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I used the Geographically Weighted Regression (GWR) to find out which independent variables lead to crime and how they vary over the space.

After a literature review, I ran a global Ordinary Least Squares (OLS) model to find the most suitable set of independent variables, then used the local GWR method. For both the OLS and GWR model I then conducted the Moran's I for the standardized residuals, to be sure that they are not spatially autocorrelated. I used ArcGIS for the analysis.

When I investigated the coefficient values within the different regions in the study area, I came to an issue that is unclear for me: For example I have included the risk factor bus stop, given as density per square kilometers, to explain burglaries (given as crime rate per 100 000 inhabitants).

The values for C1_BusDen (Coefficient 1 for bus stop density) ranges from 21 to 39. One region has a very low value with a crime rate of 1100 and bus stop density of 4.5. Another region has a very high value with a crime rate of 1250 and bus stop density of 5.07.

If the density of bus stops is high in both regions and there is also a similar crime rate, shouldn't be the relationship (value) also the same - or at least similar, but not in the lowest and highest class? Also, the region with the highest density of bus stops has at the same time the highest crime rate - the GWR coefficient value is only 26 - shouldn't it be the highest?

Do I interpret the result in the correct way, because it would make sense that a higher value in the independent variable and a higher value in the dependent varialbe lead to a stronger relationship.

  • Please expand abbreviations like GWR and OLS the first time you use them in your questions so that they can help serve future readers as learning materials. – PolyGeo Dec 2 '15 at 17:37
  • What is your citation in the lit review that indicates that model selection should be based on an OLS? Commonly GWR model selection is performed via AICc and I have not seen support for using a 1st order OLS regression for selecting parameters in a local regression. Mathematically this does not track and is a made up ESRI thing. – Jeffrey Evans Dec 2 '15 at 21:18
  • In literature I looked for the theoretical background of GWR - using a global model first was indeed provided by Esri. – the_chimp Dec 3 '15 at 7:54
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Well, it is a locally weighted regression and does not account for first-order effects very well so, this is not really surprising and inherently one of the limitations of the method. Perhaps if you changed the size of the objective kernel function (bandwidth) this would be mitigated to some degree but, I would not hold my breath. I would point out that the specified distribution and bandwidth can have notable effects on the model. If you are using the ESRI implementation, don't. There is no flexibility in specifying the distributional assumption (eg., Gaussian, Poisson, binomial, multinomial) and you are stuck with the canned regression diagnostics with no access to the objects comprising the model.

Unless you have measurable nonstationarity, GWR is a dubious method and you would be better suited by just using an OLS or GLM. Please take some time to review some of the literature relating to GWR. The Wheeler and Tiefelsdorf (2005) paper demonstrates serious bias in the coefficients due to nonsystematic localized collinearity. Páez et al., (2011) showed, through simulations, that GWR did not reliably discriminate spatially varying process and sometimes exhibited spurious correlations in the local fits. I am not saying don't use GWR but, outside of exploratory analysis, it should be used with great caution and extensively tested for validity and bias.

If 1st order spatial autocorrelation is in fact, an issue affecting residuals and iid, a spatial autoregressive or mixed effects model would be in order. You can use the Lagrange diagnostics (Anselin et al., 1996) to test for spatial dependence in linear models. This will indicate if autocorrelation is influencing the residual error. If you would like to formalize a spatial model, where nonstationarity (2nd order autocorrelation) may be an issue, two methods I would recommend investigating would be principal coordinate analysis of neighbor matrices (Dray et al., 2006) or Eigenvector-based spatial filtering (Griffith 2000).

For inferential models, I particularity like spatial filtering approaches. De Jong et al., (1984) showed an empirical relationship between Eigenvectors and Moran's-I. Building on this concept one can use Eigenvector(s), in a semi-parametric approach, to partial out the effects of spatial process on regression estimates. The concept of PCNM's is to quantify multiscale spatial process, thus directly representing the autocorrelation structure, using scaled-Eigenvector matrices. The reason I cite Dray et al., (2006), for PCNM, is that his extension allows for negative autocorrelation structures. To implement a spatial filtering approach for regression models the function SpatialFiltering in the R library spdep formalizes a brute force method that make model specification quite easy. To formally explore the spatial structure of the data, the R library pcnm provides methods for specifying and visualizing principal coordinate neighbor matrices.

References

Anselin, L., A.K. Bera, R. Florax, & M.J. Yoon (1996) Simple diagnostic tests for spatial dependence. Regional Science and Urban Economics (26)77–104.

De Jong, P., C. Sprenger, F. van Veen, (1984) On extreme values of Moran’s I and Geary’s c. Geographic Analysis. 16:17-24.

Dray, S., P. Legendre, & P.R. Peres-Neto, (2006) Spatial modeling: a comprehensive framework for principal coordinate analysis of neighbor matrices (PCNM). Ecological Modeling, 196:483-493.

Griffith, D. A., (2000) A linear regression solution to the spatial autocorrelation problem. Journal of Geographical Systems, 2:141-156.

Páez,A., S. Farber, & D. Wheeler (2011). A simulation based study of geographically weighted regression as a method for investigating spatially varying relationships. Environment and Planning 43(12):2992–3010.

Wheeler, D. & M. Tiefelsdorf, (2005). Multicollinearity and correlation among local regression coefficients in geographically weighted regression. Journal of Geographical Systems, 7:161–187.

  • Thank you for your detailed answer! I use ArcGIS for the analysis, and for both OLS and GWR models I tested spatial autocorrelation, which indicates that the residuals are randomly distributed. I do not have time to get again further into detail with GWR, I am just wondering if I can use my model and if the calculation of the coefficient values for the regions, as I indicated above, are correct? – the_chimp Dec 3 '15 at 7:58
  • If you do not have any structure in your residuals, due to autocorrelation, then there is absolutely no support for running a GWR model in the first place and you should just go with the OLS. – Jeffrey Evans Dec 3 '15 at 15:37

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