1. Robust GW summary statistics & robust GW regression are used to investigate a freshwater acidification data set. Results show that data relationships can vary across space, but this perception can depend on only a few influential outlying observations. A new robust GW regression model is used to cater for such phenomena. GW principal components analysis is used to investigate the changing local structure in multivariate spatial data sets. Using Dublin voter turnout data, the GW PCA methodology is advanced to incorporate: (i) automatic bandwidth selection, (ii) tests for its application and (iii) visualisation techniques for its output. Furthermore, a robust GW PCA is developed to detect multivariate spatial outliers. This new extension is demonstrated using the GSI SURGE project soils geochemistry data for Dublin (Fig. 1). Geographically weighted (GW) models: advances in investigating spatial heterogeneity Paul Harris, Martin Charlton & Chris Brunsdon* Studies for spatial exploration, visualisation & outlier detection Inference related problems in GW regression are investigated using new locally-compensated GW regressions and GW PCA to address strong criticisms of GW regression regarding local collinearity. A guide to fitting and interpreting GW regressions in this respect is given. Simulated and Dublin voter turnout data sets are used in these studies (Figs. 3 & 4). Bootstrap methods are also in development with regard to improved inference in GW regression. A Bayesian spatially-varying coefficient (SVC) model is also newly developed to provide an alternative to GW regression. The Bayesian SVC model is an entirely different approach to the nonparametric GW regression model and benefits from a better handle on uncertainty. The SVC model is performance tested for spatial prediction/inference using a house price data set. GW regression as a spatial predictor is assessed in three complementary studies. First, its performance is compared to: (i) standard regression; (ii) standard kriging (from geostatistics) and (iii) new hybrids (kriging combined with GW regression), using simulated data sets. Results show promise with the hybrids but standard kriging should be preferred. In a second study, GW regression with a heteroskedastic error variance is linked and compared with a corresponding kriging model. This new GW predictor is able to provide relatively accurate confidence intervals. In a third study, indicator kriging is combined with GW regression to form a second novel hybrid that also provides promisingly accurate confidence intervals. The latter studies used freshwater acidification data. Kriging with GW variograms is a novel geostatistical-nonparametric hybrid. This non-stationary variogram technique generalises moving window kriging (MWK) where classic estimators are replaced with information-rich, GW variogram estimators. Results indicate (using four pollution data sets) much promise in the new predictor (Fig. 2). Related studies visualise outputs from non-stationary variogram predictors using comap, that include a new robust MWK model. Optimal sample re-design. Initial work used GW summary statistics and a location-allocation algorithm. Current work uses the new GW predictors above or GW PCA, all with simulated annealing to achieve both univariate and multivariate optimisations. All studies use the GSI SURGE project Dublin soils data. 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. Studies for spatial prediction, its uncertainty & sample re-design Studies for model inference & statistical properties Fig. 1: Multivariate spatial outlier detection with GW PCA The GWmodel R software package The fundamental science of the above studies is transferred to studies in applied science on the StratAG project via an open source R package of statistical computing code. The R package includes existing GW models and our newly developed GW models above (including advancements from other members of the spatial heterogeneity team, not presented here). This R package will be mirrored with a set of GW modelling tools for ESRI’s ArcGIS (written in Python). * Visiting professor – University of Liverpool, UK ModelRankModelRank Std kriging4Std MWK2 Std nonlinear kriging3MWK with GW variograms1 Fig. 2: Classic local variograms (top), GW variograms (middle), global variogram (bottom left) and kriging results Fig. 4: The use of GW PCA to map matrix conditional numbers ( > 30 suggests a significant local collinearity effect). Dublin voter turnout covariate data. Fig.3. Example output from simulation studies for investigating local collinearity in GW regression. Actual parameter B1 with low levels of spatial smoothness coupled with strong correlations with B0, B2 and B3 (the other regression parameters).