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.

翻译：本文针对非规则、非网格点云 $\mathcal{V}_{n}\subset\mathbb{R}^{2}$ 上观测数据的空间独立性，提出一种非参数检验方法。对于每个位置 $v\in\mathcal{V}_{n}$，我们通过 $m$ 个最近邻观测的序数模式对局部空间结构进行编码，获得一种在严格单调变换下保持不变且对异常值稳健的符号表示。在原假设（空间独立性）下，局部序数模式独立同分布，并在对称群 $\mathcal{S}_{m}$ 上均匀分布，且与未知边际分布 $F$ 无关。利用这一特征，我们基于经验序数模式频率的加法对数比（ALR）变换构造了检验统计量 $L_{n}$。通过在图基 $\alpha$-混合条件下的图依赖过程中心极限定理，我们证明了 $L_{n}$ 依分布收敛于 $\chi^{2}_{m!-1}$ 随机变量，从而得到一种渐近枢轴量方法，无需估计冗余参数。广泛的蒙特卡罗研究证实，在中等样本量下，$\chi^{2}_{m!-1}$ 近似已足够精确；检验能有效控制名义水平下的第一类错误，且功效随空间依赖强度单调递增。值得注意的是，该方法能在线性和非线性变换的空间自回归模型中检测依赖关系，体现了序数模式方法特有的稳健性。我们的框架将空间序数模式检验范式从规则网格扩展到一般空间支撑集，为大量观测位置非规则的学科应用领域打开了序数模式推断的大门。