Page 200 - Modern Spatiotemporal Geostatistics

Modifications of BME Analysis 181

provides a poor representation of the real situation. Furthermore, the BME
confidence set size ||$^|| defined in Equation 9.37 is equal to the area sur-
rounded by the corresponding confidence probability ry-contour in Figure 9.3.
For comparison, the variation of ||$,|| vs. r\ is plotted in Figure 9.4 for both the
multipoint case of Figure 9.3a and the single-point case of Figure 9.3b. From
Figure 9.4 it is interesting to note that for any given confidence probability t],
the size of the corresponding multipoint confidence set of Figure 9.3a is con-
sistently smaller than the size of the single-point confidence set of Figure 9.3b.
In other words, in this simulation study the multipoint confidence sets seem to
offer a considerable improvement over the single-point confidence sets.
Other applications of the BME uncertainty analysis include the construc-
tion of maps displaying the space/time fluctuations in the probability that the
values of a natural process: (i.) do not exceed a certain threshold (e.g., the
concentration of a soil contaminant does not exceed a threshold derived on the
basis of environmental health standards); (ii.) are between two given bound-
aries (e.g., the temperature in a region stays within the boundaries that allow
the growth of a specified agricultural product); etc.

BME in the Context of Systems Analysis

As we have already noted, in many applications the outcomes of BME analy-
sis (predictive maps, uncertainty measures, etc.) serve as the precious inputs
to subsequent steps of scientific investigations, engineering designs, decision
making, etc. Many examples can be given, which essentially depend on the
scientific background and area of expertise of the person using modern spa-
tiotemporal geostatistics. Below, we will focus our attention on three appli-
cation areas of general systems analysis: (i.) risk analysis of natural systems;
(ii.) human-exposure assessment; and (iii.) environmental exposure-health
effect associations.

Risk analysis of natural systems

Several applications in the risk analysis of natural systems can be described in
terms of a general performance function 9{(X) such that

where the variables X = (X\, ..., X\) describe the environmental conditions
of interest. The corresponding failure probability of the system can be expressed
in terms of the BME posterior pdf and the function Pfas follows,