Forecasting Lattice and Point Spatial Data: Comparison of Unilateral and Multilateral SAR Models

Abstract

Spatial auto-regressive (SAR) models are widely used in geosciences for data analysis; their main feature is the presence of weight (W) matrices, which define the neighboring relationships between the spatial units. The statistical properties of parameter and forecast estimates strongly depend on the structure of such matrices. The least squares (LS) method is the most flexible and can estimate systems of large dimensions; however, it is biased in the presence of multilateral (sparse) matrices. Instead, the unilateral specification of SAR models provides triangular weight matrices that allow consistent LS estimates and sequential prediction functions. These two properties are strictly related and depend on the linear and recursive nature of the system. In this paper, we show the better performance in out-of-sample forecasting of unilateral SAR (estimated with LS), compared to multilateral SAR (estimated with maximum likelihood, ML). This conclusion is supported by numerical simulations and applications to real geological data, both on regular lattices and irregularly distributed points.