Non-parametric entropy tests for spatial dependence

The non-parametric testing of spatial dependence is considered, where the data are generated in a regular two-dimensional grid. Recently, spatial ordinal patterns (SOPs) and corresponding types (which imply a three-part partition of the set of all SOPs) have been proposed for this purpose, where the test statistics are linear expressions of the three type frequencies. In order to use more information from the original data while keeping the tests non-parametric, three versions of refined types are proposed that always lead to six classes of SOPs, namely rotation types, direction types, and diagonal types. In this context, we also present a novel visual representation of SOPs that allows for an intuitive understanding of their characteristics. Furthermore, to incorporate the full frequency distribution of types and refined types, our novel tests for spatial dependence rely on entropy-like statistics instead of the existing linear statistics. Their asymptotic distributions under the null of spatial independence are derived, and the finite-sample performance is analyzed within an extensive simulation study covering various unilateral and bilateral spatial processes. It is shown that the novel entropy tests have appealing power properties and help to recognize how spatial dependence propagates across the data. To illustrate possible applications in practice, two real-world data examples are analyzed, namely one from agricultural science and another one about population changes in a region of Finland.