Page 169 - Modern Spatiotemporal Geostatistics

150 Modern Spatiotemporal Geostatistics — Chapter 8

Symmetric Posteriors

For many single maximum pdf s, a measure of the accuracy of the BME esti-
mate Xk may be obtained by means of the standard deviation at each point
p k, i.e..

Equation 8.1 offers a particularly good measure of mapping accuracy in cases
of symmetric or approximately symmetric pdf. In these cases, we can also write

where Xk = Xk, mode = Xk,mean (due to symmetry), and the expectation is now
with respect to the posterior pdf rather than a realization average (as is the case
with Eq. 8.5 below). Equation 8.2 is an accuracy measure typically reported
by traditional mapping methods. As numerical simulations show, the
of BME analysis does an excellent job in approximating the actual estimation
error. On the basis of the estimation uncertainty (Eq. 8.2), confidence intervals
can be defined. In the case, e.g., of a Gaussian pdf there is a 95% confidence
that X(p k) lies in the interval

COMMENT 8.1: Traditionally, confidence intervals are taken such that the

probability of the estimated value falling on the left part of the interval is

equal to the probability of it falling on the right part. The BME confidence

intervals are as small as possible. BME confidence intervals are calculated
in the study of the Equus Beds aquifer later in this chapter (p. 155).

EXAMPLE 8.1: Consider the case of a symmetric f%(Xk)< such that a second-
order Taylor series expansion of log f K(xk) is a good approximation within a
neighborhood \Xk - Xk\ < £ around Xk = Xk, i-e.

The first derivative of the logarithm is zero (Eq. 7.1, p. 136) and, hence, it does
not appear in the Taylor expansion. By exponentiating, we find the following
approximation for the posterior pdf

where v os cpmstamt govem ny
Then, Equation 8.4 is a Gaussian pdf
with standard deviation v. Provided that the approximation range is sufficiently
large e > a x(p k), a measure of the estimation uncertainty is