An Algorithmic Description - Design and Analysis of Learning Classifier Systems - page 182

functions, their matching functions as well as their initialisation and mutation
is described.

Matching by Radial-Basis Functions

The matching function for matching by radial-basis functions is defined by
m k ( x )=exp 1
2 σ k

μ ) 2 , ( x − (8.10)

which is an unnormalised Gaussian that is parametrised by a scalar μ k and
a positive spread σ k . Thus, the probability of classifier k matching input x
decreases with the distance from μ k , where the strength of the decrease is
determined by σ k .If σ k is small, then the matching probability decreases ra-
pidly with the squared distance of x from μ k . Note that, as m k ( x ) > 0 for all
−∞ <x< ∞ , all classifiers match all inputs, even if only with a very low

probability. Thus, we always guarantee that k m k ( x n ) > 0 for all n ,thatis,
that all inputs are matched by at least one classifier, as required. Examples for

Matching by Radial Basis Functions

1

cl. 1
cl. 2
cl. 3
cl. 4

0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1

x

Fig. 8.2. Matching probability for matching by radial basis functions for different
parameters. Classifiers 1, 2, and 3 all have their matching functions centred on μ 1 =
μ 2 = μ 3 =0 . 5, but have different spreads σ 1 =0 . 1, σ 2 =0 . 01, σ 3 = 1. This visualises
how a larger spread causes the classifier to match a larger area of the input space with
higher probability. The matching function of classifier 4 is centred on μ 4 =0 . 8 and has
spread σ 4 =0 . 2, showing that μ controls the location x of the input space where the
classifier matches with probability 1.