Page 193 - Modern Spatiotemporal Geostatistics

174 Modern Spatiotemporal Geostatistics — Chapter 9

where T[-] is one-to-one. The BME posterior pdf of X(p) can be written in
terms of that of Y(p) as follows

An immediate consequence of Equation 9.23 is that the BME posterior pdf of
a natural variable can be calculated by means of the corresponding pdf of a
suitable transformation of the variable. A straightforward application of the
analysis in spatiotemporal mapping is demonstrated by Example 9.5.
EXAMPLE 9.5: Consider a map x map—which is generally characterized by a
non-Gaussian multivariate pdf—and assume that a transformation T[-] can
be established such that the y map = T~ [a^map] has a multivariate Gaussian
pdf. Then, the BME posterior pdf f K (xk) can be calculated from the Gaussian
f K(i^k) using Equation 9.23. As we shall see later in Chapter 12 ("Other sorts
of kriging," p. 249), in the case that only hard data are available as specificatory
knowledge, the multi-Gaussian kriging method is a special case of the above
procedure.

Decision making

There exist various applications where the variable Y used in the decision
problem is influenced by another variable X and, therefore, the estimation of X
directly affects the decision one makes. LetB^fc, Xk> dk', Pk) be tne benefit
if if?/, does occur at point PJ. and decision dk was made. Also, assume that
Xk is the estimate provided by BME analysis. One seeks an optimal decision
d* k such that the expected benefit is maximized. In mathematical terms, the
decision problem is written as

where d^ (xk) denotes that the optimal decision d* k is a function of the BME
estimate Xk, ar|d tne total knowledge base Stakes into account the relation
ship between X and Y. Decision analysis incorporates: (a) information about
the spatiotemporal distribution of the natural variables (represented by BME
maps, etc.), as well as (b) information about the application-specific goals and
the decision maker's preferences. The contribution of modern spatiotempo-
ral geostatistics can be considered at several levels: integrating geographical
information systems (GIS) in the decision making process, striking a reason-
able compromise among constraints (e.g., physical, economical, political, and
ecological constraints), simulating alternative decision scenarios, etc. (see also
section on "Modern Spatiotemporal Geostatistics and GIS Integration Tech-
nologies" on p. 261).