Mathematical Geosciences

, Volume 42, Issue 6, pp 657–680 | Cite as

The Use of Geographically Weighted Regression for Spatial Prediction: An Evaluation of Models Using Simulated Data Sets

  • P. Harris
  • A. S. Fotheringham
  • R. Crespo
  • M. Charlton


Increasingly, the geographically weighted regression (GWR) model is being used for spatial prediction rather than for inference. Our study compares GWR as a predictor to (a) its global counterpart of multiple linear regression (MLR); (b) traditional geostatistical models such as ordinary kriging (OK) and universal kriging (UK), with MLR as a mean component; and (c) hybrids, where kriging models are specified with GWR as a mean component. For this purpose, we test the performance of each model on data simulated with differing levels of spatial heterogeneity (with respect to data relationships in the mean process) and spatial autocorrelation (in the residual process). Our results demonstrate that kriging (in a UK form) should be the preferred predictor, reflecting its optimal statistical properties. However the GWR-kriging hybrids perform with merit and, as such, a predictor of this form may provide a worthy alternative to UK for particular (non-stationary relationship) situations when UK models cannot be reliably calibrated. GWR predictors tend to perform more poorly than their more complex GWR-kriging counterparts, but both GWR-based models are useful in that they provide extra information on the spatial processes generating the data that are being predicted.


