🤖 AI Summary
This study addresses the lack of effective nonparametric tests for spatial independence in irregularly sampled point clouds by extending the ordinal pattern test framework—originally developed for regular lattices—to arbitrary irregular spatial supports. The proposed method constructs symbolic representations of ordinal patterns from neighboring observations and, combined with an additive log-ratio (ALR) transformation, yields an asymptotically pivotal test statistic that requires no assumptions on marginal distributions and involves no redundant parameters. Leveraging the invariance of ordinal patterns under monotonic transformations and their robustness to outliers, the theoretical validity of the test is established via a central limit theorem for graph-dependent processes under α-mixing conditions. Monte Carlo experiments demonstrate that the test accurately controls Type I error rates even at moderate sample sizes and exhibits stable, monotonically increasing power against both linear and nonlinear spatial dependencies.
📝 Abstract
We propose a nonparametric test of spatial independence for data observed on irregular, non-lattice point clouds $\mathcal{V}_{n}\subset\mathbb{R}^{2}$. For each location $v\in\mathcal{V}_{n}$, we encode the local spatial configuration through the ordinal pattern of the $m$ nearest-neighbour observations, obtaining a symbolic representation that is invariant under strictly monotone transformations and robust to outliers. Under the null hypothesis of spatial independence, the local ordinal patterns are i.i.d.\ and uniformly distributed over the symmetric group $\mathcal{S}_{m}$, regardless of the unknown marginal distribution $F$. We exploit this characterisation to construct a test statistic $L_{n}$ based on the additive log-ratio (ALR) transformation of the empirical ordinal-pattern frequencies. Invoking a central limit theorem for graph-dependent processes under a graph-based $α$-mixing condition, we establish that $L_{n}$ converges in distribution to a $χ^{2}_{m!-1}$ random variable, yielding an asymptotically pivotal procedure with no nuisance parameters. An extensive Monte Carlo study confirms that the $χ^{2}_{m!-1}$ approximation is accurate already at moderate sample sizes, that the test controls size at the nominal level, and that power increases monotonically with the strength of spatial dependence. Notably, the test detects dependence in both linear and nonlinearly transformed spatial autoregressive models, illustrating the robustness that is characteristic of ordinal-pattern methods. Our framework extends the spatial ordinal-pattern testing paradigm from regular lattices to general spatial supports, opening the door to ordinal-pattern inference in the many applied settings where observations are irregularly located.