Page 138 - Modern Spatiotemporal Geostatistics

Mathematical Formulation of the BME Method 119

It may be possible that, while the multivariate pdf f s is unknown, an
equation that describes the evolution of a lower level pdf can be derived from
physical or mathematical considerations (see "Some other forms of general
knowledge," in Chapter 3, p. 81). A set of g a functions could then be properly
selected so that the stochastic expectations ~g^ with respect to / § can be cal-
culated. Under these conditions, the problem has essentially been reduced to
that described above in the section "General knowledge in the form of random
field statistics" (p. 107). Let us consider the following rather simple example.

EXAMPLE 5.8: In Chapter 3, in the case of Example 3.11 (p. 82), the operator
y g is simply given by 9£ [x; *] — fJ-i(t)gi(x)< where Hi(t) is the solution of the
equation

Depending on the form of the function g\(-), this equation may be solved
analytically or numerically.
In its current formulation, BME uses the information measure (Eq. 5.1)
and the expected information or entropy (Eq. 5.2). But it may be worth
examining situations in which one could use other measures available in the
literature (e.g., Aczel and Daroczy, 1975; Ebanks et al., 1998). New measures
could also be proposed by modern geostatisticians, as long as they are physically
meaningful and epistemically sound. In such a case, entropy may not be part
of the mapping formulation, and the letter "E" in the acronym "BME" should
be replaced by another letter denoting the new information measure.

The Meta-Prior Stage

In real-world applications the knowledge bases are continuously refined due to
further interaction with the environment. At the meta-prior stage we collect
and organize specificatory knowledge 5 in appropriate quantitative forms that
can be explicitly incorporated into the BME formulation. In many applications
a variety of case-specific data are available, which usually makes the analysis
at the meta-prior stage not a trivial task. The quality and quantity of the
hard and soft data collected is a matter of experimental and/or computational
investigations. BME's concern is that accurate spatiotemporal maps be drawn
from good-quality data and other sources of knowledge by means of acceptable
rules of reasoning, which can be adequately quantified in terms of the analytical
and computational tools available.
As we already saw in Chapter 4 ("Epistemic Geostatistics and the BME
Analysis," p. 90), the distinction between prior and meta-prior knowledge is an
important epistemic issue. Whereas at the prior stage general knowledge (3
was introduced, at the meta-prior stage the specificatory knowledge ,5 is con-
sidered, including hard and soft data. Certain of these kinds of data have been
discussed in Chapter 3 (p. 73, "The General Knowledge Base" and p. 82, "The