A Variance-Stabilizing Coding Scheme for Spatial Link Matrices

Abstract

In spatial statistics and spatial econometrics two coding schemes are used predominately. Except for some initial work, the properties of both coding schemes have not been investigated systematically. In this paper we do so for significant spatial processes specified as either a simultaneous autoregressive or a moving average process. Results show that the C-coding scheme emphasizes spatial objects with relatively large numbers of connections, such as those in the interior of a study region. In contrast, the ^-coding scheme assigns higher leverage to spatial objects with few connections, such as those on the periphery of a study region. To address this topology-induced heterogeneity, we design a novel ^-coding scheme whose properties lie in between those of the C-coding and the JK-coding schemes. To compare these three coding schemes within and across the different spatial processes, we find a set of autocorrelation parameters that makes the processes stochastically homologous via a method based on the exact conditional expectation of Moran's /. In the new S-coding scheme the topology induced heterogeneity can be removed in toto for Moran's/ as well as for moving average processes and it can be substantially alleviated for autoregressive processes.

Short-Run Regional Forecasts: Spatial Models Through Varying Cross-Sectional and Temporal Dimensions

“…Although there are different ways to model connectivity between areas, spatial weight matrices used in this paper are based on the contiguity criterion, where a weight of 1 is assigned to the cell ( , ) i j of W if the spatial units i and j share a common boundary, and 0 otherwise. The matrices are row-standardized in the case of the SVAR, and globally standardized (Tiefelsdorf et al 1999) for SF, because of the symmetry requirement for eigenvector extraction.…”

Spatial Weights: Constructing Weight-Compatible Exchange Matrices from Proximity Matrices

“…Tiefelsdorf et al (1999) for an explicit illustration. The latter authors also propose and investigate various spatial coding schemes aimed at extracting a convenient spatial weights matrix V (G) from spatial link or proximity matrices G = (g ij ), meant as an immediate, possibly rough but accessible spatial information about proximity relationships between regions i and j. Proximities G between regions may be determined by mutual contiguity, accessibility, inverse distance, flow, etc.…”

“…In particular, Tiefelsdorf et al (1999) together with other workers have considered the following coding schemes V (G):…”

Computing the Jacobian in Spatial Models: An Applied Survey

“…However, row standardisation upweights neighbour relations for observations with few neighbours, and downweights relations for those with many neighbours. Tiefelsdorf et al (1999) propose a variance-stabilising scheme instead of row standardisation, which for underlying symmetric neighbour relations also yields an asymmetric spatial weights matrix that is similar to symmetric. Further discussion of these issues may be found in Bivand et al (2008); Ward and Gleditsch (2008).…”