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GEOSTATS: Re: restrictions for rho in a SAR model

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  • Melanie Wall
    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
    Message 1 of 5 , Nov 19, 1999
      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 Wall
      Division of Biostatistics
      A460 Mayo Building Box 303
      420 Delaware Street S.E.
      Minneapolis, MN 55455
      > 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
      > $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
      > _/ Marc G. Genton
      > /_/ _/ / / _______/ Department of Mathematics, 2-390
      > / _/ _/ / / / 77 Massachusetts Avenue
      > / _/ / / / Cambridge, MA 02139-4307
      > / / / /
      > / / / / E-mail: genton@...
      > _/ _/ _/ _/ _/ _/ _/ http://www-math.mit.edu/~genton
      > Phone: (617) 253-4390
      > $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

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