Page 37 - Modern Spatiotemporal Geostatistics

18 Modern Spatiotemporal Geostatistics — Chapter 1

on a few choices. As a matter of fact, the case of the indetermination thesis
implies that one should use a more scientifically sound method in one's effort
to choose an appropriate model, as is suggested by the following postulate.
POSTULATE 1.3: In modern Spatiotemporal geostatistics, the rational
approach for choosing the appropriate model from the data is by means
of a theory that represents the physical knowledge available and satisfies
plausible logical principles of scientific reasoning.
The meaning of Postulate 1.3 is that, in addition to conforming to empiri-
cal evidence, there exist physical knowledge (scientific theories, laws, etc.) and
epistemic criteria (explanatory power, informativeness, etc.) that are capable
of singling out one model from a class of empirically equivalent models (or, at
least, capable of narrowing the choices considerably). In this sense, Postulate
1.3 is closely related to the issue of scientific content addressed in the previ-
ous section. As the following example demonstrates, instead of fitting various
mathematical models to the available data, one could fit the model that is
justified on the basis of a physical theory that produces the specific set of data.

EXAMPLE 1.14: Assume that a set of temporal burden data X(ti), i =
l,...,m on a target human organ subjected to pollutant exposure is avail-
able. If the pure induction (theory-free) approach is adopted, several valid
covariance functions

could be fitted to the burden data by means of a conventional data-fitting
technique. The question then arises: Which model best describes the temporal
burden correlations? The physics of the situation offers a realistic answer to
this question. The burden X(t) on a target organ due to exposure obeys the
stochastic first-order kinematics

where A is the transfer rate out of the organ determined from the data, U(t)
is the uptake rate (zero mean, white-noise process with a unit variance), and
X(0) — 0 is the initial condition. On the basis of the law (Eq. 1.3), the
theoretical burden covariance is given by

The process of fitting the covariance model (Eq. 1.4) to the burden data
has physical content, which was not the case with the arbitrary functions
Cfc(^i) ^2)) k = 1,2,... of Equation 1.2 above.
Other applications of the important Postulate 1.3 will be discussed in the
following chapters (see e.g.. Chapter 7, Example 7.5, p. 143).