Prediction of Occurrence of Discrete Events - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 182

Table 5.6 shows input data for the Seafloor Example. There are five binary
patterns (relative age of basaltic rocks; depth below sea-level; vicinity to contact
between youngest basaltic rocks; type of basaltic rock; and vicinity of fissures) and
2 5

10 m. Ordinary WofE weights and
contrasts are shown in the first three columns of Table 5.7 . WLR was applied to the
dataset of Table 5.6 after replacing every 1 for a map layer by its positive weight
and every 0 by its negative weight. Because presence of fissures initially resulted in
a negative value of W + , its 0s for absence were replaced by this negative value,
because it is absence (instead of presence) of fissures that is weakly positively
correlated with vent occurrence. The resulting logistic regression coefficients and
their standard deviations are shown in columns 4 and 5 of Table 5.7 . Multiplying
them by the weights yields the modified weights shown in the next two columns.
The modified WofE weights in the last columns of Table 5.7 yield S

¼ 32 unique conditions. Unit of area is 10 m

¼ 13 so that

bias due to lack of conditional independence is avoided.
Agterberg and Cheng ( 2002 ) proposed their conditional independence test using
original WofE results for the Seafloor Example. Figure 5.29 show estimated and
observed numbers of vents in the 5-layer model. The sum of posterior probabilities
based on all five map layers was S

13. Clearly,
there is significant violation of the conditional independence assumption. In a
separate experiment (Table 5.9 ) for a slightly larger study area of 3,985 km 2 a
three-layer model was analyzed. The first column of Table 5.9 shows unique
conditions with “1” for presence and “0” for absence. The area of the unit cell
was set at 0.01 km 2 in this experiment. With weights and standard deviations
similar to those listed for WofE in Table 5.7 this resulted in the posterior probabil-
ities P f with standard deviations s ( P f ) shown in columns 3 and 4 of Table 5.9 .
Multiplication of each P f by area (number of unit cells) of its unique condition
results in the eight predicted vent frequencies N IJK P f . Their sum provides the
estimate T

¼ 37.59 and much greater than N ¼

¼

14.05 which is only slightly larger their n

¼

13. The corresponding

variance s 2 ( T )(

¼

41.6511) is the sum of the eight values listed in the last column of
Table 5.9 . The square root of this number gives s ( T )

¼ 6.45. The z -test can be

applied to test the standardized difference ( T n )/ s ( T ) ¼1.05 for statistical signif-

icance. The 95 % confidence level for a one-tailed test to see if T is significantly
greater than n is 1.645. Consequently, the hypothesis that the three layers in the
model of Table 5.9 are conditionally independent can be accepted. On the contrary,
if the same z -test is applied to the five-layer model ( cf . Fig. 5.29 ), the assumption of
conditional independence is rejected. For more detailed explanations of this test,
see Agterberg and Cheng ( 2002 ).
It is interesting to compare the newly derived WLR results with those resulting
from application of WRL directly to the input data shown in Table 5.6 . In Table 5.8
it is shown that indirectly estimated coefficients divided by their standard devia-
tions are exactly equal to direct estimates divided by their standard deviations.
Repetition of the preceding experiment using the Meguma Terrain Example yielded
the results shown in Table 5.10 . In this application, the seven map layers used were
approximately conditionally independent of the 68 gold deposits, and the results
obtained by either direct or indirect application of WLR are not greatly different as
they were for the first example.