Geoscience Reference
In-Depth Information
Relationships between probabilities, odds and logits previously were discussed
in Sect.
2.2
. The result of application of Bayes' rule applied to a single map layer
can be extended by using it as prior probability input for a second map layer. This
process can be repeated by further adding additional map layers provided that there
is approximate conditional independence (CI) of map layers. The order in which
new patterns are added is immaterial.
Box 5.2: Bayes' Rule for Two Map Patterns
When there are two map patterns as in Fig
5.1
:
PD
B
C
D
PðÞ
¼
PB
\
\
C
;
PB
ð
\
C
Þ
PB\C
D
P D
P D
B
ðÞ
\
C
¼
. Conditional
independence
of
D
with
PB\C
ð
Þ
C
D
¼
respect to
B
and
C
implies:
P
(
B
\
C
|
D
)
¼
P
(
B
|
D
)
P
(
C
|
D
);
PB
\
PB
D
PC
D
PB\C
PB
D
PC
D
.
Consequently,
PD
B
¼
\
C
PD
ðÞ
;
ð
Þ
P D
PB
D
PC
D
PB\C
P D
B
¼
\
C
. From these two equations it follows
ð
Þ
that:
PD
B
\
C
PB
D
PC
D
PD
ðÞ
P D
B\C
PB
D
PC
D
. This expression is equivalent to ln
O
¼
ðÞ
P D
C
) ¼ ln
O
(
D
)+
W
1
+
W
2
.
(
D
|
B
\
The posterior logit on the left side of the final result shown in Box
5.2
is the sum
of the prior logit and the weights of the two map layers. The posterior probability
follows from the posterior logit. Similar expressions apply when either one or both
patterns are absent. Cheng (
2008
) has pointed out that, since it is based on a ratio,
the underlying assumption is somewhat weaker than assuming conditional inde-
pendence of
D
with respect to
B
1
and
B
2
. If there are
p
map layers, the final result is
based on prior logit plus the
p
weights for these map layers. A good WofE strategy
is first to achieve approximate conditional independence by pre-processing. A
common problem is that final estimated probabilities usually are biased. If there
are
N
deposits in a study area and the sum of all estimated probabilities is written as
S
, WofE often results in
S
N
. The difference
S
-
N
can be tested for statistical
significance (Agterberg and Cheng
2002
). The main advantage of WofE in compar-
ison with other methods such as WLR is transparency in that it is easy to compare
weights with one another. On the other hand, the coefficients resulting from logistic
regression generally are subject to considerable uncertainty (Sect.
5.2
).
The contrast
C
>
W
is the difference between positive and negative weight
for a binary map layer. It is a convenient measure for strength of spatial correlation
between a point pattern and the map layer (Bonham-Carter et al.
1988
;Agterberg
1989b
).
W
+
¼
It
is somewhat similar
to Yule's (
1912
) “measure of association”
e
C
.Both
C
and
Q
express strength of correlation between two
binary variables that only can assume the values 1 (for presence) or
¼
α
1
Q
αþ
1
with
α
¼
1(forabsence)
(
cf
.Bishopetal.
1975
, p. 378). Like the ordinary correlation coefficient,
Q
is confined
to the interval [
1, 1]. If the binary variables are uncorrelated, then
E
(
Q
)
¼
0.
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