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Affinity, however, will be considered as time dependent. Like in [1], it will be
defined as a function that determines the amount of affinity between two elements and
at the same time a limited region in the shape-space, the
affinity region
, which has the
form of an
-ball. The affinity function shall have positive but restricted values inside
the affinity region and be zero or negative outside.
Let
T
be the infinite set of time points. The affinity that an element
i
exerts on an
element
x
at time
t
is defined as a function
aff
:
S
ε
and is denoted by
aff
(
i
,
x
,
t
). There are two main types of affinity presented in the literature [cf. 5], one called
similarity based
and another called
complementarity based
. In both types, affinity is
defined as inversely proportional to distance, i.e. the smaller the distance between
i
and
x
the higher the affinity. The two elements are understood as similar if their
distance is small. For the second type of affinity we need the concept of
complementarity as defined in section 2. In order to distinguish between the two
forms of affinity I will use the notation
aff
s
(
i
,
x
,
t
) for the similarity based affinity and
aff
c
(
i
,
x
,
t
) for the complementarity based affinity. In the second version affinity is
defined as inversely proportional to the distance between
x
and
i
c
. The points around
i
or
i
c
respectively form an
×
S
×
T
→
ℜ
ε
-
ball
, which is known as the
recognition region
[5, 14] or
the
affinity region
[1].
In the following I will first present a formal treatment of affinity functions and then
show, that this is just a generalization of other approaches that explicitly consider
concentration of elements in the definition. There are different ways to define the
function
aff
such that it has the desired property. The simplest form of
aff
would be a
function with a constant (but time dependent) positive value inside the affinity region
and zero outside. The affinity region is simply determined by an upper bound to the
distance:
()
(
)
()
a
t
if
d
i
,
x
≤
b
t
⎩
⎨
⎧
(
)
(16)
aff
i
,
x
,
t
=
0
otherwise
a
and
b
are functions of time,
b
is used as an upper bound to the distance. The affinity
region is an
-ball depends on the
definition of
d
. However, such a definition seems not adequate because all elements in
the affinity region have the same affinity
a
(
t
) and there is no difference of affinity
between elements close to
i
and those more remote. A more adequate form seems to
be a linear function. It has the general form
ε
-ball around
i
restricted by
b
(
t
). The form of the
ε
(
)
( )
(
)
( )
(17)
aff
i
,
x
,
t
=
a
t
⋅
d
i
,
x
+
b
t
Again,
a
and
b
are functions of time. Let us consider some properties of this function,
more precisely of
aff
s
.
aff
s
(
i
,
x
,
t
) = 0 iff
a
(
t
)
⋅
d
(
i
,
x
) +
b
(
t
) = 0 or
()
()
b
t
(18)
()
d
i
,
x
=
−
a
t
