Page 167 - Modern Spatiotemporal Geostatistics

148 Modern Spatiotemporal Geostatistics — Chapter 7
A general way to search for an appropriate estimate is by optimizing
(with respect to the estimate sought) the expected value of some function of
the natural variable and its estimate at each point p k, i.e., a function of the
form

where the shape of the L-function depends on the physical, economical, and
other features of the situation considered. Assume, e.g., that L is some sort
of a loss function of the estimation error Xk — Xk, specific to the problem at
hand. Then, minimization of Equation 7.23 with respect to Xfe leads to the
following integral equation

where the required conditions for a minimum are assumed valid. A variety
of estimators can be derived from Equation 7.24, as is demonstrated in the
following example.
EXAMPLE 7.6: If the loss function L is chosen to be equal to the absolute
estimation error, the estimate obtained from Equation 7.24 is the BMEmedian
(Eq. 7.21). Assuming that L is of the quadratic error form, the corresponding
estimate is the BMEmean (Eq. 7.22). Also, if L has the form of a (0, l)-loss
function, Equation 7.24 gives the BMEmode estimate (Christakos, 1992).

Note that in the case of a symmetric unimodal posterior pdf, the estimates
(Eqs. 7.3, 7.21, and 7.22) coincide. This fact can be used to significantly
improve the computational efficiency of the BME approach. In Chapter 8 (p.
155), BMEmode and BMEmean estimates are calculated for a real data set
representing water-level elevations in the Equus Beds aquifer in the State of
Kansas.
A Matter of Coordination

By way of a summary, one can argue that the choice of a spatiotemporal esti-
mate (mode, median, mean, etc.) in BME analysis depends on the successful
coordination of four essential factors:
1. the satisfactory track record of the specific kind of estimate with similar
situations;
2. the theoretical background that promotes its use against other possible
estimates;
3. the explanatory rationale provided by the estimate; and
4. the case-specific objectives of the map.
The above approach is consistent with the fact that BME follows the
hypothetico-deductive conception of scientific mapping discussed in Chapter 5
(p. 122), rather than the linear (or pure inductive) paradigm.