Formulation of Huge Lattice Spatial Adjacency Matrices With Non-rectangular Shape of Socio-economic Grid-Cell Data for the Analysis of Sustainable Economy With High Computational Efficiency

Abstract

The advantage of using grid-cell data for socio-economic analysis should be the feasibility to incorporate satellite data that will enrich the regional analysis and has an important role to observe the relationship between socio-economics and nature. This advancement corresponds to the sustainable development goals that balance the socio-economic quality in harmony. In order to perform the analysis, formulation of a spatial adjacency matrix has an important role to project the spatial relationship within regions. However, no precedent research provided a practical formulation for the spatial adjacency matrix in grid-cell data structure (Fitrianto & Tanaka, 2017).The general process that used shapefiles solely, which store geometry and attribute information for the spatial features (ESRI, 1998) to construct the adjacency matrix is not suitable. The problem arises due to the existence of NA cells that represent non-inhabitant areas such as water bodies, yet the shapefile does not contain this information inside the municipal body. The NA cells create a non-rectangular lattice and it is important to exclude them in the analysis to correctly project the real information.This article provides a method to precisely project the real information by using Kronecker product to construct the adjacency matrix and applying a projection matrix to eliminate the NA cells (Tanaka & Nishii, 2009). It showed eminent efficiency compared with commonly used R package called spdep. Experimental results verified that this method, even for huge dimension with a trillion elements, produces more than 2000 times faster elapsed time than the package.

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

“…Based on this framework, we propose a straightforward formulation and algorithm to construct spatial 𝐖 matrix that stores interlinkages between each unit. This proposition is an extension of an existing formulation, but only for a two-dimensional case in Tanaka and Nishii (2009) and Fitrianto et al (2018).…”

“…For further consideration regarding higher computational efficiency, we advise constructing the equation ( 9) or (10) in the sparse data structure. Several studies show a higher efficiency for generating spatial 𝐖 matrix due to a lot of zero-element involvement in the structure (Pace & Barry, 1997;Pace & LeSage, 2004;Fitrianto et al, 2018;Zhang & Li. 2018;Lam & Souza, 2020).…”

“…This result can be compared to the output generated by other existing methods provided by several software analyses. For example, we can use cell2nb inside 'spdep' library in R. Even thoughFitrianto et al (2018) already shows the critical advantage of using Kronecker product, especially for huge 𝐃 R×C cases. Proof by Example of Kronecker Product Methods in ℝ 3 .…”