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214 Modern Spatiotemporal Geostatistics — Chapter 10
moments, f g shows a deviation from the Gaussian shape and Equation 5.6 can
be decomposed as follows
where the Gaussian operator 9£,o [xmapj PmaP] nas the form of Equation 10.1
and S 9£ [Xmap> Pmap] 's tne so-called non-Gaussian perturbation. In the case,
e.g., that the £-base includes fourth-order moments we find that
where the coefficients /z$ are calculated from the BME equations. The impli-
cation of Equation 10.42 is that the prior pdf of BME analysis is the product
of a Gaussian pdf and a non-Gaussian perturbation pdf. In order to derive the
shape of f g, one has to calculate the Lagrange multipliers involved in Equations
10.1 and 10.43. As usual, Equation 10.43 should be substituted into the set of
BME equations (Eq. 5.9, p. 107) which are solved for the multipliers. These
multipliers are then inserted back into Equation 10.42 to obtain the desired
shape of the pdf fg. In practice, a variety of computational techniques may be
used to solve the BME equations. Hristopulos and Christakos (2000) discuss
the implementation of modern Monte Carlo techniques in cases where the pdf
fg shows large deviations from the Gaussian shape.
In light of Equation 10.42, some interesting analytical expressions can be
derived. For example, the expectation of a functional L [ - ] is given by ,
where anc' tne (') denotes expectation with
respect to the Gaussian pdf. Equation 10.44 is a general expression that can
provide non-Gaussian space/time moments in terms of Gaussian expectations.
In Christakos et al. (1999b) and Hristopulos and Christakos (2000), space/time
moment expressions in terms of low-order and diagrammatic perturbations are
investigated. We illustrate the implementation of Equation 10.44 by means of
the following example.
EXAMPLE 10.4: The (non-centered) covariance between two points p i and p,j
can be expressed with the help of Equation 10.44 as follows
In the case of a zero-mean homogeneous/stationary random field with <59£
given by Equation 10.43, Equation 10.44 leads to