Spatial Heterogeneity

Spatial heterogeneity is a characteristic of the distribution of geographic data, that is, data where the location of observations is explicitly taken into account. It is a special form of structural instability, that is, the situation where a property of a distribution is not constant across the sample. The lack of constancy becomes spatial when it conforms to an underlying spatial structure such as the presence of subregions or the existence of a spatial trend. The spatial heterogeneity may pertain to moments of a distribution such as the mean, variance, or covariance (spatial autocorrelation); to the distribution itself; or to the parameters of a model such as the slope coefficients in a regression model. Detecting and estimating spatial heterogeneity is important because most spatial statistical methods assume homogeneity (i.e., stability or, more formally, spatial stationarity). The lack of such homogeneity not only may invalidate standard techniques but also, more important, may point to the failure of the maintained hypothesis or model to hold uniformly throughout the landscape. Proper treatment of spatial heterogeneity allows for the identification of locations or subregions that do not conform to the overall model. A focus on spatial heterogeneity also corresponds with a growing interest in local phenomena.

The spatial structure of the heterogeneity can be categorized as discrete variation or continuous variation. In the discrete case, the sample can be organized into a number of discrete and spatially contiguous subsets or subregions that often are referred to as spatial regimes. In the continuous case, the variability corresponds to a spatial surface, for example, describing the spatial change (or spatial drift) in a regression coefficient. Spatial heterogeneity is easily visualized by constructing a map of the values of parameters in subsets of the data or by showing a predicted surface that illustrates the spatial drift.

In dealing with discrete spatial heterogeneity, interest typically focuses on the degree to which the mean of the distribution is the same across subregions or on[Page 453] whether regression slope coefficients are constant. The former is investigated by means of spatial analysis of variance, that is, a special case of the familiar ANOVA technique where the “control” consists of a number of spatial subregions. Spatial analysis of variance often is complicated by the presence of spatial autocorrelation that requires the application of specialized techniques. A test for regional homogeneity assesses the degree to which the coefficients in a regression model are constant across spatial regimes. This is a special case of a test on structural stability (e.g., the familiar Chow test in econometrics), but its application is again complicated by the possible presence of spatial autocorrelation (resulting in a spatial Chow test).

Continuous spatial heterogeneity most often is encountered in regression models with spatially varying coefficients. The variation can be modeled in functions of auxiliary variables, as in the so-called spatial expansion method. This is similar to the specification of individual coefficient variability in multilevel (hierarchical) regression models. A recent approach is the geographically weighted regression (GWR), a special case of locally weighted regression, where a kernel estimate is obtained reflecting the variability of regression coefficients around each location in the sample.

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