Geographically weighted models

Project team: Martin Charlton, Chris Brunsdon, Paul Harris and Binbin Lu


Project goal: To develop a geospatial modelling service that provides an integrated framework for handling locally varying structures in a wide range of spatial (and spatio-temporal) models.

Rationale: To enable spatially informed decisions in complex dynamic environments relating to movement, structure and location. Location can be at the: (i) local micro-scale, (ii) meso-scale of the city/region and (iii) macro-scale of the nation. All involve sensing the environment within a series of application specific contexts combined with the analysis of data streams from various sensors using appropriate toolsets.

Research themes: Our basic toolset is the ability to model spatially heterogeneous processes using geographically weighted (GW) techniques (Brunsdon et al. 1996; 2002; 2007; Dykes and Brunsdon 2007; Fotheringham et al. 2002; Nakaya 2001). In particular, we are currently researching the following eight (related) GW themes:

  • 1. The development of flexible bandwidth GW regression models (Yang et al. 2011) and the continued development of GW generalised linear regression models.
  • 2. For dimension reduction and related multivariate problems, advances for GW principal component analysis (PCA) are underway (Harris et al. 2011a).
  • 3. GW regression and GW-geostatistical hybrids for spatial prediction (Harris et al. 2010a; 2010b; 2011b; Harris and Juggins, 2011) naturally lend themselves to optimal spatial sample design problems and we use their outputs to locate the nodes of a sensor network.
  • 4. All GW models are currently defined using Euclidean or Great Circle distances. Initial work on the use of alternative distance metrics in GW regression (Lu et al. 2011) can now be extended to allow all GW models a choice of metrics.
  • 5. Work on robust GW summary statistics and regression (Harris and Brunsdon 2010; Harris et al. 2010c) is extended to include that of robust GW PCA (Harris et al. 2011c). This completes a suite of nine robust techniques that we use within a 'weight of evidence' approach to detect outliers in (high dimensional) multivariate, spatio-temporal datasets. This approach allows us to indicate the nature of the outliers, where they can have any combination of aspatial, spatial, temporal or relational characteristics (Harris et al. 2011d).
  • 6. Various forms of GW spatial interaction models are in development (Kordi and Fotheringham 2011) and will be applied to 2011 census data at small area level.
  • 7. An understanding of the statistical properties of all GW models or estimators is a priority (Byrne et al. 2009; Charlton et al. 2011). This may entail adopting a Bayesian (spatially varying parameter) modelling approach in some cases.
  • 8. An open source R package encompassing all of the above GW models and techniques is in concurrent development. This feeds into a series GW model training workshops run by the research team.


24th-26th April 2013 - "Three-day Workshop on Geographically Weighted Modelling"


"Geographically weighted (GW) models: advances in investigating spatial heterogeneity" - Paul Harris, Martin Charlton & Chris Brunsdon
"Development of a Flexible Bandwidth GWR" - Wenbai Yang
"Geographically Weighted Regression Using a Non-Euclidean Distance Metric" - Binbin Lu

Example visualisations from GW models

Example outputs from a selection of GW models using a soils geochemistry data set for countries neighbouring the Baltic Sea:

(a) GW Standard Deviations for SiO2
(b) GW Correlations for SiO2 and TiO2
(c) An example GW PCA output for the first three GW principal components
(d) An example GW PCA output for the first GW principal component

Study data can be found in the R mvoutlier package, where the 10 variables (SiO2, TiO2, Al2O3, Fe2O3, MnO, MgO, CaO, Na2O, K2O and P2O5) of this study is a subset of 46 variables that are available. Sample size is 742 sites.


Brunsdon C, Fotheringham AS, Charlton ME (1996) Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity. Geographical Analysis 28(4):281-298

Brunsdon C, Fotheringham AS, Charlton ME (2002) Geographically weighted summary statistics - a framework for localised exploratory data analysis. Computers, Environment and Urban Systems 26:501-524

Brunsdon C, Fotheringham AS, Charlton ME (2007) Geographically Weighted Discriminant Analysis. Geographical Analysis 39:376-996

Byrne G, Charlton ME, Fotheringham AS (2009) Multiple dependent hypothesis tests in geographically weighted regression. Geocomputation 2009, Sydney, Australia

Charlton ME, Brunsdon C, Harris P (2011) Out of condition - exploring local collinearity in multivariate data. 17th European Colloquim on Theoretical and Quantitative Geography, Athens, Greece

Dykes J, Brunsdon C (2007) Geographically weighted visualisation: Interactive graphics for scale-varying exploratory analysis. IEEE Transactions on Visualisation and Computer Graphics 13(6):1161-1168

Fotheringham AS, Brunsdon C, Charlton ME (2002) Geographically Weighted Regression: The Analysis of Spatially Varying Relationships. Chichester: Wiley.

Harris P, Brunsdon C (2010) Exploring spatial variation and spatial relationships in a freshwater acidification critical load data set for Great Britain using geographically weighted summary statistics. Computers & Geosciences 36:54-70

Harris P, Charlton ME, Fotheringham AS (2010a) Moving window kriging with geographically weighted variograms. Stochastic Environmental Research and Risk Assessment 24:1193-1209

Harris P, Fotheringham AS, Crespo R, Charlton ME (2010b) The use of geographically weighted regression for spatial prediction: an evaluation of models using simulated data sets. Mathematical Geosciences 42:657-680

Harris P, Fotheringham AS, Juggins S (2010c) Robust geographically weighed regression: a technique for quantifying spatial relationships between freshwater acidification critical loads and catchment attributes. Annals of the Association of American Geographers 100(2): 286-306

Harris P, Brunsdon C, Charlton ME (2011a) Geographically weighted principal components analysis. International Journal of Geographical Information Science DOI:10.1080/13658816.2011.554838

Harris P, Brunsdon C, Fotheringham AS (2011b) Links, comparisons and extensions of the geographically weighted regression model when used as a spatial predictor. Stochastic Environmental Research and Risk Assessment 25:123-138

Harris P, Brunsdon C, Charlton ME (2011c) Multivariate spatial outlier detection using geographically weighted principal components analysis. 7th International Symposium on Spatial Data Quality, Coimbra, Portugal

Harris P, Charlton ME, Fotheringham AS (2011d) ESPON DB project phase 1 Final report - The identification of exceptional values in the ESPON Database - Technical report 2 (weight of evidence approach)

Harris P, Juggins S (2011) Estimating freshwater critical load exceedance data for Great Britain using space-varying relationship models. Mathematical Geosciences 43: 265-292

Kordi M, Fotheringham AS (2011) Origin- and destination-focused local spatial interaction models. 17th European Colloquim on Theoretical and Quantitative Geography, Athens, Greece

Lu B, Charlton ME, Fotheringham AS (2011) Geographically weighted regression using non-euclidean distance metrics. Spatial Statistics conference, Netherlands

Nakaya T (2001) Local spatial interaction modelling based on the geographically weighted regression approach. GeoJournal 53:347-358

Yang W, Fotheringham AS, Harris P (2011) Model selection in GWR: the development of a flexible bandwidth GWR. Geocomputation 2011, London, UK

3D Campus Model Sample

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