Information Technology Reference
In-Depth Information
We assume that
a
(
t
)
0. If
a
(
t
) = 0 then
b
(
t
) must also be zero which denotes the
extreme case of an affinity region shrunk to a point. The distance between two
elements must be a positive value, therefore
b
(
t
) and
a
(
t
) must have opposite signs.
Thus from (18) we get the equation
a
(
t
)
≠
d
(
i
,
x
) +
b
(
t
) = 0,
assuming both,
a
(
t
) and
b
(
t
), are positive. These equations describe the rim of the
affinity region (the surface of an
⋅
d
(
i
,
x
)
−
b
(
t
) = 0 or
−
a
(
t
)
⋅
-ball). Its form is determined by the distance
function
d
and its size by the functions
a
and
b
. The points on the rim are exactly
those
x
for which
d
(
i
,
x
) =
b
(
t
)/
a
(
t
).
For the points inside or outside the region the inequations that can be derived from
the two equations must be treated separately. In the second form of the inequation,
−
ε
a
(
t
)
⋅
d
(
i
,
x
) +
b
(
t
) < 0 (i.e.
aff
is negative) is equivalent to
a
(
t
)
⋅
d
(
i
,
x
) >
b
(
t
), i.e. for
the points outside the affinity region, and
−
a
(
t
)
⋅
d
(
i
,
x
) +
b
(
t
) > 0 (
aff
is positive) is
equivalent to
a
(
t
)
d
(
i
,
x
) <
b
(
t
), i.e. for the points inside the affinity region. The first
form of the inequation would have the (undesired) opposite result. The two cases are
illustrated by figure 8. Therefore the only linear version of
aff
that yields an affinity
region with positive values of the affinity function is that of equation (19):
⋅
(
)
()
(
)
()
(19)
aff
i
,
x
,
t
=
−
a
t
⋅
d
i
,
x
+
b
t
x
2
x
2
Affinity
region
Affinity
region
i
i
c
c
negative
values
positive
values
i
c
i
c
positive
values
negative
values
x
1
x
1
Case 1: a(t)
⋅
d(i, x) - b(t) < 0
Case 2: -a(t)
⋅
d(i, x) + b(t) < 0
Fig. 8.
Regions with positive or negative affinity values depending on the form of the function
Since
aff
is restricted inside the affinity region by
b
(
t
), it adopts a maximum at the
point
x
=
i
, its value is clearly
b
(
t
).
aff
can also be defined as a quadratic function:
(
)
( )
(
)
( )
2
(20)
aff
i
,
x
,
t
=
−
a
t
⋅
d
i
,
x
+
b
t
(For simplicity, the linear component of the equation is omitted.) This function is zero
if
,
i.e. inside the region, and negative outside. However, higher order functions would
have the unpleasant result that the region of positive values is not coherent, for
instance it could look like the shadowed regions in figure 8 for the complementarity
based affinity. Therefore such functions will not be considered as adequate
representations of affinity.
(
)
() ()
(the rim of the affinity region), positive if
()
()()
d
i
,
x
=
b
t
a
t
d
i
,
x
<
b
t
a
t
