Geographical weighting as a further refinement to regression modelling: An example focused on the NDVI–rainfall relationship

The regression analyses undertaken commonly in remote sensing are aspatial, ignoring the locational information associated with each sample site at which the variables under study were measured. Typically, basic ordinary least squares regression analysis is used to derive a relationship that is believed to be uniformly applicable across the study area. Although such global analyses may appear satisfactory, often with large coefficients of determination derived, they may provide an inappropriate description of the relationship between the variables under study. In particular, a global regression analysis may miss local detail that can be significant if the relationship is spatially non-stationary. Local statistical approaches, such as geographically weighted regression, include the spatial coordinates of the sample sites in the analysis and may provide a more appropriate basis for the investigation of the relationship between variables. The potential value of geographically weighted regression to the remote sensing community is illustrated with reference to the relationship between the normalised difference vegetation index (NDVI) and rainfall over north Africa and the Middle East over an 8-year period. For each year, spatial non-stationarity was evident, particularly with regard to the slope parameter of the regression model. Moreover, the conventional ordinary least squares regression models, while superficially strong (minimum R2=0.67), were relatively poor local descriptors of the relationship. Relative to this, the geographically weighted approach to regression provided considerably stronger relationships from the same data sets (minimum R2=0.96) as well as highlighting areas of local variation. The implications of the difference in the outputs from the two types of regression analysis are illustrated with reference to the use of the derived NDVI–rainfall relationships in mapping desert extent. For example, with the data relating to 1987 the southern limit of the Sahara was generally estimated to lie at a more southerly position when the relationship derived from OLS rather than geographically weighted regression was used.