Abstract
Geographically weighted regression (GWR) is a popular technique to deal with spatially varying relationships between a response variable and predictors. Problems, however, have been pointed out (see Wheeler and Tiefelsdorf in J Geogr Syst 7(2):161–187, 2005), which appear to be related to locally poor designs, with severe impact on the estimation of coefficients. Different remedies have been proposed. We propose two regularization methods. The first one is generalized ridge regression, which can also be seen as an empirical Bayes method. We show that it can be implemented using ordinary GWR software with an appropriate choice of the weights. The second one augments the local sample as needed while running GWR. We illustrate both methods along with ordinary GWR on an example of housing prices in the city of Bilbao (Spain) and using simulations.
Similar content being viewed by others
Notes
In this connection, we notice that the bi-square kernel has been replaced by a Gaussian kernel as the default option in the R function gwr of package spgwr (Bivand and Yu 2012) that we use in some of the computations below.
References
Bárcena M, Menéndez P, Palacios M, Tusell F (2011) Measuring the effect of the real estate bubble: a house price index for Bilbao. BILTOKI 2011.07, University of the Basque Country (UPV/EHU), http://hdl.handle.net/10810/5463
Bárcena M, Menéndez P, Palacios M, Tusell F (2014) A real-time property value index based on web data. In: Zhao Y, Cen Y (eds) Data mining applications in R. Academic Press, New York, pp 273–298
Belsley DA (1991) Conditioning diagnostics. Collinearity and weak data in regression. Wiley, New York
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
Bivand R, Yu D (2012) spgwr: geographically weighted regression. http://CRAN.R-project.org/package=spgwr, R package version 0.6-18
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(4):371–396
Fotheringham S, Charlton M, Brunsdon C (2002) Geographically weighted regression: the analysis of spatially varying relationships. Wiley, New York
Gelfand AE, Kim HJ, Sirmans C, Banerjee S (2003) Spatial modeling with spatially varying coefficient processes. J Am Stat Assoc 98(462):387–396
Geniaux G, Napoléone C (2008) Semi-parametric tools for spatial hedonic models: an introduction to mixed geographically weighted regression and geoadditive models. In: Hedonic methods in housing markets. Springer, Berlin, pp 101–127
Griffith DA (2008) Spatial-filtering-based contributions to a critique of geographically weighted regression (GWR). Environ Plan A 40(11):2751–2769
Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for non-orthogonal problems. Technometrics 12:55–67
Judge C, Hill RC, Griffiths WE, Lütkepohl H, Lee T (1988) Introduction to the theory and practice of econometrics, 2nd edn. Wiley, New York
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 Plan A 34(4):733–754
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
Páez A, Farber S, Wheeler DC (2011) A simulation-based study of geographically weighted regression as a method for investigating spatially varying relationships. Environ Plan A 43(12):2992–3010
Puntanen S, Styan GP, Isotalo J (2011) Matrix tricks for linear statistical models. Springer, Berlin
Robbins H (1956) Proceedings of the third Berkeley symposium on mathematical statistics and probability, vol 1: contributions to the theory of statistics. University of California Press, chap An Empirical Bayes approach to Statistics
Seber GAF, Lee AJ (1998) Linear regression analysis. Wiley, New York
Silvey SD (1969) Multicollinearity and imprecise estimation. J R Stat Soc Ser B 31:539–552
Silvey SD (1980) Optimal design. Chapman & Hall, London
Tibshirani R (1996) Regression shrinkage and selection via the LASSO. J R Stat Soc Ser B 58:267–288
Vidaurre D, Bielza C, Larrañaga P (2012) Lazy LASSO for local regression. Comput Stat 27:531–550
Wheeler D (2010) Visualizing and diagnosing coefficients from geographically weighted regression models. In: Jiang B, Yao X (eds) Geospatial analysis and modelling of urban structure and dynamics, GeoJournal library, vol 99. Springer, Dordrecht, pp 415–436
Wheeler D (2011) gwrr: Geographically weighted regression with penalties and diagnostic tools. http://CRAN.R-project.org/package=gwrr, R package version 0.1-1
Wheeler D, Calder C (2007) An assessment of coefficient accuracy in linear regression models with spatially varying coefficients. J Geogr Syst 9(2):145–166
Wheeler D, Tiefelsdorf M (2005) Multicollinearity and correlation among local regression coefficients in geographically weighted regression. J Geogr Syst 7(2):161–187
Wheeler DC (2007) Diagnostic tools and a remedial method for collinearity in geographically weighted regression. Environ Plan A 39(10):2464–2481
Wheeler DC (2009) Simultaneous coefficient penalization and model selection in geographically weighted regression: the geographically weighted LASSO. Environ Plan A 41(3):722–742
Wheeler DC, Waller L (2008) Comparing spatially varying coefficient models: a case study examining violent crime rates and their relationships to alcohol outlets and illegal drug arrests. J Geogr Syst 11(1):1–22
Wheeler DC, Páez A, Spinney J, Waller L (2013) A Bayesian approach to hedonic price analysis. In: Papers in regional science (to appear), pp 1–22
Acknowledgments
The comments of the editor of the journal and three referees have substantially improved the original manuscript and are gratefully acknowledged. Partial support from grants ECO2008-05622 (MCyT) and IT-347-10 (Basque Government) is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bárcena, M.J., Menéndez, P., Palacios, M.B. et al. Alleviating the effect of collinearity in geographically weighted regression. J Geogr Syst 16, 441–466 (2014). https://doi.org/10.1007/s10109-014-0199-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10109-014-0199-6