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The paper is organized as follows : In section 2 the real-valued shape-space
is outlined and the most commonly used Euclidean distance is presented. Sec-
tion 3 describes the abstraction of an antibody as a hypersphere. In section 3.1
the known hypersphere volume formula and the construction idea of that for-
mula is shown and properties of that formula are presented in section 4. Next,
the maximum volume of hyperspheres with respect to the dimension and the
radius is presented in section 4.1, and we highlight unexpected properties of
hyperspheres in high dimensions. In section 4.2, based on the mathematical
observations, implications on the use of hyperspheres as antibody recognition
regions are provided. We then present an algorithm for estimating, as opposed
to exactly calculating, the total space of overlapping hyperspheres (section 5).
Finally, results in sections 3.1, 4 and 5 are applied to explain in section 6 the
poor classification results shown in [6].
2
Real-Valued Shape-Space and Euclidean Distance
The notion of
shape-space
was introduced by Perelson and Oster [1] and al-
lows a quantitative anity description between antibodies and antigens. More
precisely, a shape-space is a metric space with an associated distance (anity)
function. The real-valued shape-space is the
n
-dimensional Euclidean space
n
,
where every element is represented as a
n
-dimensional point or simply as a vec-
tor represented by a list of
n
real numbers. The Euclidean distance
1
R
d
is the
n
andisdefinedas:
(standard) distance between any two vectors
x
,
y
∈
R
d
(
x
,
y
)=
(
x
1
−
y
1
)
2
+
...
+(
x
n
−
y
n
)
2
(1)
Moreover, the Euclidean distance
d
satisfies the metric properties :
0
reflexivity :
d
(
x
,
y
)=0iff
x
=
y
symmetry :
d
(
x
,
y
)=
d
(
y
,
x
)
triangle inequality :
d
(
x
,
y
)+
d
(
y
,
z
)
non-negativity :
d
(
x
,
y
)
≥
≥
d
(
x
,
z
)
n
for all vectors
x
,
y
,
z
∈
R
and therefore is frequently applied as a distance measurement in AIS algorithms.
3
Hyperspheres as Antibody Recognition Regions
In the original work by Perelson and Oster [1], real-valued shape-space is in-
troduced for estimating the probability that a randomly encountered antigen is
recognized by at least one of the antibodies. An antibody is specified by
n
param-
eters, e.g. the length, width, charge, etc. and can be described as a
n
-dimensional
1
Also termed Euclidean norm.
