Using spatial ordinal patterns for non-parametric testing of spatial dependence

We analyze data occurring in a regular two-dimensional grid for spatial dependence based on spatial ordinal patterns (SOPs). After having derived the asymptotic distribution of the SOP frequencies under the null hypothesis of spatial independence, we use the concept of the type of SOPs to define the statistics to test for spatial dependence. The proposed tests are not only implemented for real-valued random variables, but a solution for discrete-valued spatial processes in the plane is provided as well. The performances of the spatial-dependence tests are comprehensively analyzed by simulations, considering various data-generating processes. The results show that SOP-based dependence tests have good size properties and constitute an important and valuable complement to the spatial autocorrelation function. To be more specific, SOP-based tests can detect spatial dependence in non-linear processes, and they are robust with respect to outliers and zero inflation. To illustrate their application in practice, two real-world data examples from agricultural sciences are analyzed.