# Interpreting result of Geographically Weighted Regression (GWR)?

I want to use the Geographically Weighted Regression (GWR) to model local relationships between my dependent variable and a set of independent variables.

When running GWR in ArcGIS, the coefficients with the parameter estimates can be mapped, which is also recommended. But I am not sure, how to interpret these values correctly.

The following graphic shows a sample (!) how a result could look like. Assuming the dependent variable is auto theft and the visualized independent variable are parking lots. I want to understand how to interpret the parameter estimates.

I do not understand the connection between the parameter estimate values and what they mean in regard to the correlation, meaning if for higher values it can be said that "many parking lots lead to many auto thefts" and for lower values "few parking lot lead to many/few auto thefts".

How can this result be interpreted correctly?

The aim is to find out the local regions, where parking lots correlate with auto thefts.

The parameter estimates are essentially the slopes of regression lines that fit your data averaged at that location. So if the parameter is positive then there are more auto thefts in places with more parking lots, and if the parameter is negative then there are fewer thefts where there are more parking lots.

However, without looking at the uncertainty in those parameters, you can't say if the parameter is significantly different from zero. Since GWR works on a moving window system (and hence there's a tuning parameter for window size) your outputs will be autocorrelated, so its likely to look smooth. You can't make valid statements about any trend across the space without taking this into account.

Also, conclusions from GWR seem a bit pointless. You might be able to say "over here, we get more thefts with more parking lots, and over here we get fewer". All that tells you is that the number of parking lots is not the determining factor in the theft rate! Some factor not covered by the variables in your model must be responsible for the change in behaviour of car thieves from one area to the other. Its not the number of parking lots!

Spatial modelling should be the process of working towards removing all unknown spatial structure in your model, until what you are left with are known modelled terms and uncorrelated noise residuals. GWR doesn't really take you there.

• Thanks a lot for your answer and the detailed Information! So if there are only positive values, does it mean that parking lots lead to auto thefts, but with a different strength? And what values would areas get where there are no parking lots and no auto thefts? Commented Feb 23, 2016 at 9:08
• You stated "All that tells you is that the number of parking lots is not the determining factor in the theft rate" - how can you tell this? How would the result have to look, so that there is a determining factor? And if negtive values mean that parking lots do not lead to auto thefts, then I can exclude those areas for further analysis, because the assumption that in those regions the parking lots correlate with Auto thefts is wrong, right? Commented Feb 23, 2016 at 9:10
• If in some places more parking lots means more thefts, and in other places more parking lots mean fewer thefts, the conclusion is that something else is causing more thefts. Maybe its the deprivation in an area. Or the population density. If you can fit a linear model for thefts against some variables and end up with uncorrelated noise residuals, you've explained as much variation as possible, and you can draw reasonable inferences about how theft depends on X, Y, or Z based on the parameter estimates. Commented Feb 23, 2016 at 10:16
• Thanks for the explanation, it does make sense. The GWR4 software provides also t-values by which the significance can be tested. Then, the assumptions with "more parking lots more thefts (high values)", "few parking lots more thefts (low/negative values)" are only true in the significant Areas, right? Commented Feb 23, 2016 at 10:30
• Its hard to get a statistician to say anything is true, but we might say that the correlation is significantly non-zero in those areas. Commented Feb 23, 2016 at 10:47

I think I can understand why you can't find a logical answer for the relation between your outputs, you have to use another type of data presentation. For example, you can do normalization or use local clustering values in order to use this value as dependent variable. std. Residual (normalize it with std. deviation) can examine and explain the confidence of independent variables, in this case map each confidence as individual relation for your interpretation answer. mapping R2 is less useful in case that your independent variable is negative. I would suggest to redo your original dependent variable first and test it with fixed distance only as test and see how it goes. I hope this can help you.