Prediction of Occurrence of Discrete Events - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 179

frequencies. In this example, there are 43 deposits and 430 unique conditions. The
maximum difference of 0.088 is within the 95 % confidence interval that amounts to
( 0.207 in the two-tailed Kolmogorov-Smirnov test. Thus the WLR

model provides a good fit for the training area of Experiment 5.
Amineral potential map can be viewed as the contour map of a landscape covered
by disks representing mineral deposits. If the fit is good, the sum of posterior
probabilities for any subarea is equal to observed number of deposits in that subarea.
It is possible to improve the fit; e.g., by using non-linear functions of the variables.
However, the end product then may be equivalent to an empirical contour map of
deposit density derived without use of geoscience information and without predic-
tive potency. In this situation, superimposing a grid with very small cells on the
study area and random sampling (see Fig. 5.25 ) produces a representative sample of
this empirical contour map. Another way to illustrate this concept is to imagine
maintaining a constant proportion of cells that are sampled at random (e.g., 25 %)
but to steadily decrease grid spacing. In the limit (infinitely small cells), both training
and target area exactly duplicate the study area. A better fit then does not prove that
the method used provides a better predictive tool.
If the concept of random sampling from a study area is applied, one could argue
as in Agterberg ( 1992 ) that the known mineral deposits in a region constitute a
random subset of a larger population of discovered plus undiscovered deposits that
are of the same type in that they relate in the same way to the variables from which
the indicator map layers are formed. In WofE, one then simply can increase the
prior probability by a factor equal to total number of deposits divided by number of
known deposits. Of course, in practice the problem is that, generally, it is not
possible to estimate this ratio accurately. However, in a relative sense, the patterns
on a mineral potential map are not affected by such lack of knowledge.

1.36/ √ 43 ¼ )

5.3 Modified Weights-of-Evidence

The approach to be discussed in this section was originally introduced by
Spiegelhalter and Knill-Jones ( 1984 ) as a refinement of weights of evidence as
used in the GLADYS expert system. Agterberg ( 1992 ) had suggested applicability
of this indirect method to GIS-based regional mineral resources estimation but this
modification had not yet been tested until the method was applied in Agterberg
( 2011 ) and Zhang et al. ( 2013 ). Suppose that a study area is digitized as a number
( n ) of pixels and that X i ( i , p ) are a number of binary explanatory

variables used to predict a dichotomous random variable Y representing presence
( Y k ¼

¼ 1, 2, ...

0) of mineralization at the k -th pixel. Provided that n is
very large, we can redefine the situation in terms of binary sets B i corresponding to
the X i and a set D corresponding to Y . In most WofE applications, the B i 's are binary
with or without missing data, although the method also could be used with multi-
state explanatory variables.

1) or absence ( Y k ¼