Geoscience Reference
In-Depth Information
5.3.1 East Pacific Rise Seafloor Example
If there are
N
discrete events in a study area and the sum of all estimated probabil-
ities is written as
S
, WofE generally results in
S
N
. The difference
S
-
N
can be
tested for statistical significance. The main advantage of WofE in comparison with
WLR is transparency in that it is easy to compare weights with one another.
Although WLR yields
S
>
N
, WLR coefficients generally have relatively large
variances. By preprocessing it is usually possible to obtain WofE weights that
approximately result in
S
¼
N
. It is also possible to first perform WofE modeling
and to follow this by WLR applied to the weights. This method results in modified
weights with unbiased probabilities satisfying
S
¼
N
. An additional advantage of
this approach is that it automatically copes with missing data on some layers
because weights of unit areas with missing data can be set equal to zero as is
generally practiced in WofE applications.
¼
Box 5.5: Proof That the WLR Likelihood Function Results in
S
¼
N
Suppose that the probability of occurrence or non-occurrence of an event is written
in the form:
P
(
Y
e
f
(
x
)
/{1 +
e
f
(
x
)
};
P
(
Y
¼
1|
x
)
¼
ˀ
(
x
)
¼
¼
0|
x
)
¼
1
ˀ
(
x
). The likeli-
n
Y
y
i
1
y
i
hood function then becomes:
l
ðÞ
¼
ˀ
xðÞ
1
ˀ
xðÞ
g
. When there
would be a single explanatory variable with
ʲ
¼
[
‵
ʲ
0
ʲ
1
]: Logit(
ˀ
i
)
¼
ʲ
0
+
ʲ
1
x
i
,
and the Log likelihood function is
L
(
ʲ
)
¼
log
e
{
l
(
ʲ
)}
¼
∑
y
i
·log
e
{
ˀ
(
x
i
)}
+(1
y
i
)·log
e
{1
ˀ
(
x
i
)}. Differentiation with respect
to
ʲ
0
and
ʲ
1
gives:
∑
{
y
i
ˀ
(
x
i
)}
¼
∑
x
i
{
y
i
ˀ
(
x
i
)}
¼
0;
0. Consequently,
the total number of
discrete events (
N
) is equal
to the sum the estimated probabilities (
S
), or
∑
N
also applies when there are
p
explanatory
variables. It is noted here that the
S
y
i
¼
∑
p
(
x
i
). The relation
S
¼
N
also is useful as a final test on posterior
probabilities obtained by the LOGPOL program (Box
5.3
). Although various
calculations in this FORTRAN program are carried out in double precision,
it is possible, for very large databases, that the sum of the final posterior proba-
bilities is slightly less than
N
due to lack of complete convergence for some
posterior probabilities The logistic regression coefficients on which these proba-
bilities are based then can be used as input for a new LOGPOL run to check
for the
S
¼
¼
N
requirement.
The problem of obtaining unbiased posterior probabilities in Bayesian
approaches to regional mineral resource evaluation has been considered by several
authors including Caumon et al. (
2006
). These authors proposed a cross-validation
technique to cope with violation of conditional independence of explanatory vari-
ables in weights-of-evidence modeling. Their approach is a modification of a
method originally proposed by Journel (
2002
), Krishnan (
2008
) and Krishnan
et al.
(
2004
). Several WofE-based methods
to obtain unbiased posterior
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