GWR

1. 5.16% 55.83% 99.99% 0.000001 Development of a Flexible Bandwidth GWR Wenbai Yang Supervised by A. Stewart Fotheringham, Paul Harris National Centre for Geocomputation, National University of Ireland Maynooth From GWR to FBGWR 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: GWR 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. Choice of Bandwidth 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. Extension 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. Basic GWR A uniform bandwidth A certain bandwidth or INFINITY Mixed GWR Flexible Bandwidth GWR A vector of bandwidths Calibration for FBGWR Backfitting Method 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. … An initial guess … … … … … … Repeat until no changes in functions Case study After some steps, the bandwidth selected for each variable tends to be stable, so is the residual sum of square (RSS) for each calibration. At last, the backfitting algorithm converges, giving the final calibration for the whole model. . A case study employs FBGWR to examine the determinants of population change during the Irish Famine in 1840s. Dependent variable: percent of population decline. Two independent variables: 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. Bandwidth: 15.25% Bandwidth: Intercept 5.16% VALUATION 55.83% PopCrop 99.99% Conclusion • 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. RSS: 33131.04 RSS: 29987.97 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.