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some constant
L
. In this respect (23) is more general than (24) although the functions
a
(
t
) and
b
(
t
) cannot be considered as completely independent from each other. This
illustrates that an appropriate interpretation of the function
a
(
t
) is as the concentration
of
i
.
For the affinity in Hamming shape-spaces the distance function
d
XOR
is used. With
the linear version of the affinity function the two types of affinities have the form
(
)
( )
(
(
)
)
( )
(26)
aff
s
i
,
x
,
t
=
−
a
t
⋅
ones
XOR
i
,
x
+
b
t
(
)
( )
(
(
( )
)
)
( )
(27)
aff
c
i
,
x
,
t
=
−
a
t
⋅
ones
XOR
compl
i
,
x
+
b
t
Consider the complementarity based affinity. The affinity function adopts its
maximum value if
ones
(
XOR
(
compl
(
i
),
x
)) = 0 which is equivalent to
XOR
(
compl
(
i
),
x
) =
0
, and this means
compl
(
i
) =
x
. Thus, the point of maximum value is just the
vector
compl
(
i
), and this corresponds to the result in Euclidean and Manhattan shape-
spaces where it is the center of the affinity region. The affinity function is zero if
b
(
t
)/
a
(
t
) =
ones
(
XOR
(
compl
(
i
),
x
)). This holds for all vectors
x
that are different from
compl
(
i
) in exactly
b
(
t
)/
a
(
t
) components. For vectors that differ in less than
b
(
t
)/
a
(
t
)
components from
compl
(
i
) we have
ones
(
XOR
(
compl
(
i
),
x
)) <
b
(
t
)/
a
(
t
) and so
aff
c
(
i
,
x
,
t
) > 0, and for vectors with more than
b
(
t
)/
a
(
t
) different components
aff
c
(
i
,
x
,
t
) < 0.
Thus the vectors with exactly
b
(
t
)/
a
(
t
) different components form the rim of a region
in the Hamming space that can be taken as the affinity region.
aff
c
(
i
,
x
,
t
) has a linear
gradient from
compl
(
i
) to the rim of the region with respect to the sum of the digits of
the vectors. Thus, with respect to the affinity function, the Hamming shape-space can
be compared with an iceberg whose top rises up out of the zero surface and the rest is
below. This top is the affinity region.
As was shown in section 3,
R
/
T
is not a distance function. However,
R
/
T
can be
used to define an affinity function with
d
XOR
as the underlying distance function. This
affinity function is denoted by
R
/
T
-
aff
.
R
/
T
-
aff
is clearly a complementarity based
affinity because
R
/
T
(
x
,
x
) = 1 (the maximum value) and
R
/
T
(
x
,
compl
(
x
)) = 0 as stated
above.
R
/
T
-
aff
can be written in the form:
(
)
(
( )
)
R
T
−
aff
i
,
x
,
t
=
R
T
'
compl
i
,
x
,
t
(28)
()
()
(
(
()
)
)
b
t
⋅
n
−
a
t
⋅
ones
XOR
compl
i
,
x
=
()
()
(
(
()
)
)
b
t
⋅
n
+
a
t
⋅
ones
XOR
compl
i
,
x
With this modification of
R
/
T
the affinity function can adopt negative values, which
the function
R
/
T
itself in a Hamming space could not. It is zero if
ones
(
XOR
(
compl
(
i
),
x
) =
b
(
t
)
⋅
n
/
a
(
t
) and positive if
ones
(
XOR
(
compl
(
i
),
x
) <
b
(
t
)
⋅
n
/
a
(
t
) and this holds for
vectors that differ in less than
b
(
t
)
n
/
a
(
t
) components from
compl
(
i
). So, figuratively
spoken, the iceberg is elevated to a higher level.
To summarize,
R
/
T
-
aff
with the above modification is similar to the affinity
function of equation (27), except for the denominator. This is an interesting result
because Bersini on one side and Rogers and Tanimoto on the other side clearly had
rather different starting points and intended to do different things, but both developed
functions that - put down to their principles (and applied to Hamming shape-spaces) -
⋅