[R-sig-Geo] weighted spatial autoregression

[Roger Bivand]

Sam,

On Wed, 29 Aug 2007, Sam Field wrote:

> Roger,
>
> One possibility in this limited case might be to replicate the aggregate
> level cases based on their respective weights (since they are integers,
> i.e. within unit sample sizes), then run a spatial lag model.  This
> would be equivalent to recreating the individual level data from the
> aggregate data (excluding measures that vary within the aggregate
> units).  This would obviously inflate your sample size and one would
> have to correct for this somehow in the variance covariance matrix of
> the parameters estimates.
>
> You would have to do the same for your nb object as well of course.  I
> have looked into this by creating a list of neighbor ids from the
> original nb object, but nb2listw() requires an nb object not a list so I
> am stuck.
>

You could fake it with nb2blocknb, but that was not written for this case, 
but for the case when the individual level variables were observed, but 
that there was no address or coordinates, just a postal code. Here the LHS 
and RHS would be replicated, which doesn't seem desirable.

> The other problem would be that you would end up with a potentially
> large data set. In my case, 13,000 - maybe more then spautolm() could
> handle?  Maybe this whole idea if flawed.
>
>
> Thanks again for your input! The results change quite a bit with the
> weighted SAR models.

One interesting conclusion that I've reached is that while the spdep code 
in spautolm() replicates Waller and Gotway for unweighted and weighted SAR 
and CAR, S-Plus SpatialStats fails on the weighted CAR. The reason seems 
to be that W+G did the same as spautolm() (in SAS?) - find the spatial 
autoregressive coefficient first (optimise in one dimension), then use GLS 
to find the regression coefficients. But S+ seems to try to optimise all 
the coefficients at once, and gets bitten by the fact that
(I - \rho W) %*% diag(wts) in their case is not symmetric (W has to be 
symmetric, and the wts have to "balance" - see Cressie etc. Now I'm not 
sure that S+ is right here. If not, then the lag model can be given 
weights too, by simply passing them to the auxilliary regressions used to 
set up the framework for optimisation. The analytical covariance matrix of 
the coefficients remains a problem, though. We'd need to use some other 
mechanism to get there for the eigen method, though the LR tests used for 
sparse methods would be, I think, OK. I've also been playing with sampling 
from a fitted model, to generate synthetic "standard errors", like 
mcmcsamp() in lme4, but I don't know if it is sensible, or how well it 
would scale to many observations.

So I am thinking about how lagsarlm() could get weights, but it won't 
happen too fast, maybe.

Best wishes,

Roger

>
>
> Sam
>
>
>
> Roger Bivand wrote:
>> On Tue, 21 Aug 2007, Sam Field wrote:
>>
>>
>>> Thanks Roger!
>>>
>>> Sorry about omitting the subject line.  I have been working with errorsarlm() -
>>> did not know about spautolm().  Do you know if there is something analogous
>>> possible in the case of the spatial lag model,
>>>
>>> Y = pWY + XB + e ?
>>>
>>
>> I have not looked at it, but because it is a wierd animal, I don't think
>> it will be too easy to provide a theoretical foundation for it. The
>> heteroskedasticity is in the error term, but the autoregressive part
>> isn't. I don't think there are any examples anywhere, either.
>>
>> It ought to be possible, though.
>>
>> Roger
>>
>>
>>> I was going to start looking into it.
>>>
>>> thanks!
>>>
>>>
>>> Sam
>>>
>>>
>>>
>>>
>>> Quoting Roger Bivand <Roger.Bivand at nhh.no>:
>>>
>>>
>>>> On Tue, 21 Aug 2007, Sam Field wrote:
>>>>
>>>>
>>>>> List,
>>>>>
>>>>> I am looking for ways of estimating spatial autoregression models that
>>>>>
>>>> adjust
>>>>
>>>>> for a known source of heteroskedaticity and the Waller and Gotway (2004)
>>>>>
>>>> text
>>>>
>>>>> outline how this can be done in the case of the SAR model.  If I work at
>>>>>
>>>> it, I
>>>>
>>>>> think I can implement this myself in R, but I wanted to see if anybody else
>>>>>
>>>> had
>>>>
>>>>> done it. It seems like a pretty straightforward generalization of the SAR
>>>>>
>>>> model
>>>>
>>>>> and would make a very helpful addition to the spatial regression tools in
>>>>> spdep - especially given the effects of heteroskedaticity on the
>>>>>
>>>> consistency of
>>>>
>>>>> the SAR parameters!
>>>>>
>>>> ?spautolm
>>>>
>>>> The examples reproduce the results in Waller & Gotway, perhaps apart from
>>>> a flattish function to optimise in the weighted CAR case. spautolm() now
>>>> provides weighted or unweighted SAR, CAR, and SMA. Sparse matrix methods
>>>> are available for SAR and CAR, SAR when spatial weights are symmetric or
>>>> similar to symmetric (CAR weights have to be symmetric).
>>>>
>>>> Roger
>>>>
>>>>
>>>>> Sam
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>> --
>>>> 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