mixed geographically weighted regression

Dear Roger,

thank you for your quick response!

If I understand it correctly, the hat matrix is calculated using all 
explanatory variables. In my case, however, I would need to restrict the 
column space to those covariates where I assume varying coefficients (as 
in eq. (3)), and for this purpose I would need to calculate S_v by hand. 
Therefore, I would need the weight matrices for every observation. Or is 
there an easier way?

Kind regards,

Marco


-------- Original-Nachricht --------

Datum: Tue, 5 Jan 2010 18:00:50 +0100 (CET)
Von: Roger Bivand <Roger.Bivand at nhh.no>
An: Marco Helbich <marco.helbich at gmx.at>
CC: r-sig-geo at stat.math.ethz.ch
Betreff: Re: [R-sig-Geo] mixed geographically weighted regression

On Tue, 5 Jan 2010, Marco Helbich wrote:

Dear list,

I am trying to fit a mixed geographically weighted regression model

(with adaptive kernel) using the spgwr package, i.e. I want to hold some of the
coefficients fixed at the global level. Thus, I have the following
questions:

1. Which is the most efficient way to estimate such a model?
a) I found the posting

http://www.mail-archive.com/r-sig-geo at stat.math.ethz.ch/msg00984.html where Roger recommended to first fit a global model,
then the GWR using the residuals.

b) The method proposed in Mei et al. (2006,  pp. 588-589, see

http://www.envplan.com/abstract.cgi?id=a3768) first computes the projection matrix of
the locally varying part (called S_v) and uses this in a second step to
derive the fixed coefficients (this seems to me like an application of the
FWL-theorem see http://en.wikipedia.org/wiki/FWL_theorem).

2. In order to follow this method, I first have to find the kernel
weights at each point. The help-file says that these can be found in the
SpatialPointsDataFrame (SDF), but I could not get it from there. Where
can I extract them?

The sums of weights for each fit point are in the returned object, but
this is not what you (do not) want. The S_v matrix in the paper (eq. 3) is
returned as the hat matrix, I believe. Since you have S_v, you do not need
the W(u_i, v_i) weights (a diagonal matrix for each fit (and data) point
i). Given S_v, the unnumbered equation in the middle of the page gives you
\hat{\beta_c}, doesn't it? I think that I would pre-multiply X_c and Y by
(I - S_v), then use QR methods to complete, if I wanted to proceed with
this.

Because of concerns about how these things are done, and how they are
represented in the literature, I'd look for corrobotation - being able to
reproduce others' published results for example.

Hope this helps,

Roger

We are using such a code:
library(spgwr)
data(georgia)
g.adapt.gauss <- gwr.sel(PctBach ~ TotPop90 + PctRural + PctEld + PctFB

+ PctPov + PctBlack, data=gSRDF, adapt=TRUE)

res.adpt <- gwr(PctBach ~ TotPop90 + PctRural + PctEld + PctFB + PctPov

+ PctBlack, data=gSRDF, adapt=g.adapt.gauss)

res.adpt$SDF

I hope my problem is clear and appreciate every hint! Thank you!

Best regards
Marco

--
Roger Bivand
Economic Geography Section, Department of Economics, Norwegian School of
Economics and Business Administration, Helleveien 30, N-5045 Bergen,
Norway. voice: +47 55 95 93 55; fax +47 55 95 95 43
e-mail: Roger.Bivand at nhh.no