Comparative Spatial Filtering in Regression Analysis
Getis, Arthur, Griffith, Daniel A., Geographical Analysis
One approach to dealing with spatial autocorrelation in regression analysis involves the filtering of variables in order to separate spatial effects from the variables' total effects. In this paper we compare two filtering approaches, both of which allow spatial statistical analysts to use conventional linear regression models. Getis' filtering approach is based on the autocorrelation observed with the use of the [G.sub.i] local statistic. Griffith's approach uses an eigenfunction decomposition based on the geographic connectivity matrix used to compute a Moran's I statistic. Economic data are used to compare the workings of the two approaches. A final comparison with an autoregressive model strengthens the conclusion that both techniques are effective filtering devices, and that they yield similar regression models. We do note, however, that each technique should be used in its appropriate context.
The need to account for temporal and/or spatial autocorrelation is well known among regression analysts. Many texts on time series detail the way temporal autocorrelation can be identified and modeled. The literature on spatial autocorrelation is less well developed. One approach to dealing with spatial autocorrelation in regression analysis involves filtering, which seeks to undertake data analyses within the context of the study of stochastic error characteristic of time series analysis. In time series analysis, ARIMA and impulse-response functions are often employed to filter autocorrelation. In the spatial domain, Haining (1991), for example, shows that the temporal-type filtering approach is equivalent to a spatial autocorrelation adjustment for the case of bivariate correlation. This paper compares two additional filtering approaches, both of which allow spatial statistical analysts to use conventional linear regression models in which parameters are estimated by ordinary least squares (OLS). It outline s the mechanics of filtering, leaving a researcher to conceptually establish the underlying source of spatial autocorrelation (for example, spatial process, misspecification).
1. SPATIAL FILTERING AND SPATIAL AUTOREGRESSION
The motivation for this type of work is to allow spatial analysts to employ traditional linear regression techniques while insuring that regression residuals behave according to required model assumptions, such as uncorrelated errors. When georeferenced data are used, residuals are assumed not to be spatially autocorrelated; otherwise the regression model is said to be misspecified. Least squares estimators of intercept and slope parameters offer a considerable advantage over other estimators. OLS regression is simpler than generalized least squares regression, has a well-developed theory, and has available a battery of diagnostic statistics that make interpretations easy and straightforward.
Of the approaches used to include spatial effects in regression analyses, the most often cited is the family of autoregressive models of the general form discussed by Anselin (1988) and Griffith (1988). In essence, these models depend on one or more spatial structural matrices that account for spatial autocorrelation in the georeferenced data from which model parameters are estimated. The spatial autocorrelation devices are constructed from geographic weights matrices, which are used to capture the covariation among values of one or more random variables that are associated with the configuration of areal units. Spatial autoregressive models almost exclusively assume normality, and are nonlinear in nature. For these models, it is inappropriate to use OLS estimation procedures for model development and testing. In this paper, our procedures identify the spatial and nonspatial components of georeferenced variables that make up the OLS regression model. The spatial autoregressive model, on the other hand, essent ially supplies global measures of spatial dependence without uncovering individual spatial and nonspatial component contributions. …