Spatial Auto-Regressive (SAR) models are used in digital image analysis for filtering and forecasting purposes. The two basic approaches are Unilateral and Multilateral systems, depending on the fact that the "current" pixel X(i,j) is placed in relationship only with the "previous" ones: X(i-1,j-1) (following a lexicographic order) or also with the "future" ones: X(i+1,j+1). In vector form, these two schemes lead, respectively, to Triangular and Symmetrical contiguity matrices V, W, which may be used in least squares (LS) and maximumun likelihood (ML) estimators to consistently estimates the parameters of the SAR models. The scripts uploaded here compare Triangular and Rook first-oder SAR systems plus drift, by: 1) computing the contiguity matrices; 2) simulating processes in conditional (to the border conditions) and iterated form; and in AR and MA (moving average) form; 3) estimating the parameters with LS and ML methods.
Simulation and Estimation of Spatial AutoRegressive (SAR) Models on Regular Lattices
Carlo Grillenzoni
2024-01-01
Abstract
Spatial Auto-Regressive (SAR) models are used in digital image analysis for filtering and forecasting purposes. The two basic approaches are Unilateral and Multilateral systems, depending on the fact that the "current" pixel X(i,j) is placed in relationship only with the "previous" ones: X(i-1,j-1) (following a lexicographic order) or also with the "future" ones: X(i+1,j+1). In vector form, these two schemes lead, respectively, to Triangular and Symmetrical contiguity matrices V, W, which may be used in least squares (LS) and maximumun likelihood (ML) estimators to consistently estimates the parameters of the SAR models. The scripts uploaded here compare Triangular and Rook first-oder SAR systems plus drift, by: 1) computing the contiguity matrices; 2) simulating processes in conditional (to the border conditions) and iterated form; and in AR and MA (moving average) form; 3) estimating the parameters with LS and ML methods.
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