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regard to the given training elements, instead of complementary space. The
complementary (anomalous) space is exponentially large when compared to the
“normal” space in high dimensions. The real-valued negative selection technique
attempts to cover this high-dimensional space with hyperspheres, but as we have
shown, these have adverse properties in such high-dimensional spaces.
a
2
n
a
Fig. 3. Distance ratio n
a/ 2 between a line from center to a corner and a perpendicular
line from center to an edge
In [18] Verleysen discusses in detail, this curse of dimensionality problem, with
respect to artificial neural networks. He suggests in general to change the dis-
tance measure function for high-dimensional problems, for instance by applying
a higher-order norm ( h> 2)
d h ( x , y )= h
| x 1 − y 1 |
h + ... + | x n − y n |
h
(9)
instead of the standard Euclidean norm. In the context of inductive biases 8 ,Fre-
itas and Timmis [21] discussed different anity measures in artificial immune
systems. They illustrated the advantages and disadvantages of the 1-norm and
2-norm (see term (9)) and showed how one of these norm when compared to the
other norm can lead to an overemphasizing of the distance. As a final summa-
rizing sentence, the authors suggested that when developing an AIS, one should
make a careful choice of the norm, as the norm should take into account the
characteristics (in our case the dimension) of the data being analyzed. Unfortu-
nately, there seems to be no theoretical results, for correctly choosing the value
h with regard to the data dimension [18].
7Con lu on
The immune system is an impressive recognition system with many appealing
properties for the construction of artificial immune system algorithms. Abstract-
ing antibodies as hyperspheres and applying the Euclidean distance metric for
quantifying binding strengths, is an established method for modeling and simu-
lating the immune systems.
8 Effectiveness in problem domains.
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