[R-sig-Geo] How to get the Coefficient of Determination in the results of GWR
Danlin Yu
yud at mail.montclair.edu
Thu May 13 14:29:14 CEST 2010
You shall set "adpt" parameter to 0.1.
Sent from my Iphone
Dr. Danlin Yu
Assistant Professor of GIS, Urban Geography
Earth & Environmental Studies
Montclair State University
Montclair, NJ 07043
Tel: 973-655-4313
Fax: 973-655-4072
Email: yud at mail.montclair.edu
在 May 13, 2010，8:03 AM，huangykiz <huangykiz at 163.com> 写到：
> Hi,
> If I want to chang the adaptive Spatial Kernel = 10% neighbors in
> gwr()? How to chang it?
>
> Thank you very much.
>
> Cheers.
>
>
>
>
> 在2010-05-13 17:43:06，"Roger Bivand" <Roger.Bivand at nhh.no> 写道：
>> On Thu, 13 May 2010, huangykiz wrote:
>>
>>> Hi,
>>
>>> I am sorry I say that I cannot get the same R^2 between in R/spgwr
>>> and
>>> SAM in my data.
>>
>> Establish that the adaptive proportion is exactly the same.
>>
>> You haven't done that - copy and paste from SAM to gwr(), not using
>> gwr.sel(). Do it first for fixed Gaussian, then if you get a sensible
>> figure from SAM for adaptive, do the same there. I see very different
>> bandwidths chosen by SAM and by gwr.sel() and GWR3 - gwr.sel() and
>> GWR3
>> usually agree fairly well for CV fixed bandwidths, but gwr.sel()
>> typically
>> continues its search a little longer than GWR3.
>>
>> I don't know how SAM chooses its bandwidth or adaptive proportion,
>> it is
>> closed source, so only its authors know.
>>
>> Is SAM using Great Circle distances, if so, you should set
>> longlat=TRUE in
>> gwr.sel() and gwr()? Are your coordinates geographical (decimal
>> degrees)
>> or projected (metres)?
>>
>> Roger
>>
>>> In R/spgwr
>>> R^2: 0.972989;
>>> AICc (GWR p. 61, eq 2.33; p. 96, eq. 4.21): 4668.92
>>> Effective number of parameters (model: traceS): 435.7586;
>>> Effective number of parameters (residual: 2traceS - traceS'S): 582.3581
>>> ;
>>> Sigma (residual: 2traceS - traceS'S): 2.437066;
>>> Sigma (model: traceS): 1.927127;
>>> Sigma (ML): 1.325501;
>>>
>>> In SAM,
>>> Coefficient of Determination : 0.696
>>> Adjusted r-square (r?Adj): 0.693
>>> Sigma: 20.058
>>> Effective Number of Parameters: 10.002
>>> Akaike Information Criterion (AICc): 4838.299
>>> Correlation Coefficient (r): 0.834
>>> F: 207.852
>>>
>>> Here are my code:
>>> PET.adapt.gauss <- gwr.sel(SPECIES_RI ~ PET,
>>> data=Environmental_variables, coords=cbind(Environmental_variables
>>> $LONGX,
>>> Environmental_variables$LATY),adapt=TRUE)
>>>
>>> PET.gauss<- gwr(SPECIES_RI ~ PET, data=Environmental_variables,
>>> coords=cbind(Environmental_variables$LONGX,
>>> Environmental_variables$LATY),
>>> gweight=gwr.Gauss,adapt=PET.adapt.gauss,hatmatrix=TRUE)
>>>
>>> 1 - (PET.gauss$results$rss/crossprod(scale(Environmental_variables
>>> $SPECIES_RI, scale=FALSE)))
>>>
>>> In SAM, I selecte "spatial Weighting Function"=gaussian, adaptive
>>> Spatial Kernel, and compute Geographical Distances based on
>>> longitudinal
>>> coordinate(X) and latitudinal coordinate(Y). I donot select method
>>> for
>>> AIC optimisation.
>>>
>>> So I donot know where is wrong.
