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AI-GEOSTATS: Re: so-called pseudo-covariances

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  • Donald E. Myers
    One should be careful about referring to A- g(h) as a pseudo covariance, this might be taken to imply that it has some properties like a covariance such as
    Message 1 of 1 , May 22, 2001
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      One should be careful about referring to A- g(h) as a pseudo covariance, this might be taken to imply that it has some properties like a covariance such as positive definiteness. Pierre's comment is relevant to
      the question of solving the ordinary kriging equations rather than to any covariance like properties of A-g(h). If one visualizes the coefficient matrix of the ordinary kriging equations in 3-D (rows and columns
      are in the plane, entries in the matrix are the heights) then using the variogram the initial form has zeros down the diagonal, i.e., the shape is much like a valley. Now visualize the matrix in "solved" form, all
      diagonal entries will be 1's and all off diagonal entries are zeros. Hence in an intuitive sense the initial form is quite different from the desired end result. In constrast if the ordinary kriging equations are
      written in terms of the covariance function then the shape is more like a ridge down the diagonal, this change corresponds to several simple algebraic operations. Multiply both sides of all equations except the last
      equation (the unbiasedness condition, "nonbias" is not an English word, it is French), then add the constant A to both sides of all equations except the last. Because of the unbiasedness condition, i.e., the last
      equation, "A" on the left side can be written as A times the sum of the kriging weights. Then a slight re-grouping makes the "-g(h)" terms all appear to be of the form A-g(h). This does not ensure that the system
      has a unique solution, it is the conditional negative definiteness of the variogram. If g(h) is conditionally negative definite then -g(h) is conditionally positive definite, either property will ensure that the
      coefficient matrix is invertible. If g(h) has a sill A then A-g(h) is positive definite but in general A-g(h) is not positive definite (because in particular it is unbounded). This transition is purely a scheme to
      improve the efficiency of the equation solver and not one to obtain covariance like properties for A-g(h). You can't use the "pseudocovariance" in the Simple Kriging equations, the coefficient matrix will likely
      not be invertible.

      Perhaps someone is using this term to denote something other than A-g(h), otherwise there would be no particular reason to have such a function in S Plus

      DEMyers
      http://www.u.arizona.edu/~donaldm



      Pierre Goovaerts wrote:

      > Hi Celia,
      >
      > To compute the pseudo-cross covariance
      > for an unbounded model, you just substract the
      > semivariogram model g(h) from any positive value A,
      > such that A-g(h) is non-negative for all h.
      > The non-bias condition of the ordinary kriging system
      > allows the constant A to cancel out from the system
      > of equations, hence the choice of the value A has
      > no impact on the computation of kriging weights and
      > kriging variance.
      >
      > Cheers,
      >
      > Pierre
      >
      > <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
      >
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      > | \ / | Pierre Goovaerts
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      > <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
      >
      > On Tue, 22 May 2001, celia bulit wrote:
      >
      > > Hi all:
      > >
      > > does anyone know how to calculate the "pseudo covariance function" for an
      > > unbounded semivariogram?
      > > Does it exist the funtion for S-Plus?
      > >
      > > thanks in advance
      > > Celia
      > > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
      > >
      > > Celia Bulit
      > > cbulit@...
      > > Depto El Hombre y su Ambiente
      > > UAM-Xochimilco
      > > Calzada del Hueso 1100
      > > 04960 México D.F.
      > >
      > > tel. (52) 5483-7360
      > >
      > >
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