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Mathematical Formulation of the BME Method 109

possess only a limited amount of evidence (expressed by the moments of or-
der a < \). Thus, depending o n the unknown higher order moments ('i.e. ,

for a > X), different % 's might have been obtained.

An interesting circumstance arises when the calculation of the statistical
moments includes some measurement error, as was discussed in Chapter 3
(Example 3.5, p. 76). This case is revisited in the example below.
EXAMPLE 5.2: The g a functions at point p include the normalization con-
straint and the moments in Equation 3.4 (p. 76). Then, the constraints to be
considered in Equation 5.6 are obtained using the chi-square statistic, i.e..

where cr^ a(p) is the variance of v a(p). The 9£-°perator in Equation 5.6 is
given by Equation 5.13, where A is now replaced by N i.e.,
'n *h's case *ne general knowledge constraints yield

The Lagrange multipliers HQ, fi a (a — 1,..., N c) and rj are found by solving
the system of N c + 1 equations (Eq. 5.15). In most cases of practical interest,
these calculations are done numerically.

General knowledge in the form of physical laws

As is well known, empirical or statistical mapping techniques describe an ex-
isting set of data and are only locally predictive (i.e., interpolation is possible
only within the data range). The situation is different with BME analysis,
which is expressed by the following postulate (see also Chapter 3, p. 88).

POSTULATE 5.3: BME incorporates physical laws into space/time
mapping and, thus, it can have global prediction features (i.e., extrapo-
lation is possible beyond the range of observations).
Next we will discuss how physical laws of various forms are incorporated
into BME analysis. Since these laws constitute general knowledge Q, they
must be analyzed at the prior stage of the BME approach. At this stage, one
essentially seeks to transform the physical law into a set of moment equations
and then proceeds as before [i.e., formulates the prior pdf (Eq. 5.6) associated