We propose a flexible and robust nonparametric framework for testing spatial dependence in two- and three-dimensional random fields. Our approach involves converting spatial data into one-dimensional time series using space-filling Hilbert curves. We then apply ordinal pattern-based tests for serial dependence to this series. Because Hilbert curves preserve spatial locality, spatial dependence in the original field manifests as serial dependence in the transformed sequence. The approach is easy to implement, accommodates arbitrary grid sizes through generalized Hilbert (``gilbert'') curves, and naturally extends beyond three dimensions. This provides a practical and general alternative to existing methods based on spatial ordinal patterns, which are typically limited to two-dimensional settings.

翻译：我们提出了一种灵活且稳健的非参数框架，用于检验二维和三维随机场中的空间依赖性。该方法通过使用空间填充希尔伯特曲线将空间数据转换为一维时间序列，然后对该序列应用基于序数模式的序列依赖性检验。由于希尔伯特曲线保持了空间局部性，原始场中的空间依赖性在转换后的序列中表现为序列依赖性。该方法易于实现，通过广义希尔伯特（"gilbert"）曲线可适应任意网格尺寸，并能自然地推广到三维以上。这为现有基于空间序数模式的方法提供了一个实用且通用的替代方案，后者通常仅限于二维场景。