>>>
>>> Thank you very much for your great helps.
>>>
>>>
>>>
>>>
>>>
>>>
>>> 在2010-05-13 00:07:23，"Roger Bivand" <Roger.Bivand at nhh.no> 写道：
>>>> On Wed, 12 May 2010, Roger Bivand wrote:
>>>>
>>>>> On Wed, 12 May 2010, huangykiz wrote:
>>>>>
>>>>>> Hi, Is "adapt=TRUE"(spgwr) not the same as "adaptive Spatial
>>>>>> Kernel"(SAM)?The result of "adaptive Spatial Kernel" may be
>>>>>> better than
>>>>>> fixed bandwidth. If I want to ues "adaptive Spatial Kernel" in
>>>>>> spgwr, how
>>>>>> to write the code?
>>>>>
>>>>> READ THE HELP PAGES!
>>>>>
>>>>> adaptive_proportion <- gwr.sel(...)
>>>>>
>>>>> result <- gwr(..., adapt=adaptive_proportion; ...)
>>>>>
>>>>> exactly as on the example om the help page:
>>>>>
>>>>> 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
>>>>>
>>>>> Clear?
>>>>
>>>> I have now compared the same data in R/spgwr and SAM for
>>>> effective number
>>>> of parameters, sigma, and your questionable R^2, and they agree
>>>> adequately
>>>> when the kernel and the bandwidth are the same. Having the
>>>> algorithm
>>>> choose the bandwidth does obscure what is going on. You should
>>>> use SAM if
>>>> you prefer GUI and not needing to know how things work, and
>>>> remember that
>>>> GWR is a very doubtful approach for anything beyond exploring
>>>> non-stationarity, its original motivation.
>>>>
>>>>>
>>>>>>
>>>>>> Thanks a lot.
>>>>>>
>>>>>> Cheers.
>>>>>>
>>>>>>
>>>>>>> Hi,
>>>>>>> I think that I use the same bandwidth and kernel. In SAM, I
>>>>>>> use "spatial
>>>>>>> Weighting Function"=gaussian, adaptive Spatial Kernel, and
>>>>>>> compute
>>>>>>> Geographical Distances based on longitudinal coordinate(X) and
>>>>>>> latitudinal
>>>>>>> coordinate(Y). In spgwr, gweight is gwr.Gauss and adapt is TRUE.
>>>>>>>
>>>>>>> For example, this is my code:
>>>>>>
>>>>>>> PET.bw <- gwr.sel(SPECIES_RI ~ PET, data=variables,
>>>>>>> coords=cbind(variables$LONGX, variables$LATY),adapt=TRUE)
>>>>>>
>>>>>>> PET.gauss <- gwr(SPECIES_RI ~ PET, data=variables,
>>>>>>> coords=cbind(variables$LONGX, variables$LATY), bandwidth=PET.bw,
>>>>>>> gweight=gwr.Gauss,adapt=TRUE,hatmatrix=TRUE)
>>>>>>
>>>>>> So where do you pass PET.bw to the gwr() function? adapt=TRUE
>>>>>> will treat
>>>>>> the adaptive proportion as 1, so include all data points. If
>>>>>> you want to
>>>>>> compare, use a fixed bandwidth in both, with no CV selection.
>>>>>> Then you
>>>>>> compare like with like.
>>>>>>
>>>>>> Note that your messages are *not* reaching the list, they must
>>>>>> be sent to:
>>>>>>
>>>>>> r-sig-geo at stat.math.ethz.ch, not
>>>>>>
>>>>>> r-sig-geo-request at stat.math.ethz.ch
>>>>>>
>>>>>> You are not thinking carefully and are rushing into things and
>>>>>> drawing
>>>>>> wrong conclusions.
>>>>>>
>>>>>>>
>>>>>>> Thanks a lot.
>>>>>>>
>>>>>>> Cheers.
