Authors: Gunter Spöck (University of Klagenfurt, Austria), Jürgen Pilz (University of Klagenfurt, Austria), and Miodrag Lovric (Radford University, USA)
This extensive entry offers a comprehensive examination of statistical methods for analyzing areal and spatial interaction data—two key domains within spatial statistics. Areal data are observations tied to geographic regions (e.g., districts, counties, or countries), while spatial interaction data involve flows or movements between pairs of regions (e.g., migration, trade, or hospital visits).
The authors begin with classical techniques for areal data, including the construction of proximity matrices and the measurement of spatial autocorrelation using Moran’s I and Geary’s c. These global indices quantify the degree to which neighboring regions share similar values. Localized spatial patterns are visualized through LISA maps (Local Indicators of Spatial Association), providing insight into clusters and spatial heterogeneity.
The article then turns to spatial regression models, including Simultaneous Autoregressive (SAR), Spatial Lag, and Spatial Durbin models. These account for correlation in residuals or predictors across space. Also discussed is Geographically Weighted Regression (GWR), which estimates location-specific relationships, accommodating spatial non-stationarity.
The second half of the article addresses spatial interaction data. Origin-destination models such as the gravity model are described, modeling the volume of interaction between regions as a function of distance, population, and other covariates. These models may be constrained by origin, destination, or both, and estimated using Poisson regression within generalized linear models.
The discussion extends to Bayesian spatial modeling, including tools like spBayes (MCMC), INLA (Integrated Nested Laplace Approximation), and spatial packages in R such as spdep, spgwr, and spatialreg. The software section also highlights GIS platforms like ArcGIS and Python’s PySAL for spatial econometrics and visualization.
Finally, the authors review state-of-the-art techniques for spatial prediction, model comparison, and hybrid approaches that merge classical, Bayesian, and machine learning methodologies. Applications are drawn from economics, public health, environmental science, and urban planning.
For mathematical derivations, implementation tools, and extended references, see the full article in the International Encyclopedia of Statistical Science.