Hello all,
I am running a Geographically Weighted Regression and calibrating the bandwidth of my models using GW.model R functions. After some tests and playing around with the data, I encountered the following curiosity:
Often, when the number of regressors is higher than the number of observations, we say that there is no unique solution to the system. In fact, any statistical software would throw you an error.
However, I do not know how this occurs in GWR. My bandwidth selection, k-nearest, is lower than the number of regressors, meaning that each local regression has more regressors than observations (i.e., k-nearest neighbors < p regressors). I am able to estimate a solution of which I am very skeptical of because of this issue. But I am able to estimate a solution, which my institution says I should not.
Could somebody clarify what is going on here? Should I be concerned?
Noé
I assume that one of these issues is happening: i) either your initial sample size is very small to use GWR in a robust manner in the first place, ii) you are using an unnecessary large number of explanatory variables or iii) both of the above.
In general, you should have at least a few hundred samples to consider doing GWR to justify its use. Additionally, you may consider running some sort of feature selection (ie., stepwise regression) to remove some variables prior to building GWR. Also, you should pay attention to local colinearity that may arise in the GWR coefficients (check local VIF or condition indexes) as the estimates can be very unrealistic due to that. What matters is if the model solution is interpretable. A model where most of the local coefficients are not significant would not make much sense to use even if the overall R squared might be very high. In fact it is very easy to create GWR models with very high R squared values but with little to no meaning. If you are looking into more refined analysis perhaps you can check Multiscale GWR which offers more tools to deal with such issues.
You could also experiment a bit more with manually setting the bandwidth in different scales rather than taking the automatic bandwidth selection too seriously. I have had cases in the past where the 'optimal' bandwidth was the best in statistical terms (ie., minimizing AICc) but made no sense to accept it.
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