Page 173 - Modern Spatiotemporal Geostatistics

154 Modern Spatiotemporal Geostatistics — Chapter 8

probability P [xk — a < X(p k) < Xk + b]. The choice of r] clearly depends
on the situation under consideration. Given 77, the values a and 6 must be
determined so that

Then, the confidence width is given by

and, with probability rj, X(p k) e [x.k — «, Xk +b]. In modern spatiotemporal
geostatistics, confidence widths are a popular way to communicate quickly
and efficiently the amount of uncertainty associated with any estimate Xk-
Certainly, there are several choices of a and b values. A specific choice is that
which has the smallest width (Eq. 8.9) for a given rj.

PROPOSITION 8.1: Given 77, the a and 6 that have the smallest width
w x,rj(Pk) = a, + b correspond to the posterior pdf value

Proof: We are seeking the minimization of the width a + b subject to 77 =
This is equivalent to minimizing

(where \i, is a multiplier) with respect to a and b, which implies

or Similarly or
and the proposition isproven

While in the case of asymmetric pdf, we generally have a ^ b, in the
case of a symmetric pdf, Equation 8.10 gives a = b. Typical BME confidence
intervals obtained using Equation 8.8 are shown in Figure 8.4. Also, temporal
profiles of various BME confidence intervals of the Equus Beds water-level
estimates are plotted in Figures 8.8 and 8.9 in the following section (p. 159-
160). Generalizations of the above uncertainty measures in the context of
multipoint analysis will be considered in Chapter 9.