Nonparametric Testing of Spatial Dependence in 2D and 3D Random Fields

2025-10-17

AI Summary

This paper addresses the nonparametric testing of spatial dependence in two- and three-dimensional random fields. The proposed method maps spatial grid data onto a one-dimensional sequence via space-filling curves—specifically Hilbert and generalized Gilbert curves—and subsequently applies ordinal pattern-based statistical tests to detect dependence structures. Its key contribution lies in the first integration of locality-preserving space-filling curve mappings with nonparametric ordinal pattern analysis, thereby overcoming dimensional and grid-shape constraints inherent in conventional approaches. The framework supports arbitrary-size regular or irregular grids and extends naturally to higher dimensions. Experimental results demonstrate that the method is robust, computationally efficient, and achieves superior detection accuracy compared to existing spatial ordinal-pattern-based techniques, while maintaining conceptual simplicity and ease of implementation.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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.

Problem

• Testing spatial dependence in 2D and 3D random fields
• Converting spatial data into time series using Hilbert curves
• Providing a practical alternative to limited existing methods

Innovation

• Using Hilbert curves to transform spatial data
• Applying ordinal pattern tests for serial dependence
• Extending method to 3D with generalized gilbert curves