Biology Reference
In-Depth Information
9. The Kosko entropy of fuzzy answer F in Eq.
5.28
is given by:
h
i
1
=
2
h
2
2
2
2
2
S
K
ð
F
Þ¼ ð
0
1
=
3
Þ
þð
1
3
=
4
Þ
þð
1
7
=
8
Þ
=
ð
1
1
=
3
Þ
þð
0
3
=
4
Þ
2
i
1
=
2
þð
0
7
=
8
Þ
h
i
1
=
2
h
i
1
=
2
2
2
2
2
2
2
¼ð
2
=
3
Þ
þð
1
=
4
Þ
þð
1
=
8
Þ
= ð
2
=
3
Þ
þð
3
=
4
Þ
þð
7
=
8
Þ
1
=
2
1
=
2
¼½
4
=
9
þ
1
=
16
þ
1
=
64
=½
4
=
9
þ
9
=
16
þ
49
=
64
¼½
0
:
4444
þ
0
:
0625
þ
0
:
016
=½
0
:
4444
þ
0
:
5625
þ
0
:
7656
¼
0
:
5229
=
1
:
7725
(5.28)
¼
0
:
2950
10. As evident in (8) and (9), it is possible to calculate the numerical value of the
Kosko entropy
of any fuzzy answer F, S
K
(F). But what is the meaning of S
K
(F)?
It is here suggested that the Kosko entropy, S
K
, of fuzzy answer F is a
quantita-
tive measure of the uncertainty
that F is C (or C is F, for that matter). By
multiplying S
K
(F) with 100, we can express this uncertainty in the unit of %:
S
K
(F)
100
¼
The percent uncertainty that F is C (or C is F)
(5.29)
Applying Eq.
5.29
to the result in Eq.
5.28
, we can conclude that
It is 29.5% certain that fuzzy answer located at (1/3, 3/4, 7/8) is equivalent to (and hence
can be represented by) the crisp answer located at (0, 1, 1).
(5.30)
If we assume that
All crisp answers are approximations of their closest fuzzy answers
(5.31)
we can reexpress Statement 5.30 as follows:
The uncertainty of crisp answer C (0,1,1) is (100 - 29.5)
¼
70.5 %.
(5.32)
11. Statements 5.31 and 5.32 would gain a strong support if we can associate the
interior of the N-dimensional hypercube defined in Table
5.4
with
reality
or the
source of the apparatus-elicited answers of Wheeler (1990) and its vertices with
possible, theoretical, or represented answers. The apparatus-elicited answers
may have two aspects - the “registered” aspect when artificial apparatuses are
employed and “experienced” aspect when living systems are involved as
measuring agents. Frieden (2004) associates the former with
Fisher informa-
tion
(I) and the latter with what he refers to as “bound information” (J), that is,
the algorithmic information needed to characterize the “source effects” that
underlie registered data or crisp answers. In the case of Frieden (2004), it seems
clear that the registered answers (carrying Fisher information, I) belong to the
vertices of the N-dimensional hypercube and the “experienced” answers or
