Choosing the Geographic Weight

SpaceStat offers several different options for geographic weighting of the data associated with the variables in a GWR model.  Some weighting choices require the user to specify an "area of influence", or bandwidth.  One simple approach to geographic weighting is to choose a set number of neighbors, or a set distance from the focal location, and then weight all of the neighbors captured in this definition as "1" and all others as "0".   Other approaches refine this "all or none" scheme by allowing you to use a weighting function to weight the points closest to the focal point more heavily than others farther away but still in the "neighborhood."

Choosing the number of neighbors or distance range

As described above, you can standardize how many points are used in each local regression calculation (choose the number of neighbors option), or standardize the maximum distance between a focal point and the most distant neighbor that will be included in the local model calculations (the range option).  

Choosing the weighting function

As described above, you can choose to use an "all or none" scheme (all points that meet the nearest neighbor number, or range, criteria receive a weight of 1, others receive a 0), or a more complicated weighting function.  SpaceStat includes Gaussian and bi-square "falloff" in weight as a functions of distance from the focal point, using a fixed bandwidth that you set, or one that SpaceStat finds to be optimal (see below).   You can also fix the number of neighbors and choose a bi-square falloff with an adaptive bandwidth; in this case, the bandwidth for each target point is set at the distance to the farthest of the allowed neighbors (using the range option here gives a bandwidth that is close to the user-specified range).

Choosing the bandwidth

If you choose a geographic weighting scheme with a bandwidth that is constant over the entire geography, then you will need to use the "Bandwidth settings" page in the task manager.  On this page, you can choose a specific value (a distance, in the same units as your point geography), or you can allow SpaceStat to determine the optimal value.  To determine the optimal value, SpaceStat uses a cross-validation procedure that minimizes the quantity:

 

Here the predicted values are calculated from a local geographically weighted regression that does not include variable values from the target location.  The sum from each step in the optimization procedure, which tests a set of bandwidths in the range of bandwidths that you choose on the bandwidth settings page, is output to the log view.  If you do not specify a range, then the minimum bandwidth will be set to the minimum separation between points in the geography, and the maximum bandwidth will be set to the diagonal of the bounding rectangle of the geography.  When searching for the optimum bandwidth, we suggest that you start with a small number of bandwidth steps over a wide range, and then narrow down the range being explored.