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Structural Properties of Shape-Spaces
Werner Dilger
Chemnitz University of Technology
09107 Chemnitz, Germany
dilger@informatik.tu-chemnitz.de
Abstract.
General properties of distance functions and of affinity functions are
discussed in this paper. Reasons are given why a distance function for ℜn based
shape-spaces should be a metric. Several distance functions that are used in
shape-spaces are examined and it is shown that not all of them are metrics. It is
shown which impact the type of the distance function has on the shape-space, in
particular on the form of recognition or affinity regions in the shape-space.
Affinity functions should be defined in such a way that they determine an
affinity region with positive values inside that region and zero or negative
values outside. The form of an affinity function depends on the type of the
underlying distance function. This is demonstrated with several examples.
Keywords:
Shape-space, distance function, metric, affinity function, affinity
region.
1 Introduction
The mostly used definition of shape-spaces is the one introduced by Perelson and
Oster in [12]. According to this definition, the interaction between elements of the
immune system (cells, antibodies, or molecules) and antigens is determined by
properties of shape. Actually, this approach is an abstraction from the real immune
system, where the interaction is essentially based on electrical forces due to the
charge distribution on the surface of the molecules. The next step of abstraction, then,
is the representation of the shape properties by a string of parameters of certain types
of values like binary, integer, real, or symbolic.
A basic notion in the Perelson/Oster shape-space is that of complementarity, which
means that an immune element and an antigen must have complementary shapes in
order to exert affinity on each other. On the basis of the vector representation, in
many AIS realizations complementarity has been replaced with similarity (cf. [4]),
just by “changing the sign”. Different types of affinity have been defined, depending
on the type of the shape-space as a vector space, but all of them are based on some
distance measure like Euclidean distance or Hamming distance.
In [1], Bersini introduced an alternative definition of a shape-space which on first
glance departs considerably from the Perelson/Oster definition. The shape-space is
based on
n
2
. However, Bersini uses a special definition of
affinity which makes his shape-space particularly interesting. This definition
ℜ
, more precisely on
ℜ
