Page 155 - Modern Spatiotemporal Geostatistics

136 Modern Spatiotemporal Geostatistics — Chapter 7

the mean squared estimation error), and the median estimate (which minimizes
the absolute estimation error). These, as well as other choices, are discussed
in the section, "Other BME Estimates," on p. 147. Just as in Chapter 6,
the present chapter is also concerned with single-point estimation; multipoint
mapping is considered in a later chapter.

The BMEmode Estimate

Consider first the estimate Xk that maximizes the posterior pdf; this is the mode
of the BME posterior pdf. For such a choice, the spatiotemporal single-point
estimation procedure is summarized as follows.

BMEmode mapping concept: Derive estimates Xk of a natural
variable X(p) at space/time points p k given data (hard and/or
soft) at points p i (i = l,...,m; i ^ fc) such that: (a) the
expected information (Eq. 5.2, p. 105) is maximized with respect
to the pdf f s subject to the general knowledge base §; and (b)
the posterior pdf (Eq. 5.35, p. 120) is maximized with respect to
Xk = Xk-
The outcome of requirement (a) was already given in Equation 5.6 (p. 106).
Formally, maximization of the posterior pdf in requirement (b) involves solving
the equation

or, in light of Equation 5.35, solving the equation

The estimate provided by Equation 7.2 is the mode of the posterior pdf, for
short

Equation 7.2 is, therefore, a basic BMEmode equation. Strictly speaking, in
order for the solution of Equation 7.2 to represent a pdf maximum (rather
than a minimum or an inflexion point), the latter must be concave in the
i
neighborhood of Xk — Xk,mode< -e-, it must also hold that

Since concavity is a property of the logarithmic function, log f K is sometimes
used in place of simply f K.