In this study, we link and compare the geographically weighted regression (GWR) model with the kriging with an external drift (KED) model of geostatistics. This includes empirical work where models are performance tested with respect to prediction and prediction uncertainty accuracy. In basic forms, GWR and KED (specified with local neighbourhoods) both cater for nonstationary correlations (i.e. the process is heteroskedastic with respect to relationships between the variable of interest and its covariates) and as such, can predict more accurately than models that do not. Furthermore, on specification of an additional heteroskedastic term to the same models (now with respect to a process variance), locally-accurate measures of prediction uncertainty can result. These heteroskedastic extensions of GWR and KED can be preferred to basic constructions, whose measures of prediction uncertainty are only ever likely to be globally-accurate. We evaluate both basic and heteroskedastic GWR and KED models using a case study data set, where data relationships are known to vary across space. Here GWR performs well with respect to the more involved KED model and as such, GWR is considered a viable alternative to the more established model in this particular comparison. Our study adds to a growing body of empirical evidence that GWR can be a worthy predictor; complementing its more usual guise as an exploratory technique for investigating relationships in multivariate spatial data sets.
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For corresponding MLR prediction variances use: S(x) = y(x)T[Y T Y]−1 y(x).
In this case the bandwidth is a nonlinear parameter, which reflects a fixed local sample size that exerts the greatest influence on each local regression fit.
This is a pragmatic modelling decision where an alternative would be to find an optimal bandwidth at each iteration step of the H-GWR fit. Further work could address this issue.
In this respect, both GWR and KED are optimised for prediction accuracy only. For KED, a more succinct approach for neighbourhood selection is described in Rivoirard (1987), but is not used here as it would compromise our GWR to KED comparison.
Observe that the first part of the KED variance in expression 6 represents the kriging variance of the residuals and the second part is a consequence of estimating the MLR trend component.
Clear relationships can also be promoted by replacing the GW variance estimates with GW standard deviation (SD) estimates (and adapting the whole H-KED procedure accordingly).
Laslett (1994) provides a general view on this issue with respect to kriging versus splines.
Critical load data can be found at http://critloads.ceh.ac.uk/index.htm (last accessed 10 January 2009).
The random error addition was not required for our focused analysis, as our chosen KED-LN model could be reliably calibrated without it.
The H-KED-LR and H-KED-GWR models are effectively inter-changeable, where the former is now chosen as demonstration.
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Research presented in this paper was funded by a Strategic Research Cluster grant (07/SRC/I1168) by the Science Foundation Ireland under the National Development Plan. The authors gratefully acknowledge this support. Thanks are also due to the first author’s PhD studentship and S. Juggins at Newcastle University; and to M. Kernan at University College London for providing the case study data.
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Harris, P., Brunsdon, C. & Fotheringham, A.S. Links, comparisons and extensions of the geographically weighted regression model when used as a spatial predictor. Stoch Environ Res Risk Assess 25, 123–138 (2011). https://doi.org/10.1007/s00477-010-0444-6