[GIS] Geographically Weighted Regression Using the Poisson Distribution

geographically-weighted-regression

From my reading so far, GWR uses the Gaussian or bisquare distribution. My dependent and explanatory variables are rates. I was reading Chapter 12 of Spatial Statistical Data Analysis for GIS Users by Krivoruchko and saw this:

"For non-Gaussian data, gwr can be used with Poisson regression when input data are counts or rates and with logistic regression when data are binary or proportions."

How do I test my data to determine if I should use Gaussian, bisquare or the Poisson for my GWR analysis? What are the key differences between them? Since I was going to use ArcGIS for the GWR, is there a way I can select the Poisson?

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I think this answers my question after spending some time on the web. From the text below, it seems that Gaussian is appropriate since I have some negative rates and my variables are not integers.

"Choosing the form of the regression model

Three items will control the form and output from a geographically weighted regression model:

The nature of the dependent variable y:

Continuous (linear or Gaussian model)

Positive integer counts (Poisson model)

Proportions or rates (logistic model)
The nature of the explanatory, or "x" variable or variables

Continuous

Categorical
The weight function(s)

Geographical weights that control how neighboring locations influence values at specific locations

Non-spatial weights to account for the reliability of data (e.g., population size for disease rate data)"

source: http://www.biomedware.com/files/documentation/spacestat/Statistics/Multivariate_Modeling/Regression/About_Geographically_Weighted_Regression.htm

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[GIS] What are the most efficient Geographic Weighted Regression packages

From GWR by Roger Bivand:

Geographically weighted regression (GWR) is an exploratory technique mainly intended to indicate where non-stationarity is taking place on the map, that is where locally weighted regression coefficients move away from their global values. Its basis is the concern that the fitted coefficient values of a global model, fitted to all the data, may not represent detailed local variations in the data adequately – in this it follows other local regression implementations. It differs, however, in not looking for local variation in ‘data’ space, but by moving a weighted window over the data, estimating one set of coefficient values at every chosen ‘fit’ point. The fit points are very often the points at which observations were made, but do not have to be. If the local coefficients vary in space, it can be taken as an indication of non-stationarity.

The technique ... involves first selecting a bandwidth for an isotropic spatial weights kernel, typically a Gaussian kernel with a fixed bandwidth chosen by leave-one-out cross-validation. Choice of the bandwidth can be very demanding, as n regressions must be fitted at each step. Alternative techniques are available, for example for adaptive bandwidths, but they may often be even more compute-intensive.

> library(maptools)
> library(spdep)
> owd <- getwd()
> setwd(system.file("etc/shapes", package = "spdep"))
> NY8 <- readShapeSpatial("NY8_utm18")
> setwd(owd)
> library(spgwr)
> bwG <- gwr.sel(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data = NY8, gweight = gwr.Gauss,
+ verbose = FALSE)
> gwrG <- gwr(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data = NY8, bandwidth = bwG,
+ gweight = gwr.Gauss, hatmatrix = TRUE)
> gwrG

Once the bandwidth has been found, or chosen by hand, the gwr function may be used to fit the model with the chosen local kernel and bandwidth. If the data argument is passed a SpatialPolygonsDataFrame or a SpatialPointsDataFrame object, the output object will contain a component, which is an object of the same geometry populated with the local coefficient estimates. If the input objects have polygon support, the centroids of the spatial entities are taken as the basis for analysis. The function also takes a fit.points argument, which permits local coefficients to be created by geographically weighted regression for other support than the data points.

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[GIS] What are the most efficient Geographic Weighted Regression packages

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