CHAPTER 3. DEFINING DIVERSITY 44
which the target function is covered by the ensemble. A coincident error occurs when, for
a given input, more than 1 ensemble member gives an incorrect answer. Function coverage
is an indication of whether or not a test input yields a correct answer on ANY of the neural
networks in the ensemble.
Level 1 No coincident errors, the target function is covered. Majority vote always produces
the correct answer.
Level 2 Some coincident errors, but the majority is always correct and the function is
completely covered. The ensemble size must be greater than 4 for this to hold.
Level 3 A majority vote will not always yield the right answer, but the members of the
ensemble cover the function such that at least one always has the correct answer for
a given input.
Level 4 The function is not always covered by the members of the ensemble.
Sharkey acknowledges that ensembles exhibiting level 2 or 3 diversity could be “upwardly
mobile” as it is possible that an ensemble labelled as having level 2 diversity could contain a
subset of neural networks displaying level 1 diversity using the test set and a level 3 ensemble
could contain subsets of neural networks displaying level 1 and/or level 2 diversity on the
test set, thus removal of certain networks could result in a change of diversity level for the
better.
Carney and Cunningham [25] suggested an entropy-based measure, though this does
not allow calculation of an individual’s contribution to overall diversity. Zenobi and Cunningham [157] proposed a measure of classification ambiguity. The ambiguity of the ith
classifier, averaged over N patterns, is