Information Technology Reference
In-Depth Information
2
together with a fixed affinity region where the immune elements (antibodies) are
attracted with graded force. Bersini's approach has been adopted and modified in
several ways by Hart and Ross [7] who demonstrate the properties of this kind of
shape-space by a number of simulation experiments.
The properties of Bersini's shape-space and some extensions of it were discussed
in detail in [6]. The aim of this paper is to give a general framework for the definition
of shape-spaces that reveals the similarities and differences between various
approaches. Also, I argue that a shape-space defined over
incorporates complementarity as mirror image (or complementary) positions in
ℜ
n
should be a metric
space. This is mainly done in section 2. In section 3, the principles described in
section 2 are adopted for finite shape-spaces. Various approaches for defining
distance functions on Hamming spaces are examined and it is shown that not all of
them are metrics. Based on distance functions, affinity functions can be defined in
different ways which are presented in section 4.
ℜ
2 Structural Aspects of Shape-Spaces
I will make two general presuppositions about shape-spaces. First, a shape-space is a
set
S
of attribute strings of finite length. The values of the attributes can be taken from
arbitrary domains. Second, on
S
a function
d
:
S
×
S
→
ℜ
is defined, called “distance
function”, which satisfies the following conditions: If
x
,
y
∈
S
then
(i)
d
(
x
,
y
)
≥
0
(ii)
d
(
x
,
y
) = 0
⇔
x
=
y
(iii)
d
(
x
,
y
) =
d
(
y
,
x
)
There are no additional requirements on
d
, i.e. one is free to choose an arbitrary
two-dimensional function as long as it satisfies the three conditions. Therefore, even
such a strange function as the following one can serve as a distance function:
0
,
,
if
x
=
y
)
⎩
⎨
⎧
(
(1)
d
x
,
y
=
1
otherwise
n
are Euclidean and Manhattan distance:
Commonly used distance functions on
ℜ
(
)
∑
=
n
i
(
)
Euclidean
(2)
2
d
x
,
y
=
x
−
y
E
i
i
1
n
Manhattan
(
)
∑
=
(3)
d
x
,
y
=
x
−
y
M
i
i
i
1
Since the distance function
d
is a constituent part of a shape-space, I will denote a
shape-space in the following as a pair (
S
,
d
). Clearly, the function
d
induces a
structure on a shape-space depending on the form of the function, such that two
shape-spaces (
S
,
d
1
) and (
S
,
d
2
) with different distance functions are different even if
the underlying set is the same.
