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  • Marc Georges Genton
    ... 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
    Message 1 of 5 , Nov 12, 1999
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      Melanie Wall wrote:

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

      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

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



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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 2 of 5 , Nov 19, 1999
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        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
        >
        >
        >
        >
        > $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
        >
        > _/ Marc G. Genton
        >
        > /_/ _/ / / _______/ Department of Mathematics, 2-390
        > / _/ _/ / / / 77 Massachusetts Avenue
        > / _/ / / / Cambridge, MA 02139-4307
        > / / / /
        > / / / / E-mail: genton@...
        > _/ _/ _/ _/ _/ _/ _/ http://www-math.mit.edu/~genton
        > Phone: (617) 253-4390
        > MASSACHUSETTS INSTITUTE OF TECHNOLOGY Fax: (617) 253-4358
        >
        > $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
        >
        >
        >
        >


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