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4.2
Using Hyperspheres as Antibody Recognition Regions in
Artificial Immune Systems
The results and observations presented in sections 3.1, 4 and 4.1 indicate that
high-dimensional real-valued shape-spaces strongly bias the volume (recognition
space) of hyperspheres. A hypersphere, for example with radius
r
= 1 has a high
volume in relation to its radius length, up to dimension 15 (see Fig. 2). In higher
dimensions (
n>
15), for
r
= 1 the volume is nearly 0. This means that the
recognition space — or in the context of antibody recognition region (covered
space) — is nearly 0. In contrast, a radius that is too large (
r>
2) in high
dimensional spaces (
n>
10) will imply an exponential volume. This exponential
volume behavior, in combination with an unprecise volume estimation of over-
lapping hyperspheres, is the reason for the poor classification results reported in
the paper [6] and is discussed in the subsequent sections.
5
Estimating Volume of Overlapping Hyperspheres
In section 3.1 a formula for calculating the exact volume of a hypersphere given
by the dimension and the radius was shown. However, many proposed arti-
ficial immune system algorithms for solving pattern recognition, anomaly de-
tection and clustering problems using not only
one
but multiple overlapping
hyperspheres for classifying points [4,5,6,7,8,9]. Calculating analytically the to-
tal volume of overlapping hyperspheres is a very dicult task. Just the simple
2-dimensional case of three overlapping circles with different radii is a mathe-
matical challenge. In the following section we describe a method to estimate the
volume of (overlapping) hyperspheres.
5.1
Monte Carlo Integration
The Monte Carlo Integration is a method to integrate a function over a com-
plicated domain, where analytical expressions are very dicult to apply - e.g.
the calculation of the volume of overlapping hyperspheres in higher dimensions.
Given integrals of the form
I
=
X
h
(
x
)
f
(
x
)
d
x
,where
h
(
x
)and
f
(
x
) are func-
tions for which
h
(
x
)
f
(
x
) is integrable over the space
X
,and
f
(
x
) is a non-
negative valued, integrable function satisfying
X
f
(
x
)
d
x
=1.TheMonteCarlo
integration picks
N
random points
x
1
,
x
2
,...,
x
N
,over
X
and approximates the
integral as
N
1
N
I
≈
h
(
x
n
)
(5)
n
=1
The absolute error of this method is
independent
of the dimension of the space
X
and decreases as 1
/
√
N
. By applying this integration method, two fundamental
questions arise :