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> 在2010-05-12 20:28:47，"Roger Bivand" <Roger.Bivand at nhh
>>>>>>> .no> 写道：
>>>>>>>> On Wed, 12 May 2010, huangykiz wrote:
>>>>>>>>
>>>>>>>>> Hi,
>>>>>>>>> One of SAM author ("José Alexandre Felizola Diniz
>>>>>>>>> Filho"<diniz at icb.ufg.br>) say that they also base on GWR3 (the
>>>>>>>>> Fotherigham book) and the data used within each kernel may
>>>>>>>>> be some
>>>>>>>>> slight differences
>>>>>>>>
>>>>>>>> Naturally, if you are not using exactly the same kernel and
>>>>>>>> bandwidth,
>>>>>>>> you should not be surprised by differences in values. Please
>>>>>>>> make sure
>>>>>>>> that the bandwidth and kernel are the same and try again.
>>>>>>>>
>>>>>>>> Roger
>>>>>>>>
>>>>>>>>> Cheers
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> 在2010-05-12 20:28:47，"Roger Bivand" <Roger.Bivand at nhh.
>>>>>> no> 写道：
>>>>>>> On Wed, 12 May 2010, huangykiz wrote:
>>>>>>>
>>>>>>>> Hi,
>>>>>>>> One of SAM author ("José Alexandre Felizola Diniz
>>>>>>>> Filho"<diniz at icb.ufg.br>) say that they also base on GWR3 (the
>>>>>>>> Fotherigham book) and the data used within each kernel may
>>>>>>>> be some
>>>>>>>> slight differences
>>>>>>>
>>>>>>> Naturally, if you are not using exactly the same kernel and
>>>>>>> bandwidth, you
>>>>>>> should not be surprised by differences in values. Please make
>>>>>>> sure that
>>>>>>> the bandwidth and kernel are the same and try again.
>>>>>>>
>>>>>>> Roger
>>>>>>>
>>>>>>>> Cheers.
>>>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> 在2010-05-12 15:27:58，"Roger Bivand" <Roger.Bivand at nhh.
>>>>>> no> 写道：
>>>>>>> On Wed, 12 May 2010, huangykiz wrote:
>>>>>>>
>>>>>>>> Hi,
>>>>>>>>
>>>>>>>> I am sorry I donot know how to install module spgwr from
>>>>>>>> sourceforge (I
>>>>>>>> can find it on the web
>>>>>>>> http://r-spatial.cvs.sourceforge.net/viewvc/r-spatial/spgwr/R/gwr.R?view=log
>>>>>>>> ).
>>>>>>>> So I use the code sketch to calculate quasi-global R2. The
>>>>>>>> results are
>>>>>>>> different between SAM and spgwr(Attached are the results ). The
>>>>>>>> quasi-global R2 in R is 0.4515894, but in SAM is 0.696.
>>>>>>>> This is my code:
>>>>>>>>
>>>>>>>> library(spgwr)
>>>>>>>> Environmental_variables<-read.csv
>>>>>>>> ("Environmental_variables100.csv",header=TRUE)
>>>>>>>> attach(Environmental_variables)
>>>>>>>> region_PET.bw <- gwr.sel(SPECIES_RI ~ PET,
>>>>>>>> data=Environmental_variables,
>>>>>>>> coords=cbind(Environmental_variables$LONGX,
>>>>>>>> Environmental_variables$LATY),adapt=TRUE)
>>>>>>>> region_PET.gauss <- gwr(SPECIES_RI ~ PET,
>>>>>>>> data=Environmental_variables,
>>>>>>>> coords=cbind(Environmental_variables$LONGX,
>>>>>>>> Environmental_variables$LATY), bandwidth=region_PET.bw,
>>>>>>>> gweight=gwr.Gauss,adapt=TRUE,hatmatrix=TRUE)
>>>>>>>> names(region_PET.gauss$SDF)
>>>>>>>> region_PET.gauss$SDF$localR2
>>>>>>>> 1 -
>>>>>>>> (region_PET.gauss$results$rss/crossprod(scale
>>>>>>>> (Environmental_variables$SPECIES_RI,
>>>>>>>> scale=FALSE)))
>>>>>>>>
>>>>>>>> Thank you very much.
