Simple Formulation of Spatial Weights Matrix in 3D Lattice Space: Unravelling Regional Interlinkages Beyond the Flat Surface

Abstract A rigorous spatial weights matrix (W) is essential for unraveling neighboring linkages between spatial units. Precedent research only focused on how to construct a two-dimensional space by imposing a fundamental assumption that linkages between flat areas and areas with higher variance of elevation are the same. This study presents a comprehensive yet simple formulation to construct a spatial W matrix in three-dimensional space. Central to this approach is the innovative use of the Kronecker product in combination with shift matrices, which incorporating altitude dimension into the spatial W matrix formulation. This comprehensive framework benefit to captures the complexity of different spatial interactions in topographically diverse environments.