure, generalized distance correlation, which is proposed by direct definition. We conduct comprehensive simulations on various types of data, including large dimensional vectors and symmetry positive definite matrix, which shows remarkable performance. An application of microbiome dataset, characterized with high dimensionality, is implemented.