>>>>>>>
>>>>>>> SAM is closed source - ask them how they compute it. For
>>>>>>> spgwr, the code
>>>>>>> is provided, so you can read it for yourself. For the record,
>>>>>>> the current
>>>>>>> gwr() code in spgwr gives the same value as GWR3, which is
>>>>>>> also closed
>>>>>>> source, and where the Effective number of parameters (model:
>>>>>>> traceS),
>>>>>>> Sigma, and Residual sum of squares also agree. I suppose SAM
>>>>>>> has a
>>>>>>> different understanding of GWR internals than the authors of
>>>>>>> the GWR book.
>>>>>>>
>>>>>>> Once again:
>>>>>>>
>>>>>>> Please *do* write to the R-sig-geo list rather than to me
>>>>>>> directly -
>>>>>>> others can answer your question as well, perhaps better, and
>>>>>>> in a more
>>>>>>> timely way than I can. In addition, threads in the list can be
>>>>>>> searched in
>>>>>>> the archives, so others can avoid the same problem later.
>>>>>>>
>>>>>>> Please summarise to the list if this resolves the problem.
>>>>>>>
>>>>>>> Roger
>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> 在2010-05-12 01:16:18，"Roger Bivand" <Roger.Bivand at nh
>>>>>>>> h.no> 写道：
>>>>>>>>> On Wed, 12 May 2010, huangykiz wrote:
>>>>>>>>>
>>>>>>>>>> Hi, I just need one for global, not *each* fit point. In
>>>>>>>>>> this case, how
>>>>>>>>>> can I select or do? Why in other software such as SAM
>>>>>>>>>> (Spatial Analysis
>>>>>>>>>> in Macroecology) just gives one R2?
>>>>>>>>>
>>>>>>>>> If you believe theirs, good luck! The authors of the GWR
>>>>>>>>> book have local
>>>>>>>>> R^2 values in GWR3 and formulae that are wrong by their own
>>>>>>>>> admission in
>>>>>>>>> private emails. The localR2 now agrees with the as-yet
>>>>>>>>> unreleased GWR4
>>>>>>>>> from the GWR authors. How SAM can be "better", I don't know.
>>>>>>>>> What you
>>>>>>>>> are suggesting is that the model fitted with fit points at
>>>>>>>>> data points
>>>>>>>>> (but not at other fit points) might have a "quasi-global"
>>>>>>>>> R^2, based on
>>>>>>>>> the RSS of the pooled fit. For the columbus case, that might
>>>>>>>>> be:
>>>>>>>>>
>>>>>>>>> 1 - (col.gauss$results$rss/crossprod(scale(columbus$crime,
>>>>>>>>> scale=FALSE)))
>>>>>>>>>
>>>>>>>>> but I don't know whether this is in any way correct. I've
>>>>>>>>> added it as:
>>>>>>>>>
>>>>>>>>> Quasi-global R2:
>>>>>>>>>
>>>>>>>>> to the print output of a GWR model fitted with a hatmatrix,
>>>>>>>>> and have
>>>>>>>>> committed it to sourceforge, project r-spatial, module
>>>>>>>>> spgwr. Arguably,
>>>>>>>>> it ought to be adjusted by the ratio of degrees of freedom,
>>>>>>>>> but I don't
>>>>>>>>> trust the DF either. Could you please check out spgwr from
>>>>>>>>> sourceforge
>>>>>>>>> ,install it from source, and confirm that the "quasi-global
>>>>>>>>> R2" does the
>>>>>>>>> same as SAM, or use the code sketch above to do the same,
>>>>>>>>> and report
>>>>>>>>> back?
>>>>>>>>>
>>>>>>>>> Roger
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Thanks a lot.
