Hi,
Set In gwr.sel() or gwr()?. How to write the code about set "adpt" parameter to
0.1? May you tell more detail.
Thank you very much.
Cheers,
ÔÚ2010-05-13 20:29:14£¬"Danlin Yu" <y...@mail.montclair.edu> дµÀ£º
>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: y...@mail.montclair.edu
>
>ÔÚ May 13, 2010£¬8:03 AM£¬huangykiz <huangy...@163.com> дµ½£º
>
>> Hi,
>> If I want to chang the adaptive Spatial Kernel = 10% neighbors in
>> gwr()? How to chang it?
>>
>> Thank you very much.
>>
>> Cheers.
>>
>>
>>
>>
>> ÔÚ2010-05-13 17:43:06£¬"Roger Bivand" <roger.biv...@nhh.no> дµÀ£º
>>> 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.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> ÔÚ2010-05-13 00:07:23£¬"Roger Bivand" <roger.biv...@nhh.no> дµÀ£º
>>>>> 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@stat.math.ethz.ch, not
>>>>>>>
>>>>>>> r-sig-geo-requ...@stat.math.ethz.ch
>>>>>>>
>>>>>>> You are not thinking carefully and are rushing into things and
>>>>>>> drawing
>>>>>>> wrong conclusions.
>>>>>>>
>>>>>>>>
>>>>>>>> Thanks a lot.
>>>>>>>>
>>>>>>>> Cheers.
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> ÔÚ2010-05-12 20:28:47£¬"Roger Bivand" <roger.biv...@nhh
>>>>>>>> .no> дµÀ£º
>>>>>>>>> On Wed, 12 May 2010, huangykiz wrote:
>>>>>>>>>
>>>>>>>>>> Hi,
>>>>>>>>>> One of SAM author ("Jos¨¦ Alexandre Felizola Diniz
>>>>>>>>>> Filho"<di...@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
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> ÔÚ2010-05-12 20:28:47£¬"Roger Bivand" <roger.biv...@nhh.
>>>>>>> no> дµÀ£º
>>>>>>>> On Wed, 12 May 2010, huangykiz wrote:
>>>>>>>>
>>>>>>>>> Hi,
>>>>>>>>> One of SAM author ("Jos¨¦ Alexandre Felizola Diniz
>>>>>>>>> Filho"<di...@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.
>>>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> ÔÚ2010-05-12 15:27:58£¬"Roger Bivand" <roger.biv...@nhh.
>>>>>>> no> дµÀ£º
>>>>>>>> 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
>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> ÔÚ2010-05-12 01:16:18£¬"Roger Bivand" <roger.biv...@nh
>>>>>>>>> h.no> дµÀ£º
>>>>>>>>>> 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,
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> ÔÚ2010-05-11 23:59:44£¬"Roger Bivand" <Roger.Bivand@
>>>>>>>>>>> nhh.no> дµÀ£º
>>>>>>>>>>>> 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.
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> ÔÚ2010-05-11 22:33:59£¬"Roger Bivand" <Roger.Biv
>>>>>>>>>>>>> a...@nhh.no> дµÀ£º
>>>>>>>>>>>>>> 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.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> ÔÚ2010-05-11 18:28:44£¬"Roger Bivand" <Roger
>>>>>>>>>>>>>>> .biv...@nhh.no> дµÀ£º
>>>>>>>>>>>>>>>> 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.biv...@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.biv...@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.biv...@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.biv...@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.biv...@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.biv...@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.biv...@nhh.no
>>
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