[R-sig-Geo] KNN spatial weights matrix: asymmetric vs symmetric for spatial autoregressive models

[Roger Bivand]

On Thu, 16 Dec 2010, Dustin Duncan wrote:

> hi all,
>
> i have one question ***does it matter if the KNN spatial weights matrix is
> asymmetric or symmetric for linear spatial autoregressive models? *** i am
> getting slightly different findings depending on the weights matrix
> specification for the Lagrange multiplier diagnostic for spatial dependence.
> i am using Bivand's spdep package for these analysis. i've read that GeoDA
> can't fit KNN spatial models due to the asymmetry (for now), but that using
> KNN is legitimate for these spatial autoregresive models. however, i could
> use the asymmetric version or the symmetric one (though making KNN symmetric
> sounds a bit strange to me). any insight is greatly appreciated!!

The approximation to the Jacobian used in GeoDa requires that W be 
symmetric (or similar to symmetric, Ord, 1975). The MC approximation used 
in the Matlab toolbox does not. In spdep, you can use method="eigen" for 
smaller data sets, "LU" for exact Jacobian larger data sets, or "MC", with 
intrinsically asymmetric W. However, note that the differences between a 
symmetricised KNN W and the original one may not be large, as in SAR 
(simultaneous rather than CAR conditional) SSE terms, you do get W'W 
appearing, though its influence will vary. Try checking to see whether 
results vary enough to change your inferences:

library(spdep)
example(columbus)
knn4 <- knn2nb(knearneigh(coordinates(columbus), k=6))
knn4
knn4s <- make.sym.nb(knn4)
lw <- nb2listw(knn4)
lws <- nb2listw(knn4s)
summary(lagsarlm(CRIME ~ HOVAL + INC, data=columbus, listw=lw))
summary(lagsarlm(CRIME ~ HOVAL + INC, data=columbus, listw=lws))

So if you have to make the KNN weights symmetric to use - say - GeoDa, it 
may not matter that much in practice.

Hope this helps,

Roger

>
> thanks and happy holidays, dustin
>
>

-- 
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