>>>>>>>>>>
>>>>>>>>>> Cheers,
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> 在2010-05-11 23:59:44，"Roger Bivand" <Roger.Bivand@
>>>>>>>>>> nhh.no> 写道：
>>>>>>>>>>> On Tue, 11 May 2010, huangykiz wrote:
>>>>>>>>>>>
>>>>>>>>>>>> Hi,
>>>>>>>>>>>>
>>>>>>>>>>>> There are 49 localR2 in the results. Which one do I
>>>>>>>>>>>> need? The code
>>>>>>>>>>>> "look for localR2:" cannot run.
>>>>>>>>>>>
>>>>>>>>>>> Well, how many do you want? There is one for each fit
>>>>>>>>>>> point, they are
>>>>>>>>>>> *local* R2. Please do try to grasp what GWR does - it fits
>>>>>>>>>>> one moddel
>>>>>>>>>>> for *each* fit point.
>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Thans a lot
>>>>>>>>>>>>
>>>>>>>>>>>> Cheers.
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> 在2010-05-11 22:33:59，"Roger Bivand" <Roger.Biv
>>>>>>>>>>>> and at nhh.no> 写道：
>>>>>>>>>>>>> On Tue, 11 May 2010, huangykiz wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> Hi, OK. But I need it for compariation. In what some
>>>>>>>>>>>>>> contexts to
>>>>>>>>>>>>>> get it? May you tell me how to get it?
>>>>>>>>>>>>>
>>>>>>>>>>>>> library(spgwr)
>>>>>>>>>>>>> data(columbus)
>>>>>>>>>>>>> col.bw <- gwr.sel(crime ~ income + housing, data=columbus,
>>>>>>>>>>>>> coords=cbind(columbus$x, columbus$y))
>>>>>>>>>>>>> col.gauss <- gwr(crime ~ income + housing, data=columbus,
>>>>>>>>>>>>> coords=cbind(columbus$x, columbus$y), bandwidth=col.bw,
>>>>>>>>>>>>> hatmatrix=TRUE)
>>>>>>>>>>>>> names(col.gauss$SDF)
>>>>>>>>>>>>>
>>>>>>>>>>>>> look for localR2:
>>>>>>>>>>>>>
>>>>>>>>>>>>> col.gauss$SDF$localR2
>>>>>>>>>>>>>
>>>>>>>>>>>>> But do not rely on it or use it for anything at all!
>>>>>>>>>>>>> Like all GWR,
>>>>>>>>>>>>> it is most unreliable!
>>>>>>>>>>>>>
>>>>>>>>>>>>> Roger Bivand
>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Thank you very much for your great helps
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Best regards.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> 在2010-05-11 18:28:44，"Roger Bivand" <Roger
>>>>>>>>>>>>>> .Bivand at nhh.no> 写道：
>>>>>>>>>>>>>>> On Tue, 11 May 2010, huangykiz wrote:
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Dear professor Bivand,
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> I am a strudent. I recently use GWR(Geographically
>>>>>>>>>>>>>>>> weighted
>>>>>>>>>>>>>>>> regression) model. May I ask you a question? There is
>>>>>>>>>>>>>>>> not
>>>>>>>>>>>>>>>> Coefficient of Determination in the results of GWR.
>>>>>>>>>>>>>>>> How can I get
>>>>>>>>>>>>>>>> it? What is the programs to get it?
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Please address questions like this to the R-sig-geo
>>>>>>>>>>>>>>> list rather
>>>>>>>>>>>>>>> than to me directly in future.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> The local R2 values are available in some contexts
>>>>>>>>>>>>>>> when running
>>>>>>>>>>>>>>> gwr(), but are not well defined (neither in the GWR
>>>>>>>>>>>>>>> book nor in
>>>>>>>>>>>>>>> implementations). I advise against their use - they
>>>>>>>>>>>>>>> are most
>>>>>>>>>>>>>>> probably meaningless.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Hope this helps,
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Roger Bivand
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Thank you very much for your any helps.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Best regards.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Yong Huang
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> --
>>>>>>>>>>>>>>> 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
>>>>>>>>>>>
>>>>>>>>>>> --
>>>>>>>>>>> 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
>>>>>>>
>>>>>>> --
>>>>>>> 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
>>
>> --
>> 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
>
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