Geographically weighted regression (GWR) (Brunsdon et al. 1996; Fotheringham et al. 1998; Fotheringham et al. 2002) is a useful technique for modelling local spatial relationships. By allowing local parameters at every single location to be estimated, GWR reveals the spatial non-stationarity in relationships between a dependent variable and several independent variables over space.
Geographical weighting is usually achieved by a kernel function with a given bandwidth. Bandwidth determines the rate at which the weights decay around a regression point, and reflects the degree of spatial variation for each parameter. If the bandwidth tends to infinity, the model will tend to a global regression model.
Current approaches in GWR allow an optimal bandwidth to be selected for a model, all the relationships examined in the model follow this uniform bandwidth.
In an extension of basic GWR, semi-parametric GWR( mixed GWR, Brunsdon et al. 1999; Fotheringham et al. 2002) two optional bandwidths are defined in the model, one is infinity bandwidth for the globally fixed terms , the other is a certain bandwidth identified in the locally varying terms.
In reality, the weighting function for each parameter might need a different rate of distance-decay to reflect given process, this requires Flexible
Bandwidth GWR(FBGWR) to be developed where flexible multiple bandwidth is defined for each individual independent variable as well as for the intercept.
The development from GWR to FBGWR is described in the following figure:
y i
  bw 1
( u i
, v i
) x i 1
  bw 2
( u i
, v i
) x i 2
f f
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(0) f
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(1) 
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[y
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  i
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One practical approach to calibrate a FBGWR is backfitting method. The idea is to calibrate each term, assuming that all the other terms are known.
The algorithm regresses the partial residuals on each individual variable in an iteration way, each iteration gives new calibrations of each term, and eventually these should converge, provided some regularity conditions apply to all hat matrices. In this way, all the calibrations are solved simultaneously.
5.16% 55.83% 99.99%
0.000001
After some steps, the bandwidth selected f o r e a c h v a r i a b l e tends to be stable, so is the residual sum of s q u a r e ( R S S ) f o r each calibration. At last, the backfitting algorithm converges, g i v i n g t h e f i n a l calibration for the whole model.
.
A case study employs FBGWR to examine the determinants of population change during the Irish Famine in 1840s.
percent of population decline.
land value per hectare (VALUATION), population per acre of cropped land( PopCrop_41).
The kernel function used is adaptive bi-square. Bandwidth are searched by AIC optimization for each variable during each iteration as shown in above figure.
As a diagnostic, the RSS of the FBGWR model is compared with that of the basic
GWR model. The maps on the right show that FBGWR performs better here.
 FBGWR acts as a generalisation of other simpler models including global linear regression, GWR, semi-parametric GWR.
 It allows full exploratory investigation of data relationships that may vary at different spatial scales within a given model
 It can also be used as a diagnostic tool to guide model selection for more parsimonious GWR/MGWR/MLR fits
 Further work includes tests on more complex model and on simulated data.
More model fit diagnostics also need to be explored.
Bandwidth: 15.25%
Bandwidth:
Intercept 5.16%
VALUATION 55.83%
PopCrop 99.99%
Research presented in this poster was funded by a Strategic Research Cluster Grant (07/SRC/I1168) by Science Foundation Ireland under the National Development Plan. The authors gratefully acknowledge this support.