Relationship nonstationarity Relationship heterogeneity GWR Kriging Spatial interpolation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Atkinson PM (2001) Geographical information science: geocomputation and nonstationarity. Prog Phys Geogr 25:111–122 Google Scholar
  2. Bitter C, Mulligan GF, Dall’erba S (2007) Incorporating spatial variation in housing attribute prices: a comparison of geographically weighted regression and the spatial expansion method. J Geogr Syst 9:7–27 CrossRefGoogle Scholar
  3. Chilès JP, Delfiner P (1999) Geostatistics—modelling spatial uncertainty. Wiley, New York Google Scholar
  4. Cressie N (1986) Kriging nonstationary data. J Am Stat Assoc 81:625–634 CrossRefGoogle Scholar
  5. Deutsch CV (1997) Direct assessment of local accuracy and precision. In: Baafi EY, Scofield NA (eds) Geostatistics Wollongong ’96. Kluwer Academic, Dordrecht, pp 115–125 Google Scholar
  6. Deutsch CV, Journel AG (1998) GSLIB geostatistical software library and user’s guide. Oxford University Press, New York Google Scholar
  7. Farber S, Páez A (2007) A systematic investigation of cross-validation in GWR model estimation: empirical analysis and Monte Carlo simulations. J Geogr Syst 9:371–396 CrossRefGoogle Scholar
  8. Fotheringham AS, Brunsdon C, Charlton M (2002) Geographically weighted regression—the analysis of spatially varying relationships. Wiley, Chichester Google Scholar
  9. Gambolati G, Volpi G (1979) A conceptual deterministic analysis of the kriging technique in hydrology. Water Resour Res 15:625–629 CrossRefGoogle Scholar
  10. Gao X, Asami Y, Chung C (2006) An empirical evaluation of spatial regression models. Comput Geosci 32:1040–1051 CrossRefGoogle Scholar
  11. Gelfand AE, Kim HJ, Sirmans CJ, Banerjee S (2003) Spatial modeling with spatially varying coefficient processes. J Am Stat Assoc 98:387–396 CrossRefGoogle Scholar
  12. Gelfand AE, Schmidt AM, Banerjee S, Sirmans CJ (2004) Nonstationary multivariate process modeling through spatially varying coregionalisation. Test 13:266–312 CrossRefGoogle Scholar
  13. Genton MC, Furrer R (1998) Analysis of rainfall data by robust spatial statistics using S+SPATIALSTATS. J Geogr Inf Decis Anal 2:126–136 Google Scholar
  14. Goovaerts P (2001) Geostatistical modelling of uncertainty in soil science. Geoderma 103:3–26 CrossRefGoogle Scholar
  15. Haas TC (1990) Lognormal and moving window methods of estimating acid deposition. J Am Stat Assoc 85:950–963 CrossRefGoogle Scholar
  16. Haas TC (1996) Multivariate spatial prediction in the presence of non-linear trend and covariance non-stationarity. Environmetrics 7:145–165 CrossRefGoogle Scholar
  17. Hengl T, Heuvelink GBM, Rossiter DG (2007) About regression kriging: from equations to case studies. Comput Geosci 33:1301–1315 CrossRefGoogle Scholar
  18. Heuvelink GBM, Pebesma EJ (2002) Is the ordinary kriging variance a proper measure of interpolation error? In: Hunter G, Lowell K (eds) The fifth international symposium on spatial accuracy assessment in natural resources and environmental sciences. RMIT University, Melbourne, pp 179–186 Google Scholar
  19. Isaaks EH, Srivastava RM (1989) An introduction to applied geostatistics. Oxford University Press, New York Google Scholar
  20. Journel AG (1986) Geostatistics: models and tools for the earth sciences. Math Geol 18:119–140 CrossRefGoogle Scholar
  21. Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, London Google Scholar
  22. Kammann EE, Wand MP (2003) Geoadditive Models. J R Stat Soc C, Appl Stat 52(1):1–18 CrossRefGoogle Scholar
  23. Kanevski M, Maignan M (2004) Analysis and modeling of spatial environmental data. Dekker, New York Google Scholar
  24. Kitanidis PK, Shen KF (1996) Geostatistical interpolation of chemical concentration. Adv Water Resour 19:369–378 CrossRefGoogle Scholar
  25. Leung Y, Mei C, Zhang W (2000) Statistical tests for spatial nonstationarity based on the geographically weighted regression model. Environ Plann 32:9–32 CrossRefGoogle Scholar
  26. Lloyd CD (2010) Nonstationary models for exploring and mapping monthly precipitation in the United Kingdom. Int J Climatol 30:390–405 Google Scholar
  27. Matheron G (1963) Principles of geostatistics. Econ Geol 58:1246–1266 CrossRefGoogle Scholar
  28. Matheron G (1969) Le krigeage universal. Centre de Morphologie, Fontainebleau Google Scholar
  29. Matheron G (1989) Estimating and choosing: an essay on probability in practice. Springer, Berlin Google Scholar
  30. Neuman SP, Jacobson EA (1984) Analysis of nonintrinsic spatial variability by residual kriging with application to regional groundwater levels. Math Geol 16:499–521 CrossRefGoogle Scholar
  31. Páez A, Uchida T, Miyamoto K (2002) A general framework for estimation and inference of geographically weighted regression models: 1. Location-specific kernel bandwidths and a test for locational heterogeneity. Environ Plann 34:733–754 CrossRefGoogle Scholar
  32. Páez A, Long F, Farber S (2008) Moving window approaches for hedonic price estimation: an empirical comparison of modelling techniques. Urban Stud 45:1565–1581 CrossRefGoogle Scholar
  33. Pardo-Igúzquiza E, Dowd PA (1998) The second-order stationary universal kriging model revisited. Math Geol 30:347–378 CrossRefGoogle Scholar
  34. Pereira MJ, Soares A, Rosario L (2002) Characterization of forest resources with satellite spot images by using local models of co-regionalization. In: Kleingeld WJ, Krige DG (eds) Geostatistics 2000. Geostatistical Association of Southern Africa, Cape Town Google Scholar
  35. Rivoirard J (2002) On the structural link between variables in kriging with external drift. Math Geol 34:797–808 CrossRefGoogle Scholar
  36. Schabenberger O, Gotway C (2005) Statistical methods for spatial data analysis. Chapman & Hall, London Google Scholar
  37. Wang N, Mei C, Yan X (2008) Local linear estimation of spatially varying coefficient models: an improvement on the geographically weighted regression technique. Environ Plann 40:986–1005 CrossRefGoogle Scholar
  38. Wheeler D (2007) Diagnostic tools and a remedial method for collinearity in geographically weighted regression. Environ Plann 39:2464–2481 CrossRefGoogle Scholar
  39. Wheeler D (2009) Simultaneous coefficient penalization and model selection in geographically weighted regression: the geographically weighted lasso. Environ Plann 41(3):722–742 CrossRefGoogle Scholar
  40. Yamamoto JK (2000) An alternative measure of the reliability of ordinary kriging estimates. Math Geol 32:489–509 CrossRefGoogle Scholar
  41. Zhang X, Eijkeren JC, Heemink AW (1995) On the weighted least-squares method for fitting a semivariogram model. Comput Geosci 21:605–608 CrossRefGoogle Scholar
  42. Zhang L, Gove JH, Heath LS (2005) Spatial residual analysis of six modeling techniques. Ecol Model 186:154–177 CrossRefGoogle Scholar
  43. Zimmerman DL, Pavik C, Ruggles A, Armstrong MP (1999) An experimental comparison of ordinary and universal kriging and inverse distance weighting. Math Geol 31:375–390 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2010

Authors and Affiliations

  • P. Harris
    • 1
  • A. S. Fotheringham
    • 1
  • R. Crespo
    • 2
  • M. Charlton
    • 1
  1. 1.National Centre for GeocomputationNational University of Ireland MaynoothMaynoothIreland
  2. 2.Institute for Spatial and Landscape PlanningSwiss Federal Institute of TechnologyZurichSwitzerland

Personalised recommendations