Page 127 - Modern Spatiotemporal Geostatistics

108 Modern Spatiotemporal Geostatistics — Chapter 5
deals with the cases in which the prior pdf f s of the map is sought from
the general knowledge base £ = {space/time statistical moments of orders
1,..., A} and the base Q — {multiple-point statistics of orders AI, ..., A,,}.
EXAMPLE 5.1: Assume that the g a functions at a point p include the normal-
ization constraint and the spatiotemporal statistical moments, as follows

where a = 1,...,A accounts for the orders of the corresponding statistical
moments in space/time. Then, the ^-operator in Equation 5.7 is given by

where the Lagrange multipliers are found by substituting Equation 5.13 into
Equation 5.12 and solving for the /x a's at point p for all moment orders consid-
ered (in this case, N c = A). In addition to the solution in Equation 5.13, other
approaches could also be used to determine 9£ (see below section, "Possible
modifications and generalizations of the prior stage," on p. 118). The analysis
is easily extended to more complicated situations. We could, e.g., involve the
whole set of mapping points p mop, which leads to

or we could assume a series of more elaborate g a functions such as,

in which case 9£ is generally given by Equation 5.7; etc. Note that the above
BME equation includes the case of multiple-point statistics, as well. Indeed, as
we saw in previous sections, the g a functions may have the form
which implies the multiple-point statistics

of orders \j (j = 1,2,...,v)', etc. The BME approach will account for this
kind of statistical knowledge rigorously and efficiently.

COMMENT 5.3: Th e epistemic significance o f th e situation studied i n th e

above example (Eqs. 5.12 and 5.13) can best be appreciated if we look at
the familiar way in which a scientific theory fails to be constrained or de-

termined by evidence (Quine, 1970; Friedman, 1983). Lower level (or in-
ductive) indetermination (Chapter 1, p. 16) consists in the fact that we