Processes and characteristics of urban areas at the human-environment interface (e.g. social stratification, segregation, urban poverty) depend on a diverse set of socio-demographic, economic and environmental factors. Due to the heterogeneity of urban areas, it can be assumed that the strength and direction of the influence of these factors varies over space.
Special properties of geospatial data are spatial autocorrelation and spatial heterogeneity (nonstationarity). Spatial autocorrelation implies a spatial association between an attribute value at a particular location and attribute values at other locations close by. Spatial heterogeneity describes systematic spatial variation of attribute values across space. These spatial effects must be taken into account when modeling spatial relationships in a regression model.
Traditionally, global statistical regression approaches are applied to study the influence of explanatory variables on a target variable. These approaches emphasize similarities across space. In the following analysis, this global or “one fits all approach” is juxtaposed against spatial autocorrelation and nonstationarity. In particular, we explore a global non-spatial regression model and both a global and local spatial regression model of the relationship between indicators of socio-economic disadvantage and neighborhood demographic context and crime rates in Belgian municipalities in the period 2008-2012.
An exploration of the spatial patterns of crime is warranted. The causal processes driving crime may vary over space, that is, predictor variables may operate differently in different locations. This may be especially relevant in policy studies where there is growing recognition that understanding the context of crime – the where and when of criminal events – is key to understanding how crime can be controlled and prevented. Crime studies that highlight local variations – local contexts of crime – will likely have more relevance to real-world policy applications. Empirically, if these variations in causal processes do exist and are not accounted for, the statistical model will be inaccurate.
Estimations provided by a global model might be inadequate in capturing spatially varying relationships, as global statistics are only describing average relations between the dependent variable and the considered explanatory variables. With increasing spatial variation of local observations, the reliability of global model estimates decreases.
There might be spacial dependencies that refer to attribute values in one location which might depend on values of the attributers in neighboring locations.
The assumption of spatial heterogeneity can be suggested by the fact that criminality and its determinants may be distributed unevenly across space. Another source of spatial heterogeneity is the dynamics between population and location. That is, cultural differences and differences in attitudes and behaviour across locations may alter how people react to various contextual variables. Given the potential of spatial heterogeneity, it would be naïve to assume that the spatial processes between criminality and its determinants are stationary (or universal) and can be captured by a conventional “global” model.
Following Tobler’s first law of geography which states that “everything is related to everything else, but near things are more related than distant things”, GWR has to be calibrated in a way that observations near to observation i have more influence on the estimation of the parameters that data located further away from i.
GWR takes advantage of spatial dependence in the data. Spatial dependence implies that data available in locations near the focal location are more informative about the relationship between the independent and the dependent variables in the focal location. When evaluating estimates for a focal location, GWR gives more weight to data from closer locations than to data from more distant locations. It is assumed that the relative weight of the contributing locations decays at an empirically determined rate as that distance from the focal location increases.
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