- Melanie Wall wrote:

> I have a question about the parameter space of the rho

Dear Melanie,

> parameter in a SAR model.

>

> Consider the following two dispersion matrices Sigma for Spatial

> Regression Models. Let W be a symmetric neighbor matrix.

>

> CAR: Sigma = (I - rho W)^{-1}

> SAR: Sigma = [t(I - rho W) %*% (I - rho W)]^{-1}

>

> Everywhere I look, people discuss the restrictions on rho so that (I - rho

> W) is positive definite. It is very straightforward to show the

> conditions on rho that satisfy this condition as they are related to the

> eigenvalues of W. Obviously for the CAR model, we should be restricting

> ourself to (I -rho W) positive definite, but, in the SAR model there is no

> need to restrict (I - rho W) to be positive definite, instead all we need

> is invertibility (i.e. non singular). The problem with this is, it is not

> much of a restriction because all it requires is that rho does not = the

> inverse of any of the eigenvalues of W. Thus there is really no upper

> bound on what rho can be to ensure (I- rho W) to be invertible

> (but not positive definite). Has anyone ever seen any discussion of

> this?

>

> I have already run across two examples where if I use the CAR model in S+

> Spatial Stat's slm() function, I get a rho value that is less than

> 1/(max(eigenvalue of W) (as it should be to ensure (I - rho W) positive

> definite). But then, when I run a SAR model using the exact same data I

> get a rho value outside the range that would ensure positive definiteness.

> That is (I - rho W) is invertible but not positive definite. It seems to

> me that it is hard to interpret an estimated rho if there is no upper

> bound on what it could be.

>

> Does anyone have any comments about this topic.

>

> Thanks

> MW

>

> -----------------------

> Melanie Wall, Ph.D.

> Division of Biostatistics

> A460 Mayo Building Box 303

> 420 Delaware Street S.E.

> Minneapolis, MN 55455

> (612)625-2138

> melanie@...

Usually, the underlying process Z(x,y) admitting a SAR representation is assumed

to be

covariance stationary under translation of (x,y). This can be shown to be the

case if a

certain polynomial in complex variables is non-zero on the unit complex circle.

This imposes conditions on the admissible values for rho. For instance, in 2D,

if the same rho

is used in each direction, then |rho|<1/4.

Two good references are Whittle, P. (1954), ``On stationary processes in the

plane'', Biometrika,

41, 434-449, as well as Ali, M.M. (1979), ``Analysis of stationary

spatial-temporal processes:

Estimation and prediction'', Biometrika, 66, 515-518.

You might also take a look at the extended abstract of our talk:

de Luna, X., Genton, M. G., (1999): "Indirect inference for spatio-temporal

autoregression models",

Proceedings of the Workshop Spatial-temporal modeling and its application,

Leeds, UK, 61-64

(edited by K.V. Mardia, R.G. Aykroyd and I.L. Dryden).

It consists of a new simulation-based estimation procedure for the more general

STAR

models, allowing for different rho's in each space direction, and time, as well

as for robust estimates.

We are currently writing the full paper about this topic.

Regards

Marc Genton

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

_/ Marc G. Genton

/_/ _/ / / _______/ Department of Mathematics, 2-390

/ _/ _/ / / / 77 Massachusetts Avenue

/ _/ / / / Cambridge, MA 02139-4307

/ / / /

/ / / / E-mail: genton@...

_/ _/ _/ _/ _/ _/ _/ http://www-math.mit.edu/~genton

Phone: (617) 253-4390

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Fax: (617) 253-4358

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

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DO NOT SEND Subscribe/Unsubscribe requests to the list! - Thanks for the comments in response to my question about the SAR model.

One comment was about restricting rho such that the the underlying process

Z(x,y) is covariance stationary. This brings up an interesting issue

because, I think, in general it is not possible. In general, the

covariance matrix (Sigma) resulting from a SAR specification with neighbor

matrix W

Sigma = [t(I - rho W) %*% (I - rho W)]^{-1}

is NOT going to be covariance stationary no matter how we restrict

rho. Isn't it true that W has to have a very special form in order to be

able to put restrictions on rho that ensure Sigma is covariance

stationary. I believe W must be symmetric and have equal neighbors for

each location for there to be any hope of Sigma being covariance

stationary. Haining (1990 p. 82) talks about how Sigma is not usually

representing a covariance stationary process.

The question I asked in my last e-mail was about the restriction on rho

needed to make Sigma positive definite. But, when I asked that, I was not

aware of this seemingly bigger problem of lack of second order

stationarity in Sigma that occurs for most W. So now I have a new

question, Should I care? Is this a problem? This sort of model has

obviously been used for a long time, do people just ignore the fact that,

in general, Sigma is not going to represent a stationary process?

I have just recently started to look at this sort of model in the spatial

setting and it is clearly not a straightforward extension of what I know

about time series models.

Thanks for any replies.

Melanie

-----------------------

Melanie Wall

Division of Biostatistics

A460 Mayo Building Box 303

420 Delaware Street S.E.

Minneapolis, MN 55455

(612)625-2138

melanie@...>

--

> Dear Melanie,

>

> Usually, the underlying process Z(x,y) admitting a SAR representation is assumed

> to be

> covariance stationary under translation of (x,y). This can be shown to be the

> case if a

> certain polynomial in complex variables is non-zero on the unit complex circle.

> This imposes conditions on the admissible values for rho. For instance, in 2D,

> if the same rho

> is used in each direction, then |rho|<1/4.

> Two good references are Whittle, P. (1954), ``On stationary processes in the

> plane'', Biometrika,

> 41, 434-449, as well as Ali, M.M. (1979), ``Analysis of stationary

> spatial-temporal processes:

> Estimation and prediction'', Biometrika, 66, 515-518.

>

> You might also take a look at the extended abstract of our talk:

> de Luna, X., Genton, M. G., (1999): "Indirect inference for spatio-temporal

> autoregression models",

> Proceedings of the Workshop Spatial-temporal modeling and its application,

> Leeds, UK, 61-64

> (edited by K.V. Mardia, R.G. Aykroyd and I.L. Dryden).

> It consists of a new simulation-based estimation procedure for the more general

> STAR

> models, allowing for different rho's in each space direction, and time, as well

> as for robust estimates.

> We are currently writing the full paper about this topic.

>

> Regards

>

> Marc Genton

>

>

>

>

> $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

>

> _/ Marc G. Genton

>

> /_/ _/ / / _______/ Department of Mathematics, 2-390

> / _/ _/ / / / 77 Massachusetts Avenue

> / _/ / / / Cambridge, MA 02139-4307

> / / / /

> / / / / E-mail: genton@...

> _/ _/ _/ _/ _/ _/ _/ http://www-math.mit.edu/~genton

> Phone: (617) 253-4390

> MASSACHUSETTS INSTITUTE OF TECHNOLOGY Fax: (617) 253-4358

>

> $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

>

>

>

>

*To post a message to the list, send it to ai-geostats@....

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