[R-sig-Geo] mixed geographically weighted regression

[Roger Bivand]

On Tue, 5 Jan 2010, Marco Helbich wrote:

> 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?

Naturally. Use the hat matrix from a regular GWR fit with only X_v 
included, as the paper (seems to) describe.

Roger

>
> 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@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
>
>

-- 
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