Page 197 - Modern Spatiotemporal Geostatistics

178 Modern Spatiotemporal Geostatistics — Chapter 9

is a BME confidence set <!?,, with a confidence level given by

The A/? probability density sets are easily constructed from the posterior
pdf, in which case Equation 9.38 above provides the corresponding BME con-
fidence set. Note that if the inverse function for r](/3) exists, then one can
directly obtain a BME confidence set $,, for a selected level 77. In this case,
the $„ is given bv

A useful corollary of Proposition 9.1 for p = 1 is when the BME confidence
set is a single interval. In this case, the

is a BME confidence interval such that f K(xk,i) = fx.(x.k,u) and P[xk,i <
Xk < Xk,u] = f] (see also Chapter 8). A result that is useful in computational
applications is suggested by the following proposition (Serre et al, 1998).

PROPOSITION 9.2: If the multipoint posterior pdf fK(xk) is approx
imated by a Gaussian law, the probability function P[A.p] is efficientl
calculated by means of the following expression

where

COMMENT 9.1: A s mentioned in Comment 2.8 (p . 58), when dealing with

matrix multiplications, a vector x = (xi,..., x n) will be consideredas a col-

umn vector ; i.e., the tw o forms o f writing vectors ar e identical.

Then, we can also write

EXAMPLE 9.10: In the case of the two-point posterior pdf (p = 2), the simpli-
fication P [A^] = f\ K, (3 is obtained. Hence, the confidence set ^^ coincides
with the index set A^.

COMMENT 9.2: A s w e already mentioned, th e BM E confidence sets ar e a

multipoint generalization of the single-point confidence intervals (in which
several estimated values are considered simultaneously). The same concepts

can easily be extended jointly to the vector estimation of several random

fields.