Computational Intelligence - November 2012 - 42

While visualization strategies are not entirely
new to the LCS ﬁeld, to date they have been
applied only to track learning progress within
the search space [31, 32]. The key diﬀerence
here is that visualization is applied directly to
knowledge discovery for the identification of
global attribute generalizations.
This section details steps 5 and 6. All
M-LCS runs up to this point have been dedicated to obtaining test statistics and making
statistical inferences. Our ﬁrst step towards
visualization is to train the M-LCS on the entire dataset.
M-LCS algorithms are typically very adaptive with a tendency
to maintain some level of diversity in the population as it
attempts to search for better and better rules. As a result we
would expect that some proportion of the rules will be poor
classiﬁers with useless generalizations. As previously mentioned,
rule compaction [18, 19, 20, 21] and condensation [22, 23]
algorithms oﬀer a method of eliminating useless rules and
compacting the size of the rule population. While beyond the
scope of the present study, such an algorithm could be used at
this point in the analysis pipeline in an attempt to remove some
of these useless, "noisy" rules. However, based on preliminary
observations that explored rule compaction algorithms, we
would caution the reader that a dramatic reduction in the size
of the rule population may be counter-productive to successfully identifying global patterns.
Our next step includes re-encoding the rule population. The
objective of the heat-map visualization is to discriminate predictive
attributes from non-predictive attributes and to look for patterns
of attribute interaction and heterogeneity. Therefore we encoded
each rule such that any speciﬁed attribute is coded as a 1 while a
'#' is coded as a 0. Additionally, we expanded our rule population
such that there are N copies of each rule reﬂecting respective
numerosities. Similar to ordering by rule numerosity within manual inspection, this step draws greater attention to attribute patterns within rules having a higher numerosity. The last processing
step before visualization is to apply a clustering algorithm to the
coded and expanded rule population. Clustering a population of
M-LCS rules for rule compaction was pioneered in [26, 27]. In

Beyond obtaining a global perspective of the rule
population, this software could also be used to
facilitate the identification of particularly interesting
rules from the rule population by adding rule
parameters such as class, accuracy, fitness, numerosity,
and/or action set size as potential dimensions
of the visualization.
every non-redundant pair-wise combination of attributes in
the dataset. For a dataset with 20 attributes (such as the one
we examine here), we calculate 190 co-occurrence values. We
calculated co-occurrence as follows: for every pair of
attributes we go through each of the 10 CV rule populations
and sum the number of times that both attributes are
concurrently specified in a given rule. In the example from
Table 2, the Co-occurrence Sum (CoS) for the attribute pair
(X1, X2) would be 2 since the pair only cooccurs in rule 3,
and the rule has a numerosity of 2. The significance of cooccurrence scores are determined as before using the results
of permutation testing for each pair of attributes.
Table 4 organizes the top CoS statistic results for our target dataset (only significant CoSs with sums greater than
3000 are displayed). 43 out of the 190 CoSs were identified as
significantly higher than by chance. Each of these significant
pairs had at least one predictive attribute represented. Seeing
that we found the four predictive attributes to be significantly
over-represented, it follows that co-occurrence pairs including at least one of these attributes would turn up more frequently than by chance. Of particular note, we found that our
two modeled, epistatic pairs of predictive attributes yielded
the two highest CoSs. Below these two pairs, there is an
immediate drop-off in the magnitude of CoS (almost halved).
Also note that the magnitude of these two highest CoSs
are well above the SpSs for all non-predictive attributes given
in Table 3.
While additional analysis could examine higher order
co-occurrence between all 3-way combinations of attributes
(or beyond), we can use pair-wise analysis to infer higher order
attribute interactions. For example, if a predictive, 3-way interaction existed in the data (between X0, X1, and X3) we would
expect similar pair-wise CoSs between attribute pairs (X0, X1),
(X1, X2), and (X0, X2). Using this logic we could hypothesize,
given the results in Table 4, that attribute pairs (X0, X1), and
(X2, X3) each represent an interacting pair, but since the 4
other pair-wise combinations of these attributes have about
half the co-occurrence we might suspect heterogeneity as
opposed to a 3 or 4-way interaction. Section 3.0.4 will offer a
visualization of these co-occurrence results.
3.0.3 Visualization-Heat-Map

As indicated by [30], the "visualization of classification models
can create understanding and trust in data mining models."

IEEE ComputatIonal IntEllIgEnCE magazInE | noVEmBER 2012

Figure 2 3D visualization for the identiﬁcation of interesting rules.
In this ﬁgure, speciﬁed attributes are blue, while '#' attributes are yellow. The height of attributes within each row/rule is the product of
the SpS for that attribute and the numerosity of the rule.

Table of Contents for the Digital Edition of Computational Intelligence - November 2012