An Algorithmic Description - Design and Analysis of Learning Classifier Systems - page 196

first approach might be the one to prefer. From inspecting (7.99) and (7.100) it
can be seen that both parameters of the noise precision model can be updated
incrementally without any modifications.
Even though a least squares approximation could be used to train the mixing
model, analogous to Sect. 6.1.2, the results in Chap. 6 have shown that it is possi-
ble to design heuristics that outperform this approximation. Additionally, these
heuristics might not require any parameters to be updated, besides the para-
meters of the classifiers themselves. Given that similar parameter-less heuristics
exist for the Bayesian model, they can be immediately used in incremental im-
plementations, as no parameters need to be updated. Possible approaches where
already outlined in Sect. 6.4.

Incremental Model Structure Search

The GA in Michigan-style LCS has strong parallels to cooperative co-evolutionary
algorithms (for example [235]). In these, the fitness of an individual depends on
the other individuals in the population. Equally, the fitness of a classifier in a
Michigan-style LCS depends on the other classifiers in the set of classifiers as they
cooperate to form the solution. Note that while in Pittsburgh-style LCS an indi-
vidual is a set of classifiers that provides a candidate solution, in Michigan-style
each classifier is an individual and the whole population forms the solution.
Having defined a fitness for a set of classifiers by the model structure proba-
bility, the aim is to design an algorithm that is able to increase the fitness of
this population by modifying separate classifiers. Expressed differently, we want
to design a GA for which the fixed point of its operators is the optimal set of
classifiers such that p (

M|D

) is maximised. While this is not a trivial problem, an
obvious approach is to attempt to design a cooperative co-evolutionary algorithm
with such operators, or to modify existing LCS, like XCS(F), to aim at satis-
fying the optimality criterion. However, the lack of theoretical understanding of
either method does not make the approach any simpler [171].
Here, an alternative based on Replicator Dynamics (for example, [108]) is
suggested: assume that the number of possible matching function parametrisa-
tionsisgivenbyafinite P (for any finite

and a sensible representation this is
always the case) and that C 1 ,...,C P enumerate each possible type of matching
function. Each C i stands for a classifier type that is a possible replicator in a
population. Let c =( c 1 ,...,c P ) T denote the frequency of each of the classifier
types. Assuming an infinite population model, c i gives the proportion of classi-
fiers of C i in the population. As the c i 's satisfy 0

X

1and i c i =1, c is

≤ c i ≤

an element of the P -dimensional simplex S P .
The fitness f i ( c )of C i is a function of all classifiers in the population, described
by c . The rate of increase c i /c i of classifier type C i is a measure of its evolutionary
success and may be expressed as the difference between the fitness of C i and the
average fitness

f ( c )= i c i f i ( c ), which results in the replicator equation
c i = c i f i ( c )

f ( x ) .

− (8.15)