Geology Reference
In-Depth Information
3.2.1 Multidimensional Extensions of Entropy Theory
To explain multidirectional characteristics of entropy theory, consider a pair of
events (X
i
, Y
j
) and the probability of co-occurrence of both events. The probabilities
of the two pair of events are
X
n
p
i
:
¼
p
ij
ð
i
¼
1
; ...;
m
Þ
ð
3
:
21
Þ
j¼1
X
m
ð
; ...;
Þ
ð
3
:
22
Þ
p
:
j
¼
p
ij
j
¼
1
n
i¼1
Marginal entropy values of two events can be computed using following equations
[
65
]:
X
m
1
p
i
:
HX
ðÞ
¼
p
i
:
log
ð
3
:
23
Þ
i¼1
X
n
1
p
:
j
HY
ðÞ
¼
p
:
j
log
ð
3
:
24
Þ
j¼1
Combining the above two equations, the two-dimensional entropy can be obtained
and expressed as
X
m
X
n
1
p
ij
HX
ð
;
Y
Þ
¼
p
ij
log
ð
3
:
25
Þ
i¼1
j¼1
The conditional entropy value measures the uncertainty in one dimension
(e.g., X or Y), which remains when we know event Y
j
has occurred [
26
,
73
]:
X
m
p
ij
p
:
j
log
p
:
j
p
ij
ðÞ
¼
ð
3
:
26
Þ
H
Y
j
X
i¼1
X
n
p
ij
p
i
:
p
i
:
p
ij
H
X
i
Y
ðÞ
¼
log
ð
3
:
27
Þ
j
¼
1
The average conditional entropy can be derived by considering the weighted
average of the above-mentioned conditional entropies:
