Page 219 - Modern Spatiotemporal Geostatistics

200 Modern Spatiotemporal Geostatistics — Chapter 10

for j = 1, 2, . . . , m, k. Also ,

From Equation 10.12 and well-known properties of multivariate Gaussian laws
it follows that

In light of Equations 10.11 and 10.13, Equation 10.9 reduces to Equation 10.2,
and Equation 10.5 leads to Equation 10.1.

The following corollary is a straightforward consequence of Proposition
10.1 above, with particularly useful applications in practice.

COROLLARY 10.1: Given the general and specificatory knowledge
bases described in Proposition 10.1, the BME estimate Xk = Xk,mode is
the solution of the equation

where

where the 9£ 's the operator of Equation 10.1.

Proof: The BMEmode estimate is the solution of Equation 7.10 (p. 137),
where J% is given by Equation 10.1, i.e.,

or

which, in light of Equation 10.15, leads to Equation 10.14.

EXAMPLE 10.1: Example 7.2 (p. 138) is a special case of Corollary 10.1.
Indeed, Equation 7.14 (p. 140) could have been obtained directly from Equation
10.14 above. This shows the great simplifications in calculations provided by
analytical results such as Proposition 10.1 and Corollary